# Variational Principles in Classical Mechanics - Revised Second Edition, 2019

Variational Principles in Classical Mechanics - Revised Second Edition, 2019

Variational Principles in Classical Mechanics - Revised Second Edition, 2019

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VARIATIONAL PRINCIPLES<br />

IN<br />

CLASSICAL MECHANICS<br />

REVISED SECOND EDITION<br />

Douglas Cl<strong>in</strong>e<br />

University of Rochester<br />

15 January <strong>2019</strong>

ii<br />

c°<strong>2019</strong>, 2017 by Douglas Cl<strong>in</strong>e<br />

ISBN: 978-0-9988372-8-4 e-book (Adobe PDF)<br />

ISBN: 978-0-9988372-2-2 e-book (K<strong>in</strong>dle)<br />

ISBN: 978-0-9988372-9-1 pr<strong>in</strong>t (Paperback)<br />

<strong>Variational</strong> <strong>Pr<strong>in</strong>ciples</strong> <strong>in</strong> <strong>Classical</strong> <strong>Mechanics</strong>, <strong>Revised</strong> 2 edition<br />

Contributors<br />

Author: Douglas Cl<strong>in</strong>e<br />

Illustrator: Meghan Sarkis<br />

Published by University of Rochester River Campus Libraries<br />

University of Rochester<br />

Rochester, NY 14627<br />

<strong>Variational</strong> <strong>Pr<strong>in</strong>ciples</strong> <strong>in</strong> <strong>Classical</strong> <strong>Mechanics</strong>, <strong>Revised</strong> 2 edition by Douglas Cl<strong>in</strong>e is licensed under a<br />

Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International License (CC BY-NC-SA 4.0),<br />

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Version 2.1

Contents<br />

Contents<br />

Preface<br />

Prologue<br />

iii<br />

xvii<br />

xix<br />

1 A brief history of classical mechanics 1<br />

1.1 Introduction.............................................. 1<br />

1.2 Greekantiquity............................................ 1<br />

1.3 MiddleAges.............................................. 2<br />

1.4 AgeofEnlightenment ........................................ 2<br />

1.5 <strong>Variational</strong>methods<strong>in</strong>physics ................................... 5<br />

1.6 The 20 centuryrevolution<strong>in</strong>physics............................... 7<br />

2 ReviewofNewtonianmechanics 9<br />

2.1 Introduction.............................................. 9<br />

2.2 Newton’sLawsofmotion ...................................... 9<br />

2.3 Inertialframesofreference...................................... 10<br />

2.4 First-order<strong>in</strong>tegrals<strong>in</strong>Newtonianmechanics ........................... 11<br />

2.4.1 L<strong>in</strong>earMomentum ...................................... 11<br />

2.4.2 Angularmomentum ..................................... 11<br />

2.4.3 K<strong>in</strong>eticenergy ........................................ 12<br />

2.5 Conservationlaws<strong>in</strong>classicalmechanics.............................. 12<br />

2.6 Motion of f<strong>in</strong>ite-sizedandmany-bodysystems........................... 12<br />

2.7 Centerofmassofamany-bodysystem............................... 13<br />

2.8 Totall<strong>in</strong>earmomentumofamany-bodysystem.......................... 14<br />

2.8.1 Center-of-massdecomposition................................ 14<br />

2.8.2 Equationsofmotion ..................................... 14<br />

2.9 Angularmomentumofamany-bodysystem............................ 16<br />

2.9.1 Center-of-massdecomposition................................ 16<br />

2.9.2 Equationsofmotion ..................................... 16<br />

2.10Workandk<strong>in</strong>eticenergyforamany-bodysystem......................... 18<br />

2.10.1 Center-of-massk<strong>in</strong>eticenergy................................ 18<br />

2.10.2 Conservativeforcesandpotentialenergy.......................... 18<br />

2.10.3 Totalmechanicalenergy................................... 19<br />

2.10.4 Totalmechanicalenergyforconservativesystems..................... 20<br />

2.11VirialTheorem ............................................ 22<br />

2.12ApplicationsofNewton’sequationsofmotion........................... 24<br />

2.12.1 Constantforceproblems................................... 24<br />

2.12.2 L<strong>in</strong>earRestor<strong>in</strong>gForce.................................... 25<br />

2.12.3 Position-dependentconservativeforces........................... 25<br />

2.12.4 Constra<strong>in</strong>edmotion ..................................... 27<br />

2.12.5 VelocityDependentForces.................................. 28<br />

2.12.6 SystemswithVariableMass................................. 29<br />

2.12.7 Rigid-body rotation about a body-fixedrotationaxis................... 31<br />

iii

iv<br />

CONTENTS<br />

2.12.8 Timedependentforces.................................... 34<br />

2.13Solutionofmany-bodyequationsofmotion ............................ 37<br />

2.13.1 Analyticsolution....................................... 37<br />

2.13.2 Successiveapproximation .................................. 37<br />

2.13.3 Perturbationmethod..................................... 37<br />

2.14Newton’sLawofGravitation .................................... 38<br />

2.14.1 Gravitationaland<strong>in</strong>ertialmass............................... 38<br />

2.14.2 Gravitational potential energy .............................. 39<br />

2.14.3 Gravitational potential .................................. 40<br />

2.14.4 Potentialtheory ....................................... 41<br />

2.14.5 Curl of the gravitational field................................ 41<br />

2.14.6 Gauss’sLawforGravitation................................. 43<br />

2.14.7 CondensedformsofNewton’sLawofGravitation..................... 44<br />

2.15Summary ............................................... 46<br />

Problems .................................................. 48<br />

3 L<strong>in</strong>ear oscillators 51<br />

3.1 Introduction.............................................. 51<br />

3.2 L<strong>in</strong>earrestor<strong>in</strong>gforces ........................................ 51<br />

3.3 L<strong>in</strong>earityandsuperposition ..................................... 52<br />

3.4 Geometricalrepresentationsofdynamicalmotion......................... 53<br />

3.4.1 Configuration space ( ) ................................ 53<br />

3.4.2 State space, ( ˙ ) ..................................... 54<br />

3.4.3 Phase space, ( ) .................................... 54<br />

3.4.4 Planependulum ....................................... 55<br />

3.5 L<strong>in</strong>early-damped free l<strong>in</strong>ear oscillator ................................ 56<br />

3.5.1 Generalsolution ....................................... 56<br />

3.5.2 Energydissipation ...................................... 59<br />

3.6 S<strong>in</strong>usoidally-drive, l<strong>in</strong>early-damped, l<strong>in</strong>ear oscillator . . . .................... 60<br />

3.6.1 Transient response of a driven oscillator .......................... 60<br />

3.6.2 Steady state response of a driven oscillator ........................ 61<br />

3.6.3 Completesolutionofthedrivenoscillator ......................... 62<br />

3.6.4 Resonance........................................... 63<br />

3.6.5 Energyabsorption ...................................... 63<br />

3.7 Waveequation ............................................ 66<br />

3.8 Travell<strong>in</strong>gandstand<strong>in</strong>gwavesolutionsofthewaveequation................... 67<br />

3.9 Waveformanalysis .......................................... 68<br />

3.9.1 Harmonicdecomposition................................... 68<br />

3.9.2 The free l<strong>in</strong>early-damped l<strong>in</strong>ear oscillator ......................... 68<br />

3.9.3 Dampedl<strong>in</strong>earoscillatorsubjecttoanarbitraryperiodicforce ............. 69<br />

3.10Signalprocess<strong>in</strong>g ........................................... 70<br />

3.11 Wave propagation .......................................... 71<br />

3.11.1 Phase,group,andsignalvelocitiesofwavepackets .................... 72<br />

3.11.2 Fouriertransformofwavepackets ............................. 77<br />

3.11.3 Wave-packetUncerta<strong>in</strong>tyPr<strong>in</strong>ciple............................. 78<br />

3.12Summary ............................................... 80<br />

Problems .................................................. 83<br />

4 Nonl<strong>in</strong>ear systems and chaos 85<br />

4.1 Introduction.............................................. 85<br />

4.2 Weaknonl<strong>in</strong>earity .......................................... 86<br />

4.3 Bifurcation,andpo<strong>in</strong>tattractors .................................. 88<br />

4.4 Limitcycles.............................................. 89<br />

4.4.1 Po<strong>in</strong>caré-Bendixsontheorem ................................ 89<br />

4.4.2 vanderPoldampedharmonicoscillator:.......................... 90

CONTENTS<br />

v<br />

4.5 Harmonically-driven,l<strong>in</strong>early-damped,planependulum...................... 93<br />

4.5.1 Closetol<strong>in</strong>earity....................................... 93<br />

4.5.2 Weaknonl<strong>in</strong>earity ...................................... 95<br />

4.5.3 Onsetofcomplication .................................... 96<br />

4.5.4 Perioddoubl<strong>in</strong>gandbifurcation............................... 96<br />

4.5.5 Roll<strong>in</strong>g motion ........................................ 96<br />

4.5.6 Onsetofchaos ........................................ 97<br />

4.6 Differentiationbetweenorderedandchaoticmotion........................ 98<br />

4.6.1 Lyapunovexponent ..................................... 98<br />

4.6.2 Bifurcationdiagram ..................................... 99<br />

4.6.3 Po<strong>in</strong>caréSection .......................................100<br />

4.7 Wave propagation for non-l<strong>in</strong>ear systems . . . ...........................101<br />

4.7.1 Phase,group,andsignalvelocities .............................101<br />

4.7.2 Soliton wave propagation ..................................103<br />

4.8 Summary ...............................................104<br />

Problems ..................................................106<br />

5 Calculus of variations 107<br />

5.1 Introduction..............................................107<br />

5.2 Euler’s differentialequation .....................................108<br />

5.3 ApplicationsofEuler’sequation...................................110<br />

5.4 Selectionofthe<strong>in</strong>dependentvariable................................113<br />

5.5 Functions with several <strong>in</strong>dependent variables () ........................115<br />

5.6 Euler’s<strong>in</strong>tegralequation.......................................117<br />

5.7 Constra<strong>in</strong>edvariationalsystems...................................118<br />

5.7.1 Holonomicconstra<strong>in</strong>ts....................................118<br />

5.7.2 Geometric(algebraic)equationsofconstra<strong>in</strong>t .......................118<br />

5.7.3 K<strong>in</strong>ematic (differential)equationsofconstra<strong>in</strong>t ......................118<br />

5.7.4 Isoperimetric(<strong>in</strong>tegral)equationsofconstra<strong>in</strong>t ......................119<br />

5.7.5 Propertiesoftheconstra<strong>in</strong>tequations ...........................119<br />

5.7.6 Treatmentofconstra<strong>in</strong>tforces<strong>in</strong>variationalcalculus...................120<br />

5.8 Generalizedcoord<strong>in</strong>ates<strong>in</strong>variationalcalculus ..........................121<br />

5.9 Lagrangemultipliersforholonomicconstra<strong>in</strong>ts ..........................122<br />

5.9.1 Algebraicequationsofconstra<strong>in</strong>t..............................122<br />

5.9.2 Integralequationsofconstra<strong>in</strong>t...............................124<br />

5.10Geodesic................................................126<br />

5.11<strong>Variational</strong>approachtoclassicalmechanics ............................127<br />

5.12Summary ...............................................128<br />

Problems ..................................................129<br />

6 Lagrangian dynamics 131<br />

6.1 Introduction..............................................131<br />

6.2 Newtonian plausibility argument for Lagrangian mechanics ...................132<br />

6.3 Lagrangeequationsfromd’Alembert’sPr<strong>in</strong>ciple..........................134<br />

6.3.1 d’Alembert’sPr<strong>in</strong>cipleofVirtualWork ..........................134<br />

6.3.2 Transformationtogeneralizedcoord<strong>in</strong>ates.........................135<br />

6.3.3 Lagrangian ..........................................136<br />

6.4 LagrangeequationsfromHamilton’sActionPr<strong>in</strong>ciple ......................137<br />

6.5 Constra<strong>in</strong>edsystems .........................................138<br />

6.5.1 Choiceofgeneralizedcoord<strong>in</strong>ates..............................138<br />

6.5.2 M<strong>in</strong>imalsetofgeneralizedcoord<strong>in</strong>ates...........................138<br />

6.5.3 Lagrangemultipliersapproach ...............................138<br />

6.5.4 Generalizedforcesapproach.................................140<br />

6.6 Apply<strong>in</strong>gtheEuler-Lagrangeequationstoclassicalmechanics..................140<br />

6.7 Applicationstounconstra<strong>in</strong>edsystems...............................142

vi<br />

CONTENTS<br />

6.8 Applicationstosystems<strong>in</strong>volv<strong>in</strong>gholonomicconstra<strong>in</strong>ts .....................144<br />

6.9 Applications<strong>in</strong>volv<strong>in</strong>gnon-holonomicconstra<strong>in</strong>ts.........................157<br />

6.10Velocity-dependentLorentzforce ..................................164<br />

6.11Time-dependentforces........................................165<br />

6.12Impulsiveforces............................................166<br />

6.13TheLagrangianversustheNewtonianapproachtoclassicalmechanics.............168<br />

6.14Summary ...............................................169<br />

Problems ..................................................172<br />

7 Symmetries, Invariance and the Hamiltonian 175<br />

7.1 Introduction..............................................175<br />

7.2 Generalizedmomentum .......................................175<br />

7.3 InvarianttransformationsandNoether’sTheorem.........................177<br />

7.4 Rotational<strong>in</strong>varianceandconservationofangularmomentum..................179<br />

7.5 Cycliccoord<strong>in</strong>ates ..........................................180<br />

7.6 K<strong>in</strong>eticenergy<strong>in</strong>generalizedcoord<strong>in</strong>ates .............................181<br />

7.7 GeneralizedenergyandtheHamiltonianfunction.........................182<br />

7.8 Generalizedenergytheorem.....................................183<br />

7.9 Generalizedenergyandtotalenergy ................................183<br />

7.10Hamiltonian<strong>in</strong>variance........................................184<br />

7.11Hamiltonianforcycliccoord<strong>in</strong>ates .................................189<br />

7.12Symmetriesand<strong>in</strong>variance .....................................189<br />

7.13Hamiltonian<strong>in</strong>classicalmechanics .................................189<br />

7.14Summary ...............................................190<br />

Problems ..................................................192<br />

8 Hamiltonian mechanics 195<br />

8.1 Introduction..............................................195<br />

8.2 LegendreTransformationbetweenLagrangianandHamiltonianmechanics...........196<br />

8.3 Hamilton’sequationsofmotion...................................197<br />

8.3.1 Canonicalequationsofmotion ...............................198<br />

8.4 Hamiltonian <strong>in</strong> differentcoord<strong>in</strong>atesystems............................199<br />

8.4.1 Cyl<strong>in</strong>drical coord<strong>in</strong>ates ................................199<br />

8.4.2 Spherical coord<strong>in</strong>ates, .................................200<br />

8.5 ApplicationsofHamiltonianDynamics...............................201<br />

8.6 Routhianreduction..........................................206<br />

8.6.1 R -RouthianisaHamiltonianforthecyclicvariables................207<br />

8.6.2 R -RouthianisaHamiltonianforthenon-cyclicvariables ...........208<br />

8.7 Variable-masssystems ........................................212<br />

8.7.1 Rocketpropulsion:......................................212<br />

8.7.2 Mov<strong>in</strong>gcha<strong>in</strong>s: ........................................213<br />

8.8 Summary ...............................................215<br />

Problems ..................................................217<br />

9 Hamilton’s Action Pr<strong>in</strong>ciple 221<br />

9.1 Introduction..............................................221<br />

9.2 Hamilton’s Pr<strong>in</strong>ciple of Stationary Action<br />

9.2.1 Stationary-actionpr<strong>in</strong>ciple<strong>in</strong>Lagrangianmechanics ...................222<br />

9.2.2 Stationary-actionpr<strong>in</strong>ciple<strong>in</strong>Hamiltonianmechanics ..................223<br />

9.2.3 Abbreviatedaction......................................224<br />

9.2.4 Hamilton’s Pr<strong>in</strong>ciple applied us<strong>in</strong>g <strong>in</strong>itial boundary conditions . ............225<br />

9.3 Lagrangian ..............................................228<br />

9.3.1 StandardLagrangian.....................................228<br />

9.3.2 Gauge<strong>in</strong>varianceofthestandardLagrangian .......................228<br />

9.3.3 Non-standardLagrangians..................................230<br />

9.3.4 Inversevariationalcalculus .................................230

CONTENTS<br />

vii<br />

9.4 ApplicationofHamilton’sActionPr<strong>in</strong>cipletomechanics.....................231<br />

9.5 Summary ...............................................232<br />

10 Nonconservative systems 235<br />

10.1Introduction..............................................235<br />

10.2Orig<strong>in</strong>sofnonconservativemotion .................................235<br />

10.3Algebraicmechanicsfornonconservativesystems .........................236<br />

10.4Rayleigh’sdissipationfunction ...................................236<br />

10.4.1 Generalizeddissipativeforcesforl<strong>in</strong>earvelocitydependence...............237<br />

10.4.2 Generalizeddissipativeforcesfornonl<strong>in</strong>earvelocitydependence.............238<br />

10.4.3 Lagrangeequationsofmotion................................238<br />

10.4.4 Hamiltonianmechanics ...................................238<br />

10.5DissipativeLagrangians .......................................241<br />

10.6Summary ...............................................243<br />

11 Conservative two-body central forces 245<br />

11.1Introduction..............................................245<br />

11.2Equivalentone-bodyrepresentationfortwo-bodymotion.....................246<br />

11.3 Angular momentum L ........................................248<br />

11.4Equationsofmotion .........................................249<br />

11.5 Differentialorbitequation:......................................250<br />

11.6Hamiltonian..............................................251<br />

11.7Generalfeaturesoftheorbitsolutions ...............................252<br />

11.8Inverse-square,two-body,centralforce...............................253<br />

11.8.1 Boundorbits .........................................254<br />

11.8.2 Kepler’slawsforboundplanetarymotion .........................255<br />

11.8.3 Unboundorbits........................................256<br />

11.8.4 Eccentricity vector . . . ...................................257<br />

11.9Isotropic,l<strong>in</strong>ear,two-body,centralforce ..............................259<br />

11.9.1 Polarcoord<strong>in</strong>ates.......................................260<br />

11.9.2 Cartesiancoord<strong>in</strong>ates ....................................261<br />

11.9.3 Symmetry tensor A 0 .....................................262<br />

11.10Closed-orbitstability.........................................263<br />

11.11Thethree-bodyproblem.......................................268<br />

11.12Two-bodyscatter<strong>in</strong>g.........................................269<br />

11.12.1Totaltwo-bodyscatter<strong>in</strong>gcrosssection ..........................269<br />

11.12.2 Differentialtwo-bodyscatter<strong>in</strong>gcrosssection .......................270<br />

11.12.3Impactparameterdependenceonscatter<strong>in</strong>gangle ....................270<br />

11.12.4Rutherfordscatter<strong>in</strong>g ....................................272<br />

11.13Two-bodyk<strong>in</strong>ematics.........................................274<br />

11.14Summary ...............................................280<br />

Problems ..................................................282<br />

12 Non-<strong>in</strong>ertial reference frames 285<br />

12.1Introduction..............................................285<br />

12.2 Translational acceleration of a reference frame ...........................285<br />

12.3Rotat<strong>in</strong>greferenceframe.......................................286<br />

12.3.1 Spatialtimederivatives<strong>in</strong>arotat<strong>in</strong>g,non-translat<strong>in</strong>g,referenceframe .........286<br />

12.3.2 Generalvector<strong>in</strong>arotat<strong>in</strong>g,non-translat<strong>in</strong>g,referenceframe ..............287<br />

12.4 Reference frame undergo<strong>in</strong>g rotation plus translation . . . ....................288<br />

12.5Newton’slawofmotion<strong>in</strong>anon-<strong>in</strong>ertialframe ..........................288<br />

12.6Lagrangianmechanics<strong>in</strong>anon-<strong>in</strong>ertialframe ...........................289<br />

12.7Centrifugalforce ...........................................290<br />

12.8Coriolisforce .............................................291<br />

12.9Routhianreductionforrotat<strong>in</strong>gsystems ..............................295<br />

12.10EffectivegravitationalforcenearthesurfaceoftheEarth ....................298

viii<br />

CONTENTS<br />

12.11Freemotionontheearth.......................................300<br />

12.12Weathersystems ...........................................302<br />

12.12.1Low-pressuresystems: ....................................302<br />

12.12.2High-pressuresystems:....................................304<br />

12.13Foucaultpendulum..........................................304<br />

12.14Summary ...............................................306<br />

Problems ..................................................307<br />

13 Rigid-body rotation 309<br />

13.1Introduction..............................................309<br />

13.2Rigid-bodycoord<strong>in</strong>ates........................................310<br />

13.3 Rigid-body rotation about a body-fixedpo<strong>in</strong>t...........................310<br />

13.4Inertiatensor .............................................312<br />

13.5Matrixandtensorformulationsofrigid-bodyrotation ......................313<br />

13.6Pr<strong>in</strong>cipalaxissystem.........................................313<br />

13.7 Diagonalize the <strong>in</strong>ertia tensor . ...................................314<br />

13.8Parallel-axistheorem.........................................315<br />

13.9Perpendicular-axistheoremforplanelam<strong>in</strong>ae ...........................318<br />

13.10Generalpropertiesofthe<strong>in</strong>ertiatensor...............................319<br />

13.10.1Inertialequivalence......................................319<br />

13.10.2 Orthogonality of pr<strong>in</strong>cipal axes ...............................320<br />

13.11Angular momentum L and angular velocity ω vectors ......................321<br />

13.12K<strong>in</strong>eticenergyofrotat<strong>in</strong>grigidbody................................323<br />

13.13Eulerangles..............................................325<br />

13.14Angular velocity ω ..........................................327<br />

13.15K<strong>in</strong>eticenergy<strong>in</strong>termsofEulerangularvelocities ........................328<br />

13.16Rotational<strong>in</strong>variants.........................................329<br />

13.17Euler’sequationsofmotionforrigid-bodyrotation ........................330<br />

13.18Lagrangeequationsofmotionforrigid-bodyrotation.......................331<br />

13.19Hamiltonianequationsofmotionforrigid-bodyrotation.....................333<br />

13.20Torque-freerotationofan<strong>in</strong>ertially-symmetricrigidrotor ....................333<br />

13.20.1Euler’sequationsofmotion:.................................333<br />

13.20.2Lagrangeequationsofmotion: ...............................337<br />

13.21Torque-freerotationofanasymmetricrigidrotor.........................339<br />

13.22Stability of torque-free rotation of an asymmetric body . . ....................340<br />

13.23Symmetric rigid rotor subject to torque about a fixedpo<strong>in</strong>t ...................343<br />

13.24Theroll<strong>in</strong>gwheel...........................................347<br />

13.25Dynamicbalanc<strong>in</strong>gofwheels ....................................350<br />

13.26Rotationofdeformablebodies....................................351<br />

13.27Summary ...............................................352<br />

Problems ..................................................354<br />

14 Coupled l<strong>in</strong>ear oscillators 357<br />

14.1Introduction..............................................357<br />

14.2 Two coupled l<strong>in</strong>ear oscillators . ...................................357<br />

14.3Normalmodes ............................................359<br />

14.4 Center of mass oscillations . . . ...................................360<br />

14.5Weakcoupl<strong>in</strong>g ............................................361<br />

14.6Generalanalytictheoryforcoupledl<strong>in</strong>earoscillators .......................363<br />

14.6.1 K<strong>in</strong>etic energy tensor T ...................................363<br />

14.6.2 Potential energy tensor V ..................................364<br />

14.6.3 Equationsofmotion .....................................365<br />

14.6.4 Superposition.........................................366<br />

14.6.5 Eigenfunctionorthonormality................................366<br />

14.6.6 Normalcoord<strong>in</strong>ates .....................................367

CONTENTS<br />

ix<br />

14.7 Two-body coupled oscillator systems ................................368<br />

14.8 Three-body coupled l<strong>in</strong>ear oscillator systems . ...........................374<br />

14.9Molecularcoupledoscillatorsystems ................................379<br />

14.10DiscreteLatticeCha<strong>in</strong>........................................382<br />

14.10.1Longitud<strong>in</strong>almotion.....................................382<br />

14.10.2Transversemotion ......................................382<br />

14.10.3Normalmodes........................................383<br />

14.10.4 Travell<strong>in</strong>g waves .......................................386<br />

14.10.5Dispersion...........................................386<br />

14.10.6Complexwavenumber ....................................387<br />

14.11Damped coupled l<strong>in</strong>ear oscillators ..................................388<br />

14.12Collective synchronization of coupled oscillators ..........................389<br />

14.13Summary ...............................................392<br />

Problems ..................................................393<br />

15 Advanced Hamiltonian mechanics 395<br />

15.1Introduction..............................................395<br />

15.2PoissonbracketrepresentationofHamiltonianmechanics ....................397<br />

15.2.1 PoissonBrackets .......................................397<br />

15.2.2 FundamentalPoissonbrackets: ...............................397<br />

15.2.3 Poissonbracket<strong>in</strong>variancetocanonicaltransformations .................398<br />

15.2.4 CorrespondenceofthecommutatorandthePoissonBracket...............399<br />

15.2.5 Observables<strong>in</strong>Hamiltonianmechanics...........................400<br />

15.2.6 Hamilton’sequationsofmotion...............................403<br />

15.2.7 Liouville’s Theorem . . ...................................407<br />

15.3Canonicaltransformations<strong>in</strong>Hamiltonianmechanics.......................409<br />

15.3.1 Generat<strong>in</strong>gfunctions.....................................410<br />

15.3.2 Applicationsofcanonicaltransformations .........................412<br />

15.4Hamilton-Jacobitheory .......................................414<br />

15.4.1 Time-dependentHamiltonian................................414<br />

15.4.2 Time-<strong>in</strong>dependentHamiltonian...............................416<br />

15.4.3 Separationofvariables....................................417<br />

15.4.4 Visual representation of the action function . ......................424<br />

15.4.5 AdvantagesofHamilton-Jacobitheory...........................424<br />

15.5Action-anglevariables ........................................425<br />

15.5.1 Canonicaltransformation ..................................425<br />

15.5.2 Adiabatic<strong>in</strong>varianceoftheactionvariables ........................428<br />

15.6Canonicalperturbationtheory ...................................430<br />

15.7Symplecticrepresentation ......................................432<br />

15.8ComparisonoftheLagrangianandHamiltonianformulations ..................432<br />

15.9Summary ...............................................434<br />

Problems ..................................................437<br />

16 Analytical formulations for cont<strong>in</strong>uous systems 439<br />

16.1Introduction..............................................439<br />

16.2Thecont<strong>in</strong>uousuniforml<strong>in</strong>earcha<strong>in</strong> ................................439<br />

16.3TheLagrangiandensityformulationforcont<strong>in</strong>uoussystems ...................440<br />

16.3.1 Onespatialdimension....................................440<br />

16.3.2 Threespatialdimensions ..................................441<br />

16.4TheHamiltoniandensityformulationforcont<strong>in</strong>uoussystems ..................442<br />

16.5L<strong>in</strong>earelasticsolids..........................................443<br />

16.5.1 Stresstensor .........................................444<br />

16.5.2 Stra<strong>in</strong>tensor .........................................444<br />

16.5.3 Moduliofelasticity......................................445<br />

16.5.4 Equationsofmotion<strong>in</strong>auniformelasticmedia......................446

x<br />

CONTENTS<br />

16.6 Electromagnetic fieldtheory.....................................447<br />

16.6.1 Maxwellstresstensor ....................................447<br />

16.6.2 Momentum <strong>in</strong> the electromagnetic field ..........................448<br />

16.7 Ideal fluiddynamics .........................................449<br />

16.7.1 Cont<strong>in</strong>uityequation .....................................449<br />

16.7.2 Euler’shydrodynamicequation...............................449<br />

16.7.3 Irrotational flow and Bernoulli’s equation .........................450<br />

16.7.4 Gas flow............................................450<br />

16.8 Viscous fluiddynamics........................................452<br />

16.8.1 Navier-Stokesequation....................................452<br />

16.8.2 Reynoldsnumber.......................................453<br />

16.8.3 Lam<strong>in</strong>ar and turbulent fluid flow..............................453<br />

16.9Summaryandimplications .....................................455<br />

17 Relativistic mechanics 457<br />

17.1Introduction..............................................457<br />

17.2GalileanInvariance..........................................457<br />

17.3SpecialTheoryofRelativity.....................................459<br />

17.3.1 E<strong>in</strong>ste<strong>in</strong>Postulates......................................459<br />

17.3.2 Lorentztransformation ...................................459<br />

17.3.3 TimeDilation: ........................................460<br />

17.3.4 LengthContraction .....................................461<br />

17.3.5 Simultaneity .........................................461<br />

17.4Relativistick<strong>in</strong>ematics........................................464<br />

17.4.1 Velocitytransformations...................................464<br />

17.4.2 Momentum ..........................................464<br />

17.4.3 Centerofmomentumcoord<strong>in</strong>atesystem..........................465<br />

17.4.4 Force .............................................465<br />

17.4.5 Energy.............................................465<br />

17.5Geometryofspace-time .......................................467<br />

17.5.1 Four-dimensionalspace-time ................................467<br />

17.5.2 Four-vectorscalarproducts .................................468<br />

17.5.3 M<strong>in</strong>kowskispace-time ....................................469<br />

17.5.4 Momentum-energyfourvector ...............................470<br />

17.6Lorentz-<strong>in</strong>variantformulationofLagrangianmechanics......................471<br />

17.6.1 Parametricformulation ...................................471<br />

17.6.2 ExtendedLagrangian ....................................471<br />

17.6.3 Extendedgeneralizedmomenta...............................473<br />

17.6.4 ExtendedLagrangeequationsofmotion ..........................473<br />

17.7Lorentz-<strong>in</strong>variantformulationsofHamiltonianmechanics.....................476<br />

17.7.1 Extendedcanonicalformalism................................476<br />

17.7.2 ExtendedPoissonBracketrepresentation .........................478<br />

17.7.3 ExtendedcanonicaltransformationandHamilton-Jacobitheory.............478<br />

17.7.4 ValidityoftheextendedHamilton-Lagrangeformalism..................478<br />

17.8TheGeneralTheoryofRelativity..................................480<br />

17.8.1 Thefundamentalconcepts..................................480<br />

17.8.2 E<strong>in</strong>ste<strong>in</strong>’spostulatesfortheGeneralTheoryofRelativity ................481<br />

17.8.3 Experimentalevidence<strong>in</strong>supportoftheGeneralTheoryofRelativity .........481<br />

17.9Implicationsofrelativistictheorytoclassicalmechanics .....................482<br />

17.10Summary ...............................................483<br />

Problems ..................................................484<br />

18 The transition to quantum physics 485<br />

18.1Introduction..............................................485<br />

18.2Briefsummaryoftheorig<strong>in</strong>sofquantumtheory .........................485

CONTENTS<br />

xi<br />

18.2.1 Bohrmodeloftheatom...................................487<br />

18.2.2 Quantization .........................................487<br />

18.2.3 Wave-particleduality ....................................488<br />

18.3Hamiltonian<strong>in</strong>quantumtheory...................................489<br />

18.3.1 Heisenberg’smatrix-mechanicsrepresentation.......................489<br />

18.3.2 Schröd<strong>in</strong>ger’swave-mechanicsrepresentation .......................491<br />

18.4Lagrangianrepresentation<strong>in</strong>quantumtheory...........................492<br />

18.5CorrespondencePr<strong>in</strong>ciple ......................................493<br />

18.6Summary ...............................................494<br />

19 Epilogue 495<br />

Appendices<br />

A Matrix algebra 497<br />

A.1 Mathematicalmethodsformechanics................................497<br />

A.2 Matrices................................................497<br />

A.3 Determ<strong>in</strong>ants .............................................501<br />

A.4 Reduction of a matrix to diagonal form . . . ...........................503<br />

B Vector algebra 505<br />

B.1 L<strong>in</strong>earoperations...........................................505<br />

B.2 Scalarproduct ............................................505<br />

B.3 Vectorproduct ............................................506<br />

B.4 Tripleproducts............................................507<br />

C Orthogonal coord<strong>in</strong>ate systems 509<br />

C.1 Cartesian coord<strong>in</strong>ates ( ) ....................................509<br />

C.2 Curvil<strong>in</strong>ear coord<strong>in</strong>ate systems ...................................509<br />

C.2.1 Two-dimensional polar coord<strong>in</strong>ates ( ) ..........................510<br />

C.2.2 Cyl<strong>in</strong>drical Coord<strong>in</strong>ates ( ) ..............................512<br />

C.2.3 Spherical Coord<strong>in</strong>ates ( ) ................................512<br />

C.3 Frenet-Serretcoord<strong>in</strong>ates ......................................513<br />

D Coord<strong>in</strong>ate transformations 515<br />

D.1 Translationaltransformations....................................515<br />

D.2 Rotationaltransformations .....................................515<br />

D.2.1 Rotationmatrix .......................................515<br />

D.2.2 F<strong>in</strong>iterotations........................................518<br />

D.2.3 Inf<strong>in</strong>itessimalrotations....................................519<br />

D.2.4 Properandimproperrotations ...............................519<br />

D.3 Spatial<strong>in</strong>versiontransformation...................................520<br />

D.4 Timereversaltransformation ....................................521<br />

E Tensor algebra 523<br />

E.1 Tensors ................................................523<br />

E.2 Tensorproducts............................................524<br />

E.2.1 Tensorouterproduct.....................................524<br />

E.2.2 Tensor<strong>in</strong>nerproduct.....................................524<br />

E.3 Tensorproperties...........................................525<br />

E.4 Contravariantandcovarianttensors ................................526<br />

E.5 Generalized<strong>in</strong>nerproduct......................................527<br />

E.6 Transformationpropertiesofobservables..............................528<br />

F Aspects of multivariate calculus 529<br />

F.1 Partial differentiation ........................................529

xii<br />

CONTENTS<br />

F.2 L<strong>in</strong>earoperators ...........................................529<br />

F.3 TransformationJacobian.......................................531<br />

F.3.1 Transformationof<strong>in</strong>tegrals:.................................531<br />

F.3.2 Transformation of differentialequations:..........................531<br />

F.3.3 PropertiesoftheJacobian: .................................531<br />

F.4 Legendretransformation.......................................532<br />

G Vector differential calculus 533<br />

G.1 Scalar differentialoperators .....................................533<br />

G.1.1 Scalar field ..........................................533<br />

G.1.2 Vector field ..........................................533<br />

G.2 Vector differentialoperators<strong>in</strong>cartesiancoord<strong>in</strong>ates .......................533<br />

G.2.1 Scalar field ..........................................533<br />

G.2.2 Vector field ..........................................534<br />

G.3 Vector differential operators <strong>in</strong> curvil<strong>in</strong>ear coord<strong>in</strong>ates . . ....................535<br />

G.3.1 Gradient: ...........................................535<br />

G.3.2 Divergence: ..........................................536<br />

G.3.3 Curl:..............................................536<br />

G.3.4 Laplacian:...........................................536<br />

H Vector <strong>in</strong>tegral calculus 537<br />

H.1 L<strong>in</strong>e <strong>in</strong>tegral of the gradient of a scalar field............................537<br />

H.2 Divergencetheorem .........................................537<br />

H.2.1 Flux of a vector fieldforGaussiansurface.........................537<br />

H.2.2 Divergence<strong>in</strong>cartesiancoord<strong>in</strong>ates. ............................538<br />

H.3 StokesTheorem............................................540<br />

H.3.1 Thecurl............................................540<br />

H.3.2 Curl<strong>in</strong>cartesiancoord<strong>in</strong>ates................................541<br />

H.4 Potential formulations of curl-free and divergence-free fields ...................543<br />

I Waveform analysis 545<br />

I.1 Harmonicwaveformdecomposition.................................545<br />

I.1.1 PeriodicsystemsandtheFourierseries...........................545<br />

I.1.2 AperiodicsystemsandtheFourierTransform.......................547<br />

I.2 Time-sampledwaveformanalysis ..................................548<br />

I.2.1 Delta-functionimpulseresponse ..............................549<br />

I.2.2 Green’sfunctionwaveformdecomposition .........................550<br />

Bibliography 551<br />

Index 555

Examples<br />

2.1 Example: Explod<strong>in</strong>g cannon shell .................................... 15<br />

2.2 Example: Billiard-ball collisions ..................................... 15<br />

2.3 Example: Bolas thrown by gaucho ................................... 17<br />

2.4 Example: Central force ......................................... 20<br />

2.5 Example: The ideal gas law ....................................... 23<br />

2.6 Example: The mass of galaxies ..................................... 23<br />

2.7 Example: Diatomic molecule ...................................... 26<br />

2.8 Example: Roller coaster ......................................... 27<br />

2.9 Example: Vertical fall <strong>in</strong> the earth’s gravitational field. ........................ 28<br />

2.10 Example: Projectile motion <strong>in</strong> air ................................... 29<br />

2.11 Example: Moment of <strong>in</strong>ertia of a th<strong>in</strong> door .............................. 33<br />

2.12 Example: Merry-go-round ........................................ 33<br />

2.13 Example: Cue pushes a billiard ball .................................. 33<br />

2.14 Example: Center of percussion of a baseball bat ............................ 35<br />

2.15 Example: Energy transfer <strong>in</strong> charged-particle scatter<strong>in</strong>g ....................... 36<br />

2.16 Example: Field of a uniform sphere .................................. 45<br />

3.1 Example: Harmonically-driven series RLC circuit .......................... 65<br />

3.2 Example: Vibration isolation ...................................... 69<br />

3.3 Example: Water waves break<strong>in</strong>g on a beach .............................. 74<br />

3.4 Example: Surface waves for deep water ................................ 74<br />

3.5 Example: Electromagnetic waves <strong>in</strong> ionosphere ............................ 75<br />

3.6 Example: Fourier transform of a Gaussian wave packet: ....................... 77<br />

3.7 Example: Fourier transform of a rectangular wave packet: ...................... 77<br />

3.8 Example: Acoustic wave packet ..................................... 79<br />

3.9 Example: Gravitational red shift .................................... 79<br />

3.10 Example: Quantum baseball ....................................... 80<br />

4.1 Example: Non-l<strong>in</strong>ear oscillator ..................................... 87<br />

5.1 Example: Shortest distance between two po<strong>in</strong>ts ............................110<br />

5.2 Example: Brachistochrone problem ...................................110<br />

5.3 Example: M<strong>in</strong>imal travel cost ......................................112<br />

5.4 Example: Surface area of a cyl<strong>in</strong>drically-symmetric soap bubble ...................113<br />

5.5 Example: Fermat’s Pr<strong>in</strong>ciple ......................................115<br />

5.6 Example: M<strong>in</strong>imum of (∇) 2 <strong>in</strong> a volume ..............................117<br />

5.7 Example: Two dependent variables coupled by one holonomic constra<strong>in</strong>t ..............123<br />

5.8 Example: Catenary ............................................125<br />

5.9 Example: The Queen Dido problem ...................................125<br />

6.1 Example: Motion of a free particle, U=0 ...............................142<br />

6.2 Example: Motion <strong>in</strong> a uniform gravitational field ...........................142<br />

6.3 Example: Central forces .........................................143<br />

6.4 Example: Disk roll<strong>in</strong>g on an <strong>in</strong>cl<strong>in</strong>ed plane ..............................144<br />

6.5 Example: Two connected masses on frictionless <strong>in</strong>cl<strong>in</strong>ed planes ...................147<br />

6.6 Example: Two blocks connected by a frictionless bar .........................148<br />

6.7 Example: Block slid<strong>in</strong>g on a movable frictionless <strong>in</strong>cl<strong>in</strong>ed plane ...................149<br />

6.8 Example: Sphere roll<strong>in</strong>g without slipp<strong>in</strong>g down an <strong>in</strong>cl<strong>in</strong>ed plane on a frictionless floor. .....150<br />

6.9 Example: Mass slid<strong>in</strong>g on a rotat<strong>in</strong>g straight frictionless rod. ....................150<br />

xiii

xiv<br />

EXAMPLES<br />

6.10 Example: Spherical pendulum ......................................151<br />

6.11 Example: Spr<strong>in</strong>g plane pendulum ....................................152<br />

6.12 Example: The yo-yo ...........................................153<br />

6.13 Example: Mass constra<strong>in</strong>ed to move on the <strong>in</strong>side of a frictionless paraboloid ...........154<br />

6.14 Example: Mass on a frictionless plane connected to a plane pendulum ...............155<br />

6.15 Example: Two connected masses constra<strong>in</strong>ed to slide along a mov<strong>in</strong>g rod ..............156<br />

6.16 Example: Mass slid<strong>in</strong>g on a frictionless spherical shell ........................157<br />

6.17 Example: Roll<strong>in</strong>g solid sphere on a spherical shell ..........................159<br />

6.18 Example: Solid sphere roll<strong>in</strong>g plus slipp<strong>in</strong>g on a spherical shell ...................161<br />

6.19 Example: Small body held by friction on the periphery of a roll<strong>in</strong>g wheel ..............162<br />

6.20 Example: Plane pendulum hang<strong>in</strong>g from a vertically-oscillat<strong>in</strong>g support ..............165<br />

6.21 Example: Series-coupled double pendulum subject to impulsive force .................167<br />

7.1 Example: Feynman’s angular-momentum paradox ...........................176<br />

7.2 Example: Atwoods mach<strong>in</strong>e .......................................178<br />

7.3 Example: Conservation of angular momentum for rotational <strong>in</strong>variance: ..............179<br />

7.4 Example: Diatomic molecules and axially-symmetric nuclei .....................180<br />

7.5 Example: L<strong>in</strong>ear harmonic oscillator on a cart mov<strong>in</strong>g at constant velocity ............185<br />

7.6 Example: Isotropic central force <strong>in</strong> a rotat<strong>in</strong>g frame .........................186<br />

7.7 Example: The plane pendulum .....................................187<br />

7.8 Example: Oscillat<strong>in</strong>g cyl<strong>in</strong>der <strong>in</strong> a cyl<strong>in</strong>drical bowl ..........................187<br />

8.1 Example: Motion <strong>in</strong> a uniform gravitational field ...........................201<br />

8.2 Example: One-dimensional harmonic oscillator ............................201<br />

8.3 Example: Plane pendulum ........................................202<br />

8.4 Example: Hooke’s law force constra<strong>in</strong>ed to the surface of a cyl<strong>in</strong>der .................203<br />

8.5 Example: Electron motion <strong>in</strong> a cyl<strong>in</strong>drical magnetron ........................204<br />

8.6 Example: Spherical pendulum us<strong>in</strong>g Hamiltonian mechanics .....................209<br />

8.7 Example: Spherical pendulum us<strong>in</strong>g ( ˙ ˙ ) .....................210<br />

8.8 Example: Spherical pendulum us<strong>in</strong>g ( ˙) ...................211<br />

8.9 Example: S<strong>in</strong>gle particle mov<strong>in</strong>g <strong>in</strong> a vertical plane under the <strong>in</strong>fluence of an <strong>in</strong>verse-square<br />

central force ...............................................212<br />

8.10 Example: Folded cha<strong>in</strong> ..........................................213<br />

8.11 Example: Fall<strong>in</strong>g cha<strong>in</strong> .........................................214<br />

9.1 Example: Gauge <strong>in</strong>variance <strong>in</strong> electromagnetism ...........................229<br />

10.1 Example: Driven, l<strong>in</strong>early-damped, coupled l<strong>in</strong>ear oscillators .....................239<br />

10.2 Example: Kirchhoff’s rules for electrical circuits ...........................240<br />

10.3 Example: The l<strong>in</strong>early-damped, l<strong>in</strong>ear oscillator: ...........................241<br />

11.1 Example: Central force lead<strong>in</strong>g to a circular orbit =2 cos ...................250<br />

11.2 Example: Orbit equation of motion for a free body ..........................252<br />

11.3 Example: L<strong>in</strong>ear two-body restor<strong>in</strong>g force ...............................265<br />

11.4 Example: Inverse square law attractive force ..............................265<br />

11.5 Example: Attractive <strong>in</strong>verse cubic central force ............................266<br />

11.6 Example: Spirall<strong>in</strong>g mass attached by a str<strong>in</strong>g to a hang<strong>in</strong>g mass ..................267<br />

11.7 Example: Two-body scatter<strong>in</strong>g by an <strong>in</strong>verse cubic force .......................273<br />

12.1 Example: Accelerat<strong>in</strong>g spr<strong>in</strong>g plane pendulum .............................291<br />

12.2 Example: Surface of rotat<strong>in</strong>g liquid ...................................293<br />

12.3 Example: The pirouette .........................................294<br />

12.4 Example: Cranked plane pendulum ...................................296<br />

12.5 Example: Nucleon orbits <strong>in</strong> deformed nuclei ..............................297<br />

12.6 Example: Free fall from rest .......................................301<br />

12.7 Example: Projectile fired vertically upwards ..............................301<br />

12.8 Example: Motion parallel to Earth’s surface ..............................301<br />

13.1 Example: Inertia tensor of a solid cube rotat<strong>in</strong>g about the center of mass. .............316<br />

13.2 Example: Inertia tensor of about a corner of a solid cube. ......................317<br />

13.3 Example: Inertia tensor of a hula hoop ................................319<br />

13.4 Example: Inertia tensor of a th<strong>in</strong> book .................................319

EXAMPLES<br />

xv<br />

13.5 Example: Rotation about the center of mass of a solid cube .....................321<br />

13.6 Example: Rotation about the corner of the cube ............................322<br />

13.7 Example: Euler angle transformation ..................................327<br />

13.8 Example: Rotation of a dumbbell ....................................332<br />

13.9 Example: Precession rate for torque-free rotat<strong>in</strong>g symmetric rigid rotor ..............338<br />

13.10Example: Tennis racquet dynamics ...................................341<br />

13.11Example: Rotation of asymmetrically-deformed nuclei ........................342<br />

13.12Example: The Sp<strong>in</strong>n<strong>in</strong>g “Jack” ....................................345<br />

13.13Example: The Tippe Top ........................................346<br />

13.14Example: Tipp<strong>in</strong>g stability of a roll<strong>in</strong>g wheel .............................349<br />

13.15Example: Forces on the bear<strong>in</strong>gs of a rotat<strong>in</strong>g circular disk .....................350<br />

14.1 Example: The Grand Piano .......................................362<br />

14.2 Example: Two coupled l<strong>in</strong>ear oscillators ................................368<br />

14.3 Example: Two equal masses series-coupled by two equal spr<strong>in</strong>gs ...................370<br />

14.4 Example: Two parallel-coupled plane pendula .............................371<br />

14.5 Example: The series-coupled double plane pendula ..........................373<br />

14.6 Example: Three plane pendula; mean-field l<strong>in</strong>ear coupl<strong>in</strong>g ......................374<br />

14.7 Example: Three plane pendula; nearest-neighbor coupl<strong>in</strong>g ......................376<br />

14.8 Example: System of three bodies coupled by six spr<strong>in</strong>gs ........................378<br />

14.9 Example: L<strong>in</strong>ear triatomic molecular CO 2 ......................379<br />

14.10Example: Benzene r<strong>in</strong>g .........................................381<br />

14.11Example: Two l<strong>in</strong>early-damped coupled l<strong>in</strong>ear oscillators .......................388<br />

14.12Example: Collective motion <strong>in</strong> nuclei .................................391<br />

15.1 Example: Check that a transformation is canonical ..........................398<br />

15.2 Example: Angular momentum: .....................................401<br />

15.3 Example: Lorentz force <strong>in</strong> electromagnetism ..............................404<br />

15.4 Example: Wavemotion: .........................................404<br />

15.5 Example: Two-dimensional, anisotropic, l<strong>in</strong>ear oscillator ......................405<br />

15.6 Example: The eccentricity vector ....................................406<br />

15.7 Example: The identity canonical transformation ...........................412<br />

15.8 Example: The po<strong>in</strong>t canonical transformation .............................412<br />

15.9 Example: The exchange canonical transformation ...........................412<br />

15.10Example: Inf<strong>in</strong>itessimal po<strong>in</strong>t canonical transformation .......................412<br />

15.11Example: 1-D harmonic oscillator via a canonical transformation ..................413<br />

15.12Example: Free particle ..........................................417<br />

15.13Example: Po<strong>in</strong>t particle <strong>in</strong> a uniform gravitational field .......................418<br />

15.14Example: One-dimensional harmonic oscillator ...........................419<br />

15.15Example: The central force problem ..................................419<br />

15.16Example: L<strong>in</strong>early-damped, one-dimensional, harmonic oscillator ..................421<br />

15.17Example:Adiabatic<strong>in</strong>varianceforthesimplependulum .......................428<br />

15.18Example: Harmonic oscillator perturbation ..............................430<br />

15.19Example: L<strong>in</strong>dblad resonance <strong>in</strong> planetary and galactic motion ...................431<br />

16.1 Example: Acoustic waves <strong>in</strong> a gas ...................................451<br />

17.1 Example: Muon lifetime .........................................462<br />

17.2 Example: Relativistic Doppler Effect ..................................463<br />

17.3 Example: Tw<strong>in</strong> paradox .........................................463<br />

17.4 Example: Rocket propulsion .......................................466<br />

17.5 Example: Lagrangian for a relativistic free particle ..........................474<br />

17.6 Example: Relativistic particle <strong>in</strong> an external electromagnetic field ..................475<br />

17.7 Example: The Bohr-Sommerfeld hydrogen atom ............................479<br />

A.1 Example: Eigenvalues and eigenvectors of a real symmetric matrix .................504<br />

D.1 Example: Rotation matrix: .......................................517<br />

D.2 Example: Proof that a rotation matrix is orthogonal .........................518<br />

E.1 Example: Displacement gradient tensor ................................524<br />

F.1 Example: Jacobian for transform from cartesian to spherical coord<strong>in</strong>ates ..............531

xvi<br />

EXAMPLES<br />

H.1 Example: Maxwell’s Flux Equations ..................................539<br />

H.2 Example: Buoyancy forces <strong>in</strong> fluids ..................................540<br />

H.3 Example: Maxwell’s circulation equations ...............................542<br />

H.4 Example: Electromagnetic fields: ....................................543<br />

I.1 Example: Fourier transform of a s<strong>in</strong>gle isolated square pulse: ....................548<br />

I.2 Example: Fourier transform of the Dirac delta function: .......................548

Preface<br />

The goal of this book is to <strong>in</strong>troduce the reader to the <strong>in</strong>tellectual beauty, and philosophical implications,<br />

of the fact that nature obeys variational pr<strong>in</strong>ciples plus Hamilton’s Action Pr<strong>in</strong>ciple which underlie the<br />

Lagrangian and Hamiltonian analytical formulations of classical mechanics. These variational methods,<br />

which were developed for classical mechanics dur<strong>in</strong>g the 18 − 19 century, have become the preem<strong>in</strong>ent<br />

formalisms for classical dynamics, as well as for many other branches of modern science and eng<strong>in</strong>eer<strong>in</strong>g.<br />

The ambitious goal of this book is to lead the reader from the <strong>in</strong>tuitive Newtonian vectorial formulation, to<br />

<strong>in</strong>troduction of the more abstract variational pr<strong>in</strong>ciples that underlie Hamilton’s Pr<strong>in</strong>ciple and the related<br />

Lagrangian and Hamiltonian analytical formulations. This culm<strong>in</strong>ates <strong>in</strong> discussion of the contributions of<br />

variational pr<strong>in</strong>ciples to classical mechanics and the development of relativistic and quantum mechanics.<br />

The broad scope of this book attempts to unify the undergraduate physics curriculum by bridg<strong>in</strong>g the<br />

chasm that divides the Newtonian vector-differential formulation, and the <strong>in</strong>tegral variational formulation of<br />

classical mechanics, as well as the correspond<strong>in</strong>g philosophical approaches adopted <strong>in</strong> classical and quantum<br />

mechanics. This book <strong>in</strong>troduces the powerful variational techniques <strong>in</strong> mathematics, and their application to<br />

physics. Application of the concepts of the variational approach to classical mechanics is ideal for illustrat<strong>in</strong>g<br />

the power and beauty of apply<strong>in</strong>g variational pr<strong>in</strong>ciples.<br />

The development of this textbook was <strong>in</strong>fluenced by three textbooks: The <strong>Variational</strong> <strong>Pr<strong>in</strong>ciples</strong> of<br />

<strong>Mechanics</strong> by Cornelius Lanczos (1949) [La49], <strong>Classical</strong> <strong>Mechanics</strong> (1950) by Herbert Goldste<strong>in</strong>[Go50],<br />

and <strong>Classical</strong> Dynamics of Particles and Systems (1965) by Jerry B. Marion[Ma65]. Marion’s excellent<br />

textbook was unusual <strong>in</strong> partially bridg<strong>in</strong>g the chasm between the outstand<strong>in</strong>g graduate texts by Goldste<strong>in</strong><br />

and Lanczos, and a bevy of <strong>in</strong>troductory texts based on Newtonian mechanics that were available at that<br />

time. The present textbook was developed to provide a more modern presentation of the techniques and<br />

philosophical implications of the variational approaches to classical mechanics, with a breadth and depth<br />

close to that provided by Goldste<strong>in</strong> and Lanczos, but <strong>in</strong> a format that better matches the needs of the<br />

undergraduate student. An additional goal is to bridge the gap between classical and modern physics <strong>in</strong> the<br />

undergraduate curriculum. The underly<strong>in</strong>g philosophical approach adopted by this book was espoused by<br />

Galileo Galilei “You cannot teach a man anyth<strong>in</strong>g; you can only help him f<strong>in</strong>d it with<strong>in</strong> himself.”<br />

This book was written <strong>in</strong> support of the physics junior/senior undergraduate course P235W entitled<br />

“<strong>Variational</strong> <strong>Pr<strong>in</strong>ciples</strong> <strong>in</strong> <strong>Classical</strong> <strong>Mechanics</strong>” that the author taught at the University of Rochester between<br />

1993−2015. Initially the lecture notes were distributed to students to allow pre-lecture study, facilitate<br />

accurate transmission of the complicated formulae, and m<strong>in</strong>imize note tak<strong>in</strong>g dur<strong>in</strong>g lectures. These lecture<br />

notes evolved <strong>in</strong>to the present textbook. The target audience of this course typically comprised ≈ 70% junior/senior<br />

undergraduates, ≈ 25% sophomores, ≤ 5% graduate students, and the occasional well-prepared<br />

freshman. The target audience was physics and astrophysics majors, but the course attracted a significant<br />

fraction of majors from other discipl<strong>in</strong>es such as mathematics, chemistry, optics, eng<strong>in</strong>eer<strong>in</strong>g, music, and the<br />

humanities. As a consequence, the book <strong>in</strong>cludes appreciable <strong>in</strong>troductory level physics, plus mathematical<br />

review material, to accommodate the diverse range of prior preparation of the students. This textbook<br />

<strong>in</strong>cludes material that extends beyond what reasonably can be covered dur<strong>in</strong>g a one-term course. This supplemental<br />

material is presented to show the importance and broad applicability of variational concepts to<br />

classical mechanics. The book <strong>in</strong>cludes 161 worked examples, plus 158 assigned problems, to illustrate the<br />

concepts presented. Advanced group-theoretic concepts are m<strong>in</strong>imized to better accommodate the mathematical<br />

skills of the typical undergraduate physics major. To conform with modern literature <strong>in</strong> this field,<br />

this book follows the widely-adopted nomenclature used <strong>in</strong> “<strong>Classical</strong> <strong>Mechanics</strong>” by Goldste<strong>in</strong>[Go50], with<br />

recent additions by Johns[Jo05] and this textbook.<br />

The second edition of this book revised the presentation and <strong>in</strong>cludes recent developments <strong>in</strong> the field.<br />

xvii

xviii<br />

PREFACE<br />

The book is broken <strong>in</strong>to four major sections, the first of which presents a brief historical <strong>in</strong>troduction<br />

(chapter 1), followed by a review of the Newtonian formulation of mechanics plus gravitation (chapter<br />

2), l<strong>in</strong>ear oscillators and wave motion (chapter 3), and an <strong>in</strong>troduction to non-l<strong>in</strong>ear dynamics and chaos<br />

(chapter 4). The second section <strong>in</strong>troduces the variational pr<strong>in</strong>ciples of analytical mechanics that underlie<br />

this book. It <strong>in</strong>cludes an <strong>in</strong>troduction to the calculus of variations (chapter 5), the Lagrangian formulation of<br />

mechanics with applications to holonomic and non-holonomic systems (chapter 6), a discussion of symmetries,<br />

<strong>in</strong>variance, plus Noether’s theorem (chapter 7). This book presents an <strong>in</strong>troduction to the Hamiltonian, the<br />

Hamiltonian formulation of mechanics, the Routhian reduction technique, and a discussion of the subtleties<br />

<strong>in</strong>volved <strong>in</strong> apply<strong>in</strong>g variational pr<strong>in</strong>ciples to variable-mass problems.(Chapter 8). The second edition of<br />

this book presents a unified <strong>in</strong>troduction to Hamiltons Pr<strong>in</strong>ciple, <strong>in</strong>troduces a new approach for apply<strong>in</strong>g<br />

Hamilton’s Pr<strong>in</strong>ciple to systems subject to <strong>in</strong>itial boundary conditions, and discusses how best to exploit the<br />

hierarchy of related formulations based on action, Lagrangian/Hamiltonian, and equations of motion, when<br />

solv<strong>in</strong>g problems subject to symmetries (chapter 9). A consolidated <strong>in</strong>troduction to the application of the<br />

variational approach to nonconservative systems is presented (chapter 10). The third section of the book,<br />

applies Lagrangian and Hamiltonian formulations of classical dynamics to central force problems (chapter 11),<br />

motion <strong>in</strong> non-<strong>in</strong>ertial frames (chapter 12), rigid-body rotation (chapter 13), and coupled l<strong>in</strong>ear oscillators<br />

(chapter 14). The fourth section of the book <strong>in</strong>troduces advanced applications of Hamilton’s Action Pr<strong>in</strong>ciple,<br />

Lagrangian mechanics and Hamiltonian mechanics. These <strong>in</strong>clude Poisson brackets, Liouville’s theorem,<br />

canonical transformations, Hamilton-Jacobi theory, the action-angle technique (chapter 15), and classical<br />

mechanics <strong>in</strong> the cont<strong>in</strong>ua (chapter 16). This is followed by a brief review of the revolution <strong>in</strong> classical<br />

mechanics <strong>in</strong>troduced by E<strong>in</strong>ste<strong>in</strong>’s theory of relativistic mechanics. The extended theory of Lagrangian and<br />

Hamiltonian mechanics is used to apply variational techniques to the Special Theory of Relativity, followed<br />

by a discussion of the use of variational pr<strong>in</strong>ciples <strong>in</strong> the development of the General Theory of Relativity<br />

(chapter 17). The book f<strong>in</strong>ishes with a brief review of the role of variational pr<strong>in</strong>ciples <strong>in</strong> bridg<strong>in</strong>g the gap<br />

between classical mechanics and quantum mechanics, (chapter 18). These advanced topics extend beyond<br />

the typical syllabus for an undergraduate classical mechanics course. They are <strong>in</strong>cluded to stimulate student<br />

<strong>in</strong>terest <strong>in</strong> physics by giv<strong>in</strong>g them a glimpse of the physicsatthesummitthattheyhavealreadystruggled<br />

to climb. This glimpse illustrates the breadth of classical mechanics, and the pivotal role that variational<br />

pr<strong>in</strong>ciples have played <strong>in</strong> the development of classical, relativistic, quantal, and statistical mechanics.<br />

The front cover picture of this book shows a sailplane soar<strong>in</strong>g high above the Italian Alps. This picture<br />

epitomizes the unlimited horizon of opportunities provided when the full dynamic range of variational pr<strong>in</strong>ciples<br />

are applied to classical mechanics. The adjacent pictures of the galaxy, and the skier, represent the wide<br />

dynamic range of applicable topics that span from the orig<strong>in</strong> of the universe, to everyday life. These cover<br />

pictures reflect the beauty and unity of the foundation provided by variational pr<strong>in</strong>ciples to the development<br />

of classical mechanics.<br />

Information regard<strong>in</strong>g the associated P235 undergraduate course at the University of Rochester is available<br />

on the web site at http://www.pas.rochester.edu/~cl<strong>in</strong>e/P235/<strong>in</strong>dex.shtml. Information about the<br />

author is available at the Cl<strong>in</strong>e home web site: http://www.pas.rochester.edu/~cl<strong>in</strong>e/<strong>in</strong>dex.html.<br />

The author thanks Meghan Sarkis who prepared many of the illustrations, Joe Easterly who designed<br />

the book cover plus the webpage, and Moriana Garcia who organized the open-access publication. Andrew<br />

Sifa<strong>in</strong> developed the diagnostic problems <strong>in</strong>cluded <strong>in</strong> the book. The author appreciates the permission,<br />

granted by Professor Struckmeier, to quote his published article on the extended Hamilton-Lagrangian<br />

formalism. The author acknowledges the feedback and suggestions made by many students who have taken<br />

this course, as well as helpful suggestions by his colleagues; Andrew Abrams, Adam Hayes, Connie Jones,<br />

Andrew Melchionna, David Munson, Alice Quillen, Richard Sarkis, James Schneeloch, Steven Torrisi, Dan<br />

Watson, and Frank Wolfs. These lecture notes were typed <strong>in</strong> LATEX us<strong>in</strong>g Scientific WorkPlace (MacKichan<br />

Software, Inc.), while Adobe Illustrator, Photoshop, Orig<strong>in</strong>, Mathematica, and MUPAD, were used to prepare<br />

the illustrations.<br />

Douglas Cl<strong>in</strong>e,<br />

University of Rochester, <strong>2019</strong>

Prologue<br />

Two dramatically different philosophical approaches to science were developed <strong>in</strong> the field of classical mechanics<br />

dur<strong>in</strong>g the 17 - 18 centuries. This time period co<strong>in</strong>cided with the Age of Enlightenment <strong>in</strong> Europe<br />

dur<strong>in</strong>g which remarkable <strong>in</strong>tellectual and philosophical developments occurred. This was a time when both<br />

philosophical and causal arguments were equally acceptable <strong>in</strong> science, <strong>in</strong> contrast with current convention<br />

where there appears to be tacit agreement to discourage use of philosophical arguments <strong>in</strong> science.<br />

Snell’s Law: The genesis of two contrast<strong>in</strong>g philosophical approaches<br />

to science relates back to early studies of the reflection<br />

and refraction of light. The velocity of light <strong>in</strong> a medium of refractive<br />

<strong>in</strong>dex equals = <br />

. Thus a light beam <strong>in</strong>cident at an<br />

angle 1 to the normal of a plane <strong>in</strong>terface between medium 1<br />

and medium 2 isrefractedatanangle 2 <strong>in</strong> medium 2 where the<br />

angles are related by Snell’s Law.<br />

s<strong>in</strong> 1<br />

= 1<br />

= 2<br />

(Snell’s Law)<br />

s<strong>in</strong> 2 2 1<br />

Ibn Sahl of Bagdad (984) first described the refraction of light,<br />

while Snell (1621) derived his law mathematically. Both of these<br />

scientists used the “vectorial approach” where the light velocity <br />

is considered to be a vector po<strong>in</strong>t<strong>in</strong>g <strong>in</strong> the direction of propagation.<br />

Fermat’s Pr<strong>in</strong>ciple: Fermat’s pr<strong>in</strong>ciple of least time (1657),<br />

which is based on the work of Hero of Alexandria (∼ 60) and Ibn<br />

al-Haytham (1021), states that “light travels between two given<br />

po<strong>in</strong>ts along the path of shortest time”. The transit time of a<br />

light beam between two locations and <strong>in</strong> a medium with<br />

position-dependent refractive <strong>in</strong>dex () is given by<br />

=<br />

Z <br />

<br />

= 1 <br />

Z <br />

<br />

()<br />

(Fermat’s Pr<strong>in</strong>ciple)<br />

Fermat’s Pr<strong>in</strong>ciple leads to the derivation of Snell’s Law.<br />

Philosophically the physics underly<strong>in</strong>g the contrast<strong>in</strong>g vectorial<br />

and Fermat’s Pr<strong>in</strong>ciple derivations of Snell’s Law are dramatically<br />

different. The vectorial approach is based on differential relations<br />

between the velocity vectors <strong>in</strong> the two media, whereas Fermat’s<br />

variational approach is based on the fact that the light preferentially<br />

selects a path for which the <strong>in</strong>tegral of the transit time<br />

Figure 1: Vectorial and variational representations<br />

of Snell’s Law for refraction of light.<br />

between the <strong>in</strong>itial location and the f<strong>in</strong>al location is m<strong>in</strong>imized.<br />

That is, the first approach is based on “vectorial mechanics” whereas Fermat’s approach is based on<br />

variational pr<strong>in</strong>ciples <strong>in</strong> that the path between the <strong>in</strong>itial and f<strong>in</strong>al locations is varied to f<strong>in</strong>d the path that<br />

m<strong>in</strong>imizes the transit time. Fermat’s enunciation of variational pr<strong>in</strong>ciples <strong>in</strong> physics played a key role <strong>in</strong> the<br />

historical development, and subsequent exploitation, of the pr<strong>in</strong>ciple of least action <strong>in</strong> analytical formulations<br />

of classical mechanics as discussed below.<br />

xix

xx<br />

PROLOGUE<br />

Newtonian mechanics: Momentum and force are vectors that underlie the Newtonian formulation of<br />

classical mechanics. Newton’s monumental treatise, entitled “Philosophiae Naturalis Pr<strong>in</strong>cipia Mathematica”,<br />

published <strong>in</strong> 1687, established his three universal laws of motion, the universal theory of gravitation,<br />

the derivation of Kepler’s three laws of planetary motion, and the development of calculus. Newton’s three<br />

universal laws of motion provide the most <strong>in</strong>tuitive approach to classical mechanics <strong>in</strong> that they are based on<br />

vector quantities like momentum, and the rate of change of momentum, which are related to force. Newton’s<br />

equation of motion<br />

F = p<br />

(Newton’s equation of motion)<br />

<br />

is a vector differential relation between the <strong>in</strong>stantaneous forces and rate of change of momentum, or equivalent<br />

<strong>in</strong>stantaneous acceleration, all of which are vector quantities. Momentum and force are easy to visualize,<br />

and both cause and effect are embedded <strong>in</strong> Newtonian mechanics. Thus, if all of the forces, <strong>in</strong>clud<strong>in</strong>g the<br />

constra<strong>in</strong>t forces, act<strong>in</strong>g on the system are known, then the motion is solvable for two body systems. The<br />

mathematics for handl<strong>in</strong>g Newton’s “vectorial mechanics” approach to classical mechanics is well established.<br />

Analytical mechanics: <strong>Variational</strong> pr<strong>in</strong>ciples apply to many aspects of our daily life. Typical examples<br />

<strong>in</strong>clude; select<strong>in</strong>g the optimum compromise <strong>in</strong> quality and cost when shopp<strong>in</strong>g, select<strong>in</strong>g the fastest route<br />

to travel from home to work, or select<strong>in</strong>g the optimum compromise to satisfy the disparate desires of the<br />

<strong>in</strong>dividuals compris<strong>in</strong>g a family. <strong>Variational</strong> pr<strong>in</strong>ciples underlie the analytical formulation of mechanics. It<br />

is astonish<strong>in</strong>g that the laws of nature are consistent with variational pr<strong>in</strong>ciples <strong>in</strong>volv<strong>in</strong>g the pr<strong>in</strong>ciple of<br />

least action. M<strong>in</strong>imiz<strong>in</strong>g the action <strong>in</strong>tegral led to the development of the mathematical field of variational<br />

calculus, plus the analytical variational approaches to classical mechanics, by Euler, Lagrange, Hamilton,<br />

and Jacobi.<br />

Leibniz, who was a contemporary of Newton, <strong>in</strong>troduced methods based on a quantity called “vis viva”,<br />

which is Lat<strong>in</strong> for “liv<strong>in</strong>g force” and equals twice the k<strong>in</strong>etic energy. Leibniz believed <strong>in</strong> the philosophy<br />

that God created a perfect world where nature would be thrifty <strong>in</strong> all its manifestations. In 1707, Leibniz<br />

proposed that the optimum path is based on m<strong>in</strong>imiz<strong>in</strong>g the time <strong>in</strong>tegral of the vis viva, which is equivalent<br />

to the action <strong>in</strong>tegral of Lagrangian/Hamiltonian mechanics. In 1744 Euler derived the Leibniz result<br />

us<strong>in</strong>g variational concepts while Maupertuis restated the Leibniz result based on teleological arguments.<br />

The development of Lagrangian mechanics culm<strong>in</strong>ated <strong>in</strong> the 1788 publication of Lagrange’s monumental<br />

treatise entitled “Mécanique Analytique”. Lagrange used d’Alembert’s Pr<strong>in</strong>ciple to derive Lagrangian mechanics<br />

provid<strong>in</strong>g a powerful analytical approach to determ<strong>in</strong>e the magnitude and direction of the optimum<br />

trajectories, plus the associated forces.<br />

The culm<strong>in</strong>ation of the development of analytical mechanics occurred <strong>in</strong> 1834 when Hamilton proposed<br />

his Pr<strong>in</strong>ciple of Least Action, as well as develop<strong>in</strong>g Hamiltonian mechanics which is the premier variational<br />

approach <strong>in</strong> science. Hamilton’s concept of least action is def<strong>in</strong>ed to be the time <strong>in</strong>tegral of the Lagrangian.<br />

Hamilton’s Action Pr<strong>in</strong>ciple (1834) m<strong>in</strong>imizes the action <strong>in</strong>tegral def<strong>in</strong>ed by<br />

=<br />

Z <br />

<br />

(q ˙q)<br />

(Hamilton’s Pr<strong>in</strong>ciple)<br />

In the simplest form, the Lagrangian (q ˙q) equals the difference between the k<strong>in</strong>etic energy and the<br />

potential energy . Hamilton’s Least Action Pr<strong>in</strong>ciple underlies Lagrangian mechanics. This Lagrangian is<br />

a function of generalized coord<strong>in</strong>ates plus their correspond<strong>in</strong>g velocities ˙ . Hamilton also developed<br />

the premier variational approach, called Hamiltonian mechanics, that is based on the Hamiltonian (q p)<br />

which is a function of the fundamental position plus the conjugate momentum variables. In 1843<br />

Jacobi provided the mathematical framework required to fully exploit the power of Hamiltonian mechanics.<br />

Note that the Lagrangian, Hamiltonian, and the action <strong>in</strong>tegral, all are scalar quantities which simplifies<br />

derivation of the equations of motion compared with the vector calculus used by Newtonian mechanics.<br />

Figure 2 presents a philosophical roadmap illustrat<strong>in</strong>g the hierarchy of philosophical approaches based on<br />

Hamilton’s Action Pr<strong>in</strong>ciple, that are available for deriv<strong>in</strong>g the equations of motion of a system. The primary<br />

Stage1 uses Hamilton’s Action functional, = R <br />

<br />

(q ˙q) to derive the Lagrangian, and Hamiltonian<br />

functionals which provide the most fundamental and sophisticated level of understand<strong>in</strong>g. Stage1 <strong>in</strong>volves<br />

specify<strong>in</strong>g all the active degrees of freedom, as well as the <strong>in</strong>teractions <strong>in</strong>volved. Stage2 uses the Lagrangian<br />

or Hamiltonian functionals, derived at Stage1, <strong>in</strong> order to derive the equations of motion for the system of

xxi<br />

Hamilton’s action pr<strong>in</strong>ciple<br />

Stage 1<br />

Stage 2<br />

Hamiltonian<br />

Lagrangian<br />

d’ Alembert’s Pr<strong>in</strong>ciple<br />

Equations of motion<br />

Newtonian mechanics<br />

Stage 3<br />

Solution for motion<br />

Initial conditions<br />

Figure 2: Philosophical road map of the hierarchy of stages <strong>in</strong>volved <strong>in</strong> analytical mechanics. Hamilton’s<br />

Action Pr<strong>in</strong>ciple is the foundation of analytical mechanics. Stage 1 uses Hamilton’s Pr<strong>in</strong>ciple to derive the<br />

Lagranian and Hamiltonian. Stage 2 uses either the Lagrangian or Hamiltonian to derive the equations<br />

of motion for the system. Stage 3 uses these equations of motion to solve for the actual motion us<strong>in</strong>g<br />

the assumed <strong>in</strong>itial conditions. The Lagrangian approach can be derived directly based on d’Alembert’s<br />

Pr<strong>in</strong>ciple. Newtonian mechanics can be derived directly based on Newton’s Laws of Motion. The advantages<br />

and power of Hamilton’s Action Pr<strong>in</strong>ciple are unavailable if the Laws of Motion are derived us<strong>in</strong>g either<br />

d’Alembert’s Pr<strong>in</strong>ciple or Newton’s Laws of Motion.<br />

<strong>in</strong>terest. Stage3 then uses these derived equations of motion to solve for the motion of the system subject to<br />

a given set of <strong>in</strong>itial boundary conditions. Note that Lagrange first derived Lagrangian mechanics based on<br />

d’ Alembert’s Pr<strong>in</strong>ciple, while Newton’s Laws of Motion specify the equations of motion used <strong>in</strong> Newtonian<br />

mechanics.<br />

The analytical approach to classical mechanics appeared contradictory to Newton’s <strong>in</strong>tuitive vectorial<br />

treatment of force and momentum. There is a dramatic difference <strong>in</strong> philosophy between the vectordifferential<br />

equations of motion derived by Newtonian mechanics, which relate the <strong>in</strong>stantaneous force to<br />

the correspond<strong>in</strong>g <strong>in</strong>stantaneous acceleration, and analytical mechanics, where m<strong>in</strong>imiz<strong>in</strong>g the scalar action<br />

<strong>in</strong>tegral <strong>in</strong>volves <strong>in</strong>tegrals over space and time between specified <strong>in</strong>itial and f<strong>in</strong>al states. Analytical mechanics<br />

uses variational pr<strong>in</strong>ciples to determ<strong>in</strong>e the optimum trajectory, from a cont<strong>in</strong>uum of tentative possibilities,<br />

by requir<strong>in</strong>g that the optimum trajectory m<strong>in</strong>imizes the action <strong>in</strong>tegral between specified <strong>in</strong>itial and f<strong>in</strong>al<br />

conditions.<br />

Initially there was considerable prejudice and philosophical opposition to use of the variational pr<strong>in</strong>ciples<br />

approach which is based on the assumption that nature follows the pr<strong>in</strong>ciples of economy. The variational<br />

approach is not <strong>in</strong>tuitive, and thus it was considered to be speculative and “metaphysical”, but it was<br />

tolerated as an efficient tool for exploit<strong>in</strong>g classical mechanics. This opposition to the variational pr<strong>in</strong>ciples<br />

underly<strong>in</strong>g analytical mechanics, delayed full appreciation of the variational approach until the start of the<br />

20 century. As a consequence, the <strong>in</strong>tuitive Newtonian formulation reigned supreme <strong>in</strong> classical mechanics<br />

for over two centuries, even though the remarkable problem-solv<strong>in</strong>g capabilities of analytical mechanics were<br />

recognized and exploited follow<strong>in</strong>g the development of analytical mechanics by Lagrange.<br />

The full significance and superiority of the analytical variational formulations of classical mechanics<br />

became well recognised and accepted follow<strong>in</strong>g the development of the Special Theory of Relativity <strong>in</strong> 1905.<br />

The Theory of Relativity requires that the laws of nature be <strong>in</strong>variant to the reference frame. This is not<br />

satisfied by the Newtonian formulation of mechanics which assumes one absolute frame of reference and a<br />

separation of space and time. In contrast, the Lagrangian and Hamiltonian formulations of the pr<strong>in</strong>ciple of<br />

least action rema<strong>in</strong> valid <strong>in</strong> the Theory of Relativity, if the Lagrangian is written <strong>in</strong> a relativistically-<strong>in</strong>variant

xxii<br />

PROLOGUE<br />

form <strong>in</strong> space-time. The complete <strong>in</strong>variance of the variational approach to coord<strong>in</strong>ate frames is precisely<br />

the formalism necessary for handl<strong>in</strong>g relativistic mechanics.<br />

Hamiltonian mechanics, which is expressed <strong>in</strong> terms of the conjugate variables (q p), relates classical<br />

mechanics directly to the underly<strong>in</strong>g physics of quantum mechanics and quantum field theory. As a consequence,<br />

the philosophical opposition to exploit<strong>in</strong>g variational pr<strong>in</strong>ciples no longer exists, and Hamiltonian<br />

mechanics has become the preem<strong>in</strong>ent formulation of modern physics. The reader is free to draw their own<br />

conclusions regard<strong>in</strong>g the philosophical question “is the pr<strong>in</strong>ciple of economy a fundamental law of classical<br />

mechanics, or is it a fortuitous consequence of the fundamental laws of nature?”<br />

From the late seventeenth century, until the dawn of modern physics at the start of the twentieth century,<br />

classical mechanics rema<strong>in</strong>ed a primary driv<strong>in</strong>g force <strong>in</strong> the development of physics. <strong>Classical</strong> mechanics<br />

embraces an unusually broad range of topics spann<strong>in</strong>g motion of macroscopic astronomical bodies to microscopic<br />

particles <strong>in</strong> nuclear and particle physics, at velocities rang<strong>in</strong>g from zero to near the velocity of<br />

light, from one-body to statistical many-body systems, as well as hav<strong>in</strong>g extensions to quantum mechanics.<br />

Introduction of the Special Theory of Relativity <strong>in</strong> 1905, and the General Theory of Relativity <strong>in</strong> 1916,<br />

necessitated modifications to classical mechanics for relativistic velocities, and can be considered to be an<br />

extended theory of classical mechanics. S<strong>in</strong>ce the 1920 0 s, quantal physics has superseded classical mechanics<br />

<strong>in</strong> the microscopic doma<strong>in</strong>. Although quantum physics has played the lead<strong>in</strong>g role <strong>in</strong> the development of<br />

physics dur<strong>in</strong>g much of the past century, classical mechanics still is a vibrant field of physics that recently<br />

has led to excit<strong>in</strong>g developments associated with non-l<strong>in</strong>ear systems and chaos theory. This has spawned<br />

new branches of physics and mathematics as well as chang<strong>in</strong>g our notion of causality.<br />

Goals: The primary goal of this book is to <strong>in</strong>troduce the reader to the powerful variational-pr<strong>in</strong>ciples<br />

approaches that play such a pivotal role <strong>in</strong> classical mechanics and many other branches of modern science<br />

and eng<strong>in</strong>eer<strong>in</strong>g. This book emphasizes the <strong>in</strong>tellectual beauty of these remarkable developments, as well as<br />

stress<strong>in</strong>g the philosophical implications that have had a tremendous impact on modern science. A secondary<br />

goal is to apply variational pr<strong>in</strong>ciples to solve advanced applications <strong>in</strong> classical mechanics <strong>in</strong> order to<br />

<strong>in</strong>troduce many sophisticated and powerful mathematical techniques that underlie much of modern physics.<br />

This book starts with a review of Newtonian mechanics plus the solutions of the correspond<strong>in</strong>g equations<br />

of motion. This is followed by an <strong>in</strong>troduction to Lagrangian mechanics, based on d’Alembert’s Pr<strong>in</strong>ciple,<br />

<strong>in</strong> order to develop familiarity <strong>in</strong> apply<strong>in</strong>g variational pr<strong>in</strong>ciples to classical mechanics. This leads to <strong>in</strong>troduction<br />

of the more fundamental Hamilton’s Action Pr<strong>in</strong>ciple, plus Hamiltonian mechanics, to illustrate the<br />

power provided by exploit<strong>in</strong>g the full hierarchy of stages available for apply<strong>in</strong>g variational pr<strong>in</strong>ciples to classical<br />

mechanics. F<strong>in</strong>ally the book illustrates how variational pr<strong>in</strong>ciples <strong>in</strong> classical mechanics were exploited<br />

dur<strong>in</strong>g the development of both relativisitic mechanics and quantum physics. The connections and applications<br />

of classical mechanics to modern physics, are emphasized throughout the book <strong>in</strong> an effort to span the<br />

chasm that divides the Newtonian vector-differential formulation, and the <strong>in</strong>tegral variational formulation, of<br />

classical mechanics. This chasm is especially applicable to quantum mechanics which is based completely on<br />

variational pr<strong>in</strong>ciples. Note that variational pr<strong>in</strong>ciples, developed <strong>in</strong> the field of classical mechanics, now are<br />

used <strong>in</strong> a diverse and wide range of fields outside of physics, <strong>in</strong>clud<strong>in</strong>g economics, meteorology, eng<strong>in</strong>eer<strong>in</strong>g,<br />

and comput<strong>in</strong>g.<br />

This study of classical mechanics <strong>in</strong>volves climb<strong>in</strong>g a vast mounta<strong>in</strong> of knowledge, and the pathway to the<br />

top leads to elegant and beautiful theories that underlie much of modern physics. This book exploits variational<br />

pr<strong>in</strong>ciples applied to four major topics <strong>in</strong> classical mechanics to illustrate the power and importance of<br />

variational pr<strong>in</strong>ciples <strong>in</strong> physics. Be<strong>in</strong>g so close to the summit provides the opportunity to take a few extra<br />

steps beyond the normal <strong>in</strong>troductory classical mechanics syllabus to glimpse the excit<strong>in</strong>g physics found at<br />

the summit. This new physics <strong>in</strong>cludes topics such as quantum, relativistic, and statistical mechanics.

Chapter 1<br />

A brief history of classical mechanics<br />

1.1 Introduction<br />

This chapter reviews the historical evolution of classical mechanics s<strong>in</strong>ce considerable <strong>in</strong>sight can be ga<strong>in</strong>ed<br />

from study of the history of science. There are two dramatically different approaches used <strong>in</strong> classical<br />

mechanics. The first is the vectorial approach of Newton which is based on vector quantities like momentum,<br />

force, and acceleration. The second is the analytical approach of Lagrange, Euler, Hamilton, and Jacobi,<br />

that is based on the concept of least action and variational calculus. The more <strong>in</strong>tuitive Newtonian picture<br />

reigned supreme <strong>in</strong> classical mechanics until the start of the twentieth century. <strong>Variational</strong> pr<strong>in</strong>ciples, which<br />

were developed dur<strong>in</strong>g the n<strong>in</strong>eteenth century, never aroused much enthusiasm <strong>in</strong> scientific circles due to<br />

philosophical objections to the underly<strong>in</strong>g concepts;thisapproachwasmerelytoleratedasanefficient tool<br />

for exploit<strong>in</strong>g classical mechanics. A dramatic advance <strong>in</strong> the philosophy of science occurred at the start of<br />

the 20 century lead<strong>in</strong>g to widespread acceptance of the superiority of us<strong>in</strong>g variational pr<strong>in</strong>ciples.<br />

1.2 Greek antiquity<br />

The great philosophers <strong>in</strong> ancient Greece played a key role by us<strong>in</strong>g the astronomical work of the Babylonians<br />

to develop scientific theories of mechanics. Thales of Miletus (624 - 547BC), thefirst of the seven<br />

great greek philosophers, developed geometry, and is hailed as the first true mathematician. Pythagorus<br />

(570 - 495BC) developed mathematics, and postulated that the earth is spherical. Democritus (460 -<br />

370BC) has been called the father of modern science, while Socrates (469 - 399BC) is renowned for his<br />

contributions to ethics. Plato (427-347 B.C.) who was a mathematician and student of Socrates, wrote<br />

important philosophical dialogues. He founded the Academy <strong>in</strong> Athens which was the first <strong>in</strong>stitution of<br />

higher learn<strong>in</strong>g <strong>in</strong> the Western world that helped lay the foundations of Western philosophy and science.<br />

Aristotle (384-322 B.C.) is an important founder of Western philosophy encompass<strong>in</strong>g ethics, logic,<br />

science, and politics. His views on the physical sciences profoundly <strong>in</strong>fluenced medieval scholarship that<br />

extended well <strong>in</strong>to the Renaissance. He presented the first implied formulation of the pr<strong>in</strong>ciple of virtual<br />

work <strong>in</strong> statics, and his statement that “what is lost <strong>in</strong> velocity is ga<strong>in</strong>ed <strong>in</strong> force” is a veiled reference to<br />

k<strong>in</strong>etic and potential energy. He adopted an Earth centered model of the universe. Aristarchus (310 - 240<br />

B.C.) argued that the Earth orbited the Sun and used measurements to imply the relative distances of the<br />

Moon and the Sun. The greek philosophers were relatively advanced <strong>in</strong> logic and mathematics and developed<br />

concepts that enabled them to calculate areas and perimeters. Unfortunately their philosophical approach<br />

neglected collect<strong>in</strong>g quantitative and systematic data that is an essential <strong>in</strong>gredient to the advancement of<br />

science.<br />

Archimedes (287-212 B.C.) represented the culm<strong>in</strong>ation of science <strong>in</strong> ancient Greece. As an eng<strong>in</strong>eer<br />

he designed mach<strong>in</strong>es of war, while as a scientist he made significant contributions to hydrostatics and<br />

the pr<strong>in</strong>ciple of the lever. As a mathematician, he applied <strong>in</strong>f<strong>in</strong>itessimals <strong>in</strong> a way that is rem<strong>in</strong>iscent of<br />

modern <strong>in</strong>tegral calculus, which he used to derive a value for Unfortunately much of the work of the<br />

brilliant Archimedes subsequently fell <strong>in</strong>to oblivion. Hero of Alexandria (10 - 70 A.D.) described the<br />

pr<strong>in</strong>ciple of reflection that light takes the shortest path. This is an early illustration of variational pr<strong>in</strong>ciple<br />

1

2 CHAPTER 1. A BRIEF HISTORY OF CLASSICAL MECHANICS<br />

of least time. Ptolemy (83 - 161 A.D.) wrote several scientific treatises that greatly <strong>in</strong>fluenced subsequent<br />

philosophers. Unfortunately he adopted the <strong>in</strong>correct geocentric solar system <strong>in</strong> contrast to the heliocentric<br />

model of Aristarchus and others.<br />

1.3 Middle Ages<br />

Thedecl<strong>in</strong>eandfalloftheRomanEmpire<strong>in</strong>∼410 A.D. marks the end of <strong>Classical</strong> Antiquity, and the<br />

beg<strong>in</strong>n<strong>in</strong>g of the Dark Ages <strong>in</strong> Western Europe (Christendom), while the Muslim scholars <strong>in</strong> Eastern Europe<br />

cont<strong>in</strong>ued to make progress <strong>in</strong> astronomy and mathematics. For example, <strong>in</strong> Egypt, Alhazen (965 - 1040<br />

A.D.) expanded the pr<strong>in</strong>ciple of least time to reflection and refraction. The Dark Ages <strong>in</strong>volved a long<br />

scientific decl<strong>in</strong>e <strong>in</strong> Western Europe that languished for about 900 years. Science was dom<strong>in</strong>ated by religious<br />

dogma, all western scholars were monks, and the important scientific achievements of Greek antiquity were<br />

forgotten. The works of Aristotle were re<strong>in</strong>troduced to Western Europe by Arabs <strong>in</strong> the early 13 century<br />

lead<strong>in</strong>g to the concepts of forces <strong>in</strong> static systems which were developed dur<strong>in</strong>g the fourteenth century.<br />

This <strong>in</strong>cluded concepts of the work done by a force, and the virtual work <strong>in</strong>volved <strong>in</strong> virtual displacements.<br />

Leonardo da V<strong>in</strong>ci (1452-1519) was a leader <strong>in</strong> mechanics at that time. He made sem<strong>in</strong>al contributions<br />

to science, <strong>in</strong> addition to his well known contributions to architecture, eng<strong>in</strong>eer<strong>in</strong>g, sculpture, and art.<br />

Nicolaus Copernicus (1473-1543) rejected the geocentric theory of Ptolomy and formulated a scientificallybased<br />

heliocentric cosmology that displaced the Earth from the center of the universe. The Ptolomic view<br />

was that heaven represented the perfect unchang<strong>in</strong>g div<strong>in</strong>e while the earth represented change plus chaos,<br />

and the celestial bodies moved relative to the fixed heavens. The book, De revolutionibus orbium coelestium<br />

(On the Revolutions of the Celestial Spheres), published by Copernicus <strong>in</strong> 1543, is regarded as the start<strong>in</strong>g<br />

po<strong>in</strong>t of modern astronomy and the def<strong>in</strong><strong>in</strong>g epiphany that began the Scientific Revolution. ThebookDe<br />

Magnete written <strong>in</strong> 1600 by the English physician William Gilbert (1540-1603) presented the results of<br />

well-planned studies of magnetism and strongly <strong>in</strong>fluenced the <strong>in</strong>tellectual-scientific evolution at that time.<br />

Johannes Kepler (1571-1630), a German mathematician, astronomer and astrologer, was a key<br />

figure <strong>in</strong> the 17th century Scientific Revolution. He is best known for recogniz<strong>in</strong>g the connection between the<br />

motions <strong>in</strong> the sky and physics. His laws of planetary motion were developed by later astronomers based on<br />

his written work Astronomia nova, Harmonices Mundi, andEpitome of Copernican Astrononomy. Kepler<br />

was an assistant to Tycho Brahe (1546-1601) who for many years recorded accurate astronomical data<br />

that played a key role <strong>in</strong> the development of Kepler’s theory of planetary motion. Kepler’s work provided<br />

the foundation for Isaac Newton’s theory of universal gravitation. Unfortunately Kepler did not recognize<br />

the true nature of the gravitational force.<br />

Galileo Galilei (1564-1642) built on the Aristotle pr<strong>in</strong>ciple by recogniz<strong>in</strong>g the law of <strong>in</strong>ertia, the<br />

persistence of motion if no forces act, and the proportionality between force and acceleration. This amounts<br />

to recognition of work as the product of force times displacement <strong>in</strong> the direction of the force. He applied<br />

virtual work to the equilibrium of a body on an <strong>in</strong>cl<strong>in</strong>ed plane. He also showed that the same pr<strong>in</strong>ciple<br />

applies to hydrostatic pressure that had been established by Archimedes, but he did not apply his concepts<br />

<strong>in</strong> classical mechanics to the considerable knowledge base on planetary motion. Galileo is famous for the<br />

apocryphal story that he dropped two cannon balls of differentmassesfromtheTowerofPisatodemonstrate<br />

that their speed of descent was <strong>in</strong>dependent of their mass.<br />

1.4 Age of Enlightenment<br />

The Age of Enlightenment is a term used to describe a phase <strong>in</strong> Western philosophy and cultural life <strong>in</strong><br />

which reason was advocated as the primary source and legitimacy for authority. It developed simultaneously<br />

<strong>in</strong> Germany, France, Brita<strong>in</strong>, the Netherlands, and Italy around the 1650’s and lasted until the French<br />

Revolution <strong>in</strong> 1789. The <strong>in</strong>tellectual and philosophical developments led to moral, social, and political<br />

reforms. The pr<strong>in</strong>ciples of <strong>in</strong>dividual rights, reason, common sense, and deism were a revolutionary departure<br />

from the exist<strong>in</strong>g theocracy, autocracy, oligarchy, aristocracy, and the div<strong>in</strong>e right of k<strong>in</strong>gs. It led to political<br />

revolutions <strong>in</strong> France and the United States. It marks a dramatic departure from the Early Modern period<br />

which was noted for religious authority, absolute state power, guild-based economic systems, and censorship<br />

of ideas. It opened a new era of rational discourse, liberalism, freedom of expression, and scientific method.<br />

This new environment led to tremendous advances <strong>in</strong> both science and mathematics <strong>in</strong> addition to music,

1.4. AGE OF ENLIGHTENMENT 3<br />

literature, philosophy, and art. Scientific development dur<strong>in</strong>g the 17 century <strong>in</strong>cluded the pivotal advances<br />

made by Newton and Leibniz at the beg<strong>in</strong>n<strong>in</strong>g of the revolutionary Age of Enlightenment, culm<strong>in</strong>at<strong>in</strong>g <strong>in</strong> the<br />

development of variational calculus and analytical mechanics by Euler and Lagrange. The scientific advances<br />

of this age <strong>in</strong>clude publication of two monumental books Philosophiae Naturalis Pr<strong>in</strong>cipia Mathematica by<br />

Newton <strong>in</strong> 1687 and Mécanique analytique by Lagrange <strong>in</strong> 1788. These are the def<strong>in</strong>itive two books upon<br />

which classical mechanics is built.<br />

René Descartes (1596-1650) attempted to formulate the laws of motion <strong>in</strong> 1644. He talked about<br />

conservation of motion (momentum) <strong>in</strong> a straight l<strong>in</strong>e but did not recognize the vector character of momentum.<br />

Pierre de Fermat (1601-1665) and René Descartes were two lead<strong>in</strong>g mathematicians <strong>in</strong> the first<br />

half of the 17 century. Independently they discovered the pr<strong>in</strong>ciples of analytic geometry and developed<br />

some <strong>in</strong>itial concepts of calculus. Fermat and Blaise Pascal (1623-1662) were the founders of the theory<br />

of probability.<br />

Isaac Newton (1642-1727) made pioneer<strong>in</strong>g contributions to physics and mathematics as well as<br />

be<strong>in</strong>g a theologian. At 18 he was admitted to Tr<strong>in</strong>ity College Cambridge where he read the writ<strong>in</strong>gs of<br />

modern philosophers like Descartes, and astronomers like Copernicus, Galileo, and Kepler. By 1665 he had<br />

discovered the generalized b<strong>in</strong>omial theorem, and began develop<strong>in</strong>g <strong>in</strong>f<strong>in</strong>itessimal calculus. Due to a plague,<br />

the university closed for two years <strong>in</strong> 1665 dur<strong>in</strong>g which Newton worked at home develop<strong>in</strong>g the theory<br />

of calculus that built upon the earlier work of Barrow and Descartes. He was elected Lucasian Professor<br />

of Mathematics <strong>in</strong> 1669 at the age of 26. From 1670 Newton focussed on optics lead<strong>in</strong>g to his Hypothesis<br />

of Light published <strong>in</strong> 1675 and his book Opticks <strong>in</strong> 1704. Newton described light as be<strong>in</strong>g made up of a<br />

flow of extremely subtle corpuscles that also had associated wavelike properties to expla<strong>in</strong> diffraction and<br />

optical <strong>in</strong>terference that he studied. Newton returned to mechanics <strong>in</strong> 1677 by study<strong>in</strong>g planetary motion<br />

and gravitation that applied the calculus he had developed. In 1687 he published his monumental treatise<br />

entitled Philosophiae Naturalis Pr<strong>in</strong>cipia Mathematica which established his three universal laws of motion,<br />

the universal theory of gravitation, derivation of Kepler’s three laws of planetary motion, and was his first<br />

publication of the development of calculus which he called “the science of fluxions”. Newton’s laws of motion<br />

are based on the concepts of force and momentum, that is, force equals the rate of change of momentum.<br />

Newton’s postulate of an <strong>in</strong>visible force able to act over vast distances led him to be criticized for <strong>in</strong>troduc<strong>in</strong>g<br />

“occult agencies” <strong>in</strong>to science. In a remarkable achievement, Newton completely solved the laws of mechanics.<br />

His theory of classical mechanics and of gravitation reigned supreme until the development of the Theory<br />

of Relativity <strong>in</strong> 1905. The followers of Newton envisioned the Newtonian laws to be absolute and universal.<br />

This dogmatic reverence of Newtonian mechanics prevented physicists from an unprejudiced appreciation of<br />

the analytic variational approach to mechanics developed dur<strong>in</strong>g the 17 through 19 centuries. Newton<br />

was the first scientist to be knighted and was appo<strong>in</strong>ted president of the Royal Society.<br />

Gottfried Leibniz (1646-1716) was a brilliant German philosopher, a contemporary of Newton, who<br />

worked on both calculus and mechanics. Leibniz started development of calculus <strong>in</strong> 1675, ten years after<br />

Newton, but Leibniz published his work <strong>in</strong> 1684, which was three years before Newton’s Pr<strong>in</strong>cipia. Leibniz<br />

made significant contributions to <strong>in</strong>tegral calculus and developed the notation currently used <strong>in</strong> calculus.<br />

He <strong>in</strong>troduced the name calculus based on the Lat<strong>in</strong> word for the small stone used for count<strong>in</strong>g. Newton<br />

and Leibniz were <strong>in</strong>volved <strong>in</strong> a protracted argument over who orig<strong>in</strong>ated calculus. It appears that Leibniz<br />

saw drafts of Newton’s work on calculus dur<strong>in</strong>g a visit to England. Throughout their argument Newton<br />

was the ghost writer of most of the articles <strong>in</strong> support of himself and he had them published under nonde-plume<br />

of his friends. Leibniz made the tactical error of appeal<strong>in</strong>g to the Royal Society to <strong>in</strong>tercede on<br />

his behalf. Newton, as president of the Royal Society, appo<strong>in</strong>ted his friends to an “impartial” committee to<br />

<strong>in</strong>vestigate this issue, then he wrote the committee’s report that accused Leibniz of plagiarism of Newton’s<br />

work on calculus, after which he had it published by the Royal Society. Still unsatisfied he then wrote an<br />

anonymous review of the report <strong>in</strong> the Royal Society’s own periodical. This bitter dispute lasted until the<br />

death of Leibniz. When Leibniz died his work was largely discredited. The fact that he falsely claimed to be<br />

a nobleman and added the prefix “von” to his name, coupled with Newton’s vitriolic attacks, did not help<br />

his credibility. Newton is reported to have declared that he took great satisfaction <strong>in</strong> “break<strong>in</strong>g Leibniz’s<br />

heart.” Studies dur<strong>in</strong>g the 20 century have largely revived the reputation of Leibniz and he is recognized<br />

to have made major contributions to the development of calculus.

4 CHAPTER 1. A BRIEF HISTORY OF CLASSICAL MECHANICS<br />

Figure 1.1: Chronological roadmap of the parallel development of the Newtonian and <strong>Variational</strong>-pr<strong>in</strong>ciples<br />

approaches to classical mechanics.

1.5. VARIATIONAL METHODS IN PHYSICS 5<br />

1.5 <strong>Variational</strong> methods <strong>in</strong> physics<br />

Pierre de Fermat (1601-1665) revived the pr<strong>in</strong>ciple of least time, which states that light travels between<br />

two given po<strong>in</strong>ts along the path of shortest time and was used to derive Snell’s law <strong>in</strong> 1657. This enunciation<br />

of variational pr<strong>in</strong>ciples <strong>in</strong> physics played a key role <strong>in</strong> the historical development of the variational pr<strong>in</strong>ciple<br />

of least action that underlies the analytical formulations of classical mechanics.<br />

Gottfried Leibniz (1646-1716) made significant contributions to the development of variational pr<strong>in</strong>ciples<br />

<strong>in</strong> classical mechanics. In contrast to Newton’s laws of motion, which are based on the concept of<br />

momentum, Leibniz devised a new theory of dynamics based on k<strong>in</strong>etic and potential energy that anticipates<br />

the analytical variational approach of Lagrange and Hamilton. Leibniz argued for a quantity called the “vis<br />

viva”, which is Lat<strong>in</strong> for liv<strong>in</strong>g force, that equals twice the k<strong>in</strong>etic energy. Leibniz argued that the change<br />

<strong>in</strong> k<strong>in</strong>etic energy is equal to the work done. In 1687 Leibniz proposed that the optimum path is based on<br />

m<strong>in</strong>imiz<strong>in</strong>g the time <strong>in</strong>tegral of the vis viva, which is equivalent to the action <strong>in</strong>tegral. Leibniz used both<br />

philosophical and causal arguments <strong>in</strong> his work which were acceptable dur<strong>in</strong>g the Age of Enlightenment. Unfortunately<br />

for Leibniz, his analytical approach based on energies, which are scalars, appeared contradictory<br />

to Newton’s <strong>in</strong>tuitive vectorial treatment of force and momentum. There was considerable prejudice and<br />

philosophical opposition to the variational approach which assumes that nature is thrifty <strong>in</strong> all of its actions.<br />

The variational approach was considered to be speculative and “metaphysical” <strong>in</strong> contrast to the causal<br />

arguments support<strong>in</strong>g Newtonian mechanics. This opposition delayed full appreciation of the variational<br />

approach until the start of the 20 century.<br />

Johann Bernoulli (1667-1748) was a Swiss mathematician who was a student of Leibniz’s calculus, and<br />

sided with Leibniz <strong>in</strong> the Newton-Leibniz dispute over the credit for develop<strong>in</strong>g calculus. Also Bernoulli sided<br />

with the Descartes’ vortex theory of gravitation which delayed acceptance of Newton’s theory of gravitation<br />

<strong>in</strong> Europe. Bernoulli pioneered development of the calculus of variations by solv<strong>in</strong>g the problems of the<br />

catenary, the brachistochrone, and Fermat’s pr<strong>in</strong>ciple. Johann Bernoulli’s son Daniel played a significant<br />

role <strong>in</strong> the development of the well-known Bernoulli Pr<strong>in</strong>ciple <strong>in</strong> hydrodynamics.<br />

Pierre Louis Maupertuis (1698-1759) was a student of Johann Bernoulli and conceived the universal<br />

hypothesis that <strong>in</strong> nature there is a certa<strong>in</strong> quantity called action which is m<strong>in</strong>imized. Although this bold<br />

assumption correctly anticipates the development of the variational approach to classical mechanics, he<br />

obta<strong>in</strong>ed his hypothesis by an entirely <strong>in</strong>correct method. He was a dilettante whose mathematical prowess<br />

was beh<strong>in</strong>d the high standards of that time, and he could not establish satisfactorily the quantity to be<br />

m<strong>in</strong>imized. His teleological 1 argument was <strong>in</strong>fluenced by Fermat’s pr<strong>in</strong>ciple and the corpuscle theory of light<br />

that implied a close connection between optics and mechanics.<br />

Leonhard Euler (1707-1783) was the preem<strong>in</strong>ent Swiss mathematician of the 18 century and was<br />

a student of Johann Bernoulli. Euler developed, with full mathematical rigor, the calculus of variations<br />

follow<strong>in</strong>g <strong>in</strong> the footsteps of Johann Bernoulli. Euler used variational calculus to solve m<strong>in</strong>imum/maximum<br />

isoperimetric problems that had attracted and challenged the early developers of calculus, Newton, Leibniz,<br />

and Bernoulli. Euler also was the first to solve the rigid-body rotation problem us<strong>in</strong>g the three components<br />

of the angular velocity as k<strong>in</strong>ematical variables. Euler became bl<strong>in</strong>d <strong>in</strong> both eyes by 1766 but that did not<br />

h<strong>in</strong>der his prolific output <strong>in</strong> mathematics due to his remarkable memory and mental capabilities. Euler’s<br />

contributions to mathematics are remarkable <strong>in</strong> quality and quantity; for example dur<strong>in</strong>g 1775 he published<br />

one mathematical paper per week <strong>in</strong> spite of be<strong>in</strong>g bl<strong>in</strong>d. Euler implicitly implied the pr<strong>in</strong>ciple of least<br />

action us<strong>in</strong>g vis visa which is not the exact form explicitly developed by Lagrange.<br />

Jean le Rond d’Alembert (1717-1785) was a French mathematician and physicist who had the<br />

clever idea of extend<strong>in</strong>g use of the pr<strong>in</strong>ciple of virtual work from statics to dynamics. d’Alembert’s Pr<strong>in</strong>ciple<br />

rewrites the pr<strong>in</strong>ciple of virtual work <strong>in</strong> the form<br />

X<br />

(F − ṗ )r =0<br />

=1<br />

where the <strong>in</strong>ertial reaction force ṗ is subtracted from the correspond<strong>in</strong>g force F. This extension of the<br />

pr<strong>in</strong>ciple of virtual work applies equally to both statics and dynamics lead<strong>in</strong>g to a s<strong>in</strong>gle variational pr<strong>in</strong>ciple.<br />

Joseph Louis Lagrange (1736-1813) was an Italian mathematician and a student of Leonhard Euler.<br />

In 1788 Lagrange published his monumental treatise on analytical mechanics entitled Mécanique Analytique<br />

1 Teleology is any philosophical account that holds that f<strong>in</strong>al causes exist <strong>in</strong> nature, analogous to purposes found <strong>in</strong> human<br />

actions, nature <strong>in</strong>herently tends toward def<strong>in</strong>ite ends.

6 CHAPTER 1. A BRIEF HISTORY OF CLASSICAL MECHANICS<br />

which <strong>in</strong>troduces his Lagrangian mechanics analytical technique which is based on d’Alembert’s Pr<strong>in</strong>ciple of<br />

Virtual Work. Lagrangian mechanics is a remarkably powerful technique that is equivalent to m<strong>in</strong>imiz<strong>in</strong>g<br />

the action <strong>in</strong>tegral def<strong>in</strong>ed as<br />

=<br />

Z 2<br />

1<br />

The Lagrangian frequently is def<strong>in</strong>ed to be the difference between the k<strong>in</strong>etic energy and potential<br />

energy . His theory only required the analytical form of these scalar quantities. In the preface of his<br />

book he refers modestly to his extraord<strong>in</strong>ary achievements with the statement “The reader will f<strong>in</strong>d no<br />

figures <strong>in</strong> the work. The methods which I set forth do not require either constructions or geometrical or<br />

mechanical reason<strong>in</strong>gs: but only algebraic operations, subject to a regular and uniform rule of procedure.”<br />

Lagrange also <strong>in</strong>troduced the concept of undeterm<strong>in</strong>ed multipliers to handle auxiliary conditions which<br />

plays a vital part of theoretical mechanics. William Hamilton, an outstand<strong>in</strong>g figure <strong>in</strong> the analytical<br />

formulation of classical mechanics, called Lagrange the “Shakespeare of mathematics,” on account of the<br />

extraord<strong>in</strong>ary beauty, elegance, and depth of the Lagrangian methods. Lagrange also pioneered numerous<br />

significant contributions to mathematics. For example, Euler, Lagrange, and d’Alembert developed much of<br />

the mathematics of partial differential equations. Lagrange survived the French Revolution, and, <strong>in</strong> spite of<br />

be<strong>in</strong>g a foreigner, Napoleon named Lagrange to the Legion of Honour and made him a Count of the Empire<br />

<strong>in</strong> 1808. Lagrange was honoured by be<strong>in</strong>g buried <strong>in</strong> the Pantheon.<br />

Carl Friedrich Gauss (1777-1855) was a German child prodigy who made many significant contributions<br />

to mathematics, astronomy and physics. He did not work directly on the variational approach, but<br />

Gauss’s law, the divergence theorem, and the Gaussian statistical distribution are important examples of<br />

concepts that he developed and which feature prom<strong>in</strong>ently <strong>in</strong> classical mechanics as well as other branches<br />

of physics, and mathematics.<br />

Simeon Poisson (1781-1840), was a brilliant mathematician who was a student of Lagrange. He<br />

developed the Poisson statistical distribution as well as the Poisson equation that features prom<strong>in</strong>ently <strong>in</strong><br />

electromagnetic and other field theories. His major contribution to classical mechanics is development, <strong>in</strong><br />

1809, of the Poisson bracket formalism which featured prom<strong>in</strong>ently <strong>in</strong> development of Hamiltonian mechanics<br />

and quantum mechanics.<br />

The zenith <strong>in</strong> development of the variational approach to classical mechanics occurred dur<strong>in</strong>g the 19 <br />

century primarily due to the work of Hamilton and Jacobi.<br />

William Hamilton (1805-1865) was a brilliant Irish physicist, astronomer and mathematician who was<br />

appo<strong>in</strong>ted professor of astronomy at Dubl<strong>in</strong> when he was barely 22 years old. He developed the Hamiltonian<br />

mechanics formalism of classical mechanics which now plays a pivotal role <strong>in</strong> modern classical and quantum<br />

mechanics. He opened an entirely new world beyond the developments of Lagrange. Whereas the Lagrange<br />

equations of motion are complicated second-order differential equations, Hamilton succeeded <strong>in</strong> transform<strong>in</strong>g<br />

them <strong>in</strong>to a set of first-order differential equations with twice as many variables that consider momenta and<br />

their conjugate positions as <strong>in</strong>dependent variables. The differential equations of Hamilton are l<strong>in</strong>ear, have<br />

separated derivatives, and represent the simplest and most desirable form possible for differential equations to<br />

be used <strong>in</strong> a variational approach. Hence the name “canonical variables” given by Jacobi. Hamilton exploited<br />

the d’Alembert pr<strong>in</strong>ciple to give the first exact formulation of the pr<strong>in</strong>ciple of least action which underlies the<br />

variational pr<strong>in</strong>ciples used <strong>in</strong> analytical mechanics. The form derived by Euler and Lagrange employed the<br />

pr<strong>in</strong>ciple <strong>in</strong> a way that applies only for conservative (scleronomic) cases. A significant discovery of Hamilton<br />

is his realization that classical mechanics and geometrical optics can be handled from one unified viewpo<strong>in</strong>t.<br />

In both cases he uses a “characteristic” function that has the property that, by mere differentiation, the<br />

path of the body, or light ray, can be determ<strong>in</strong>ed by the same partial differential equations. This solution is<br />

equivalent to the solution of the equations of motion.<br />

Carl Gustave Jacob Jacobi (1804-1851), a Prussian mathematician and contemporary of Hamilton,<br />

made significant developments <strong>in</strong> Hamiltonian mechanics. He immediately recognized the extraord<strong>in</strong>ary importance<br />

of the Hamiltonian formulation of mechanics. Jacobi developed canonical transformation theory<br />

and showed that the function, used by Hamilton, is only one special case of functions that generate suitable<br />

canonical transformations. He proved that any complete solution of the partial differential equation,<br />

without the specific boundary conditions applied by Hamilton, is sufficient for the complete <strong>in</strong>tegration of<br />

the equations of motion. This greatly extends the usefulness of Hamilton’s partial differential equations.<br />

In 1843 JacobidevelopedboththePoissonbrackets,andtheHamilton-Jacobi, formulations of Hamiltonian<br />

mechanics. The latter gives a s<strong>in</strong>gle, first-order partial differential equation for the action function <strong>in</strong> terms

1.6. THE 20 CENTURY REVOLUTION IN PHYSICS 7<br />

of the generalized coord<strong>in</strong>ates which greatly simplifies solution of the equations of motion. He also derived<br />

a pr<strong>in</strong>ciple of least action for time-<strong>in</strong>dependent cases that had been studied by Euler and Lagrange.<br />

Jacobi developed a superior approach to the variational <strong>in</strong>tegral that, by elim<strong>in</strong>at<strong>in</strong>g time from the <strong>in</strong>tegral,<br />

determ<strong>in</strong>ed the path without say<strong>in</strong>g anyth<strong>in</strong>g about how the motion occurs <strong>in</strong> time.<br />

James Clerk Maxwell (1831-1879) was a Scottish theoretical physicist and mathematician. His most<br />

prom<strong>in</strong>ent achievement was formulat<strong>in</strong>g a classical electromagnetic theory that united previously unrelated<br />

observations, plus equations of electricity, magnetism and optics, <strong>in</strong>to one consistent theory. Maxwell’s<br />

equations demonstrated that electricity, magnetism and light are all manifestations of the same phenomenon,<br />

namely the electromagnetic field. Consequently, all other classic laws and equations of electromagnetism<br />

were simplified cases of Maxwell’s equations. Maxwell’s achievements concern<strong>in</strong>g electromagnetism have<br />

been called the “second great unification <strong>in</strong> physics”. Maxwell demonstrated that electric and magnetic<br />

fields travel through space <strong>in</strong> the form of waves, and at a constant speed of light. In 1864 Maxwell wrote “A<br />

Dynamical Theory of the Electromagnetic Field” which proposed that light was <strong>in</strong> fact undulations <strong>in</strong> the<br />

same medium that is the cause of electric and magnetic phenomena. His work <strong>in</strong> produc<strong>in</strong>g a unified model<br />

of electromagnetism is one of the greatest advances <strong>in</strong> physics. Maxwell, <strong>in</strong> collaboration with Ludwig<br />

Boltzmann (1844-1906), also helped develop the Maxwell—Boltzmann distribution, which is a statistical<br />

means of describ<strong>in</strong>g aspects of the k<strong>in</strong>etic theory of gases. These two discoveries helped usher <strong>in</strong> the era of<br />

modern physics, lay<strong>in</strong>g the foundation for such fields as special relativity and quantum mechanics. Boltzmann<br />

founded the field of statistical mechanics and was an early staunch advocate of the existence of atoms and<br />

molecules.<br />

Henri Po<strong>in</strong>caré (1854-1912) was a French theoretical physicist and mathematician. He was the first to<br />

present the Lorentz transformations <strong>in</strong> their modern symmetric form and discovered the rema<strong>in</strong><strong>in</strong>g relativistic<br />

velocity transformations. Although there is similarity to E<strong>in</strong>ste<strong>in</strong>’s Special Theory of Relativity, Po<strong>in</strong>caré and<br />

Lorentz still believed <strong>in</strong> the concept of the ether and did not fully comprehend the revolutionary philosophical<br />

change implied by E<strong>in</strong>ste<strong>in</strong>. Po<strong>in</strong>caré worked on the solution of the three-body problem <strong>in</strong> planetary motion<br />

and was the first to discover a chaotic determ<strong>in</strong>istic system which laid the foundations of modern chaos<br />

theory. It rejected the long-held determ<strong>in</strong>istic view that if the position and velocities of all the particles are<br />

known at one time, then it is possible to predict the future for all time.<br />

The last two decades of the 19 century saw the culm<strong>in</strong>ation of classical physics and several important<br />

discoveries that led to a revolution <strong>in</strong> science that toppled classical physics from its throne. The end of the<br />

19 century was a time dur<strong>in</strong>g which tremendous technological progress occurred; flight, the automobile,<br />

and turb<strong>in</strong>e-powered ships were developed, Niagara Falls was harnessed for power, etc. Dur<strong>in</strong>g this period,<br />

He<strong>in</strong>rich Hertz (1857-1894) produced electromagnetic waves confirm<strong>in</strong>g their derivation us<strong>in</strong>g Maxwell’s<br />

equations. Simultaneously he discovered the photoelectric effect which was crucial evidence <strong>in</strong> support of<br />

quantum physics. Technical developments, such as photography, the <strong>in</strong>duction spark coil, and the vacuum<br />

pumpplayedasignificant role <strong>in</strong> scientific discoveries made dur<strong>in</strong>g the 1890’s. At the end of the 19 century,<br />

scientists thought that the basic laws were understood and worried that future physics would be <strong>in</strong> the fifth<br />

decimal place; some scientists worried that little was left for them to discover. However, there rema<strong>in</strong>ed a<br />

few, presumed m<strong>in</strong>or, unexpla<strong>in</strong>ed discrepancies plus new discoveries that led to the revolution <strong>in</strong> science<br />

that occurred at the beg<strong>in</strong>n<strong>in</strong>g of the 20 century.<br />

1.6 The 20 century revolution <strong>in</strong> physics<br />

The two greatest achievements of modern physics occurred at the beg<strong>in</strong>n<strong>in</strong>g of the 20 century. The first<br />

was E<strong>in</strong>ste<strong>in</strong>’s development of the Theory of Relativity; the Special Theory of Relativity <strong>in</strong> 1905 and the<br />

General Theory of Relativity <strong>in</strong> 1915. This was followed <strong>in</strong> 1925 by the development of quantum mechanics.<br />

Albert E<strong>in</strong>ste<strong>in</strong> (1879-1955) developed the Special Theory of Relativity <strong>in</strong> 1905 and the General Theory<br />

of Relativity <strong>in</strong> 1915; both of these revolutionary theories had a profound impact on classical mechanics<br />

and the underly<strong>in</strong>g philosophy of physics. The Newtonian formulation of mechanics was shown to be an<br />

approximation that applies only at low velocities, while the General Theory of Relativity superseded Newton’s<br />

Law of Gravitation and expla<strong>in</strong>ed the Equivalence Pr<strong>in</strong>ciple. The Newtonian concepts of an absolute<br />

frame of reference, plus the assumption of the separation of time and space, were shown to be <strong>in</strong>valid at<br />

relativistic velocities. E<strong>in</strong>ste<strong>in</strong>’s postulate that the laws of physics are the same <strong>in</strong> all <strong>in</strong>ertial frames requires<br />

a revolutionary change <strong>in</strong> the philosophy of time, space and reference frames which leads to a breakdown<br />

<strong>in</strong> the Newtonian formalism of classical mechanics. By contrast, the Lagrange and Hamiltonian variational

8 CHAPTER 1. A BRIEF HISTORY OF CLASSICAL MECHANICS<br />

formalisms of mechanics, plus the pr<strong>in</strong>ciple of least action, rema<strong>in</strong> <strong>in</strong>tact us<strong>in</strong>g a relativistically <strong>in</strong>variant<br />

Lagrangian. The <strong>in</strong>dependence of the variational approach to reference frames is precisely the formalism<br />

necessary for relativistic mechanics. The <strong>in</strong>variance to coord<strong>in</strong>ate frames of the basic field equations also<br />

must rema<strong>in</strong> <strong>in</strong>variant for the General Theory of Relativity which also can be derived <strong>in</strong> terms of a relativistic<br />

action pr<strong>in</strong>ciple. Thus the development of the Theory of Relativity unambiguously demonstrated the<br />

superiority of the variational formulation of classical mechanics over the vectorial Newtonian formulation,<br />

and thus the considerable effort made by Euler, Lagrange, Hamilton, Jacobi, and others <strong>in</strong> develop<strong>in</strong>g the<br />

analytical variational formalism of classical mechanics f<strong>in</strong>ally came to fruition at the start of the 20 century.<br />

Newton’s two crown<strong>in</strong>g achievements, the Laws of Motion and the Laws of Gravitation, that had reigned<br />

supreme s<strong>in</strong>ce published <strong>in</strong> the Pr<strong>in</strong>cipia <strong>in</strong> 1687, were toppled from the throne by E<strong>in</strong>ste<strong>in</strong>.<br />

Emmy Noether (1882-1935) has been described as “the greatest ever woman mathematician”. In<br />

1915 she proposed a theorem that a conservation law is associated with any differentiable symmetry of a<br />

physical system. Noether’s theorem evolves naturally from Lagrangian and Hamiltonian mechanics and<br />

she applied it to the four-dimensional world of general relativity. Noether’s theorem has had an important<br />

impact <strong>in</strong> guid<strong>in</strong>g the development of modern physics.<br />

Other profound developments that had revolutionary impacts on classical mechanics were quantum<br />

physics and quantum field theory. The 1913 model of atomic structure by Niels Bohr (1885-1962) and<br />

the subsequent enhancements by Arnold Sommerfeld (1868-1951), were based completely on classical<br />

Hamiltonian mechanics. The proposal of wave-particle duality by Louis de Broglie (1892-1987), made<br />

<strong>in</strong> his 1924 thesis, was the catalyst lead<strong>in</strong>g to the development of quantum mechanics. In 1925 Werner<br />

Heisenberg (1901-1976), and Max Born (1882-1970) developed a matrix representation of quantum<br />

mechanics us<strong>in</strong>g non-commut<strong>in</strong>g conjugate position and momenta variables.<br />

Paul Dirac (1902-1984) showed <strong>in</strong> his Ph.D. thesis that Heisenberg’s matrix representation of quantum<br />

physics is based on the Poisson Bracket generalization of Hamiltonian mechanics, which, <strong>in</strong> contrast to<br />

Hamilton’s canonical equations, allows for non-commut<strong>in</strong>g conjugate variables. In 1926 Erw<strong>in</strong> Schröd<strong>in</strong>ger<br />

(1887-1961) <strong>in</strong>dependently <strong>in</strong>troduced the operational viewpo<strong>in</strong>t and re<strong>in</strong>terpreted the partial differential<br />

equation of Hamilton-Jacobi as a wave equation. His start<strong>in</strong>g po<strong>in</strong>t was the optical-mechanical analogy of<br />

Hamilton that is a built-<strong>in</strong> feature of the Hamilton-Jacobi theory. Schröd<strong>in</strong>ger then showed that the wave<br />

mechanics he developed, and the Heisenberg matrix mechanics, are equivalent representations of quantum<br />

mechanics. In 1928 Dirac developed his relativistic equation of motion for the electron and pioneered the<br />

field of quantum electrodynamics. Dirac also <strong>in</strong>troduced the Lagrangian and the pr<strong>in</strong>ciple of least action to<br />

quantum mechanics, and these ideas were developed <strong>in</strong>to the path-<strong>in</strong>tegral formulation of quantum mechanics<br />

and the theory of electrodynamics by Richard Feynman(1918-1988).<br />

The concepts of wave-particle duality, and quantization of observables, both are beyond the classical<br />

notions of <strong>in</strong>f<strong>in</strong>ite subdivisions <strong>in</strong> classical physics. In spite of the radical departure of quantum mechanics<br />

from earlier classical concepts, the basic feature of the differential equations of quantal physics is their selfadjo<strong>in</strong>t<br />

character which means that they are derivable from a variational pr<strong>in</strong>ciple. Thus both the Theory of<br />

Relativity, and quantum physics are consistent with the variational pr<strong>in</strong>ciple of mechanics, and <strong>in</strong>consistent<br />

with Newtonian mechanics. As a consequence Newtonian mechanics has been dislodged from the throne<br />

it occupied s<strong>in</strong>ce 1687, and the <strong>in</strong>tellectually beautiful and powerful variational pr<strong>in</strong>ciples of analytical<br />

mechanics have been validated.<br />

The 2015 observation of gravitational waves is a remarkable recent confirmation of E<strong>in</strong>ste<strong>in</strong>’s General<br />

Theory of Relativity and the validity of the underly<strong>in</strong>g variational pr<strong>in</strong>ciples <strong>in</strong> physics. Another advance <strong>in</strong><br />

physics is the understand<strong>in</strong>g of the evolution of chaos <strong>in</strong> non-l<strong>in</strong>ear systems that have been made dur<strong>in</strong>g the<br />

past four decades. This advance is due to the availability of computers which has reopened this <strong>in</strong>terest<strong>in</strong>g<br />

branch of classical mechanics, that was pioneered by Henri Po<strong>in</strong>caré about a century ago. Although classical<br />

mechanics is the oldest and most mature branch of physics, there still rema<strong>in</strong> new research opportunities <strong>in</strong><br />

this field of physics.<br />

The focus of this book is to <strong>in</strong>troduce the general pr<strong>in</strong>ciples of the mathematical variational pr<strong>in</strong>ciple<br />

approach, and its applications to classical mechanics. It will be shown that the variational pr<strong>in</strong>ciples, that<br />

were developed <strong>in</strong> classical mechanics, now play a crucial role <strong>in</strong> modern physics and mathematics, plus<br />

many other fields of science and technology.<br />

References:<br />

Excellent sources of <strong>in</strong>formation regard<strong>in</strong>g the history of major players <strong>in</strong> the field of classical mechanics<br />

can be found on Wikipedia, and the book “<strong>Variational</strong> Pr<strong>in</strong>ciple of <strong>Mechanics</strong>” by Lanczos.[La49]

Chapter 2<br />

Review of Newtonian mechanics<br />

2.1 Introduction<br />

It is assumed that the reader has been <strong>in</strong>troduced to Newtonian mechanics applied to one or two po<strong>in</strong>t objects.<br />

This chapter reviews Newtonian mechanics for motion of many-body systems as well as for macroscopic<br />

sized bodies. Newton’s Law of Gravitation also is reviewed. The purpose of this review is to ensure that the<br />

reader has a solid foundation of elementary Newtonian mechanics upon which to build the powerful analytic<br />

Lagrangian and Hamiltonian approaches to classical dynamics.<br />

Newtonian mechanics is based on application of Newton’s Laws of motion which assume that the concepts<br />

of distance, time, and mass, are absolute, that is, motion is <strong>in</strong> an <strong>in</strong>ertial frame. The Newtonian idea of<br />

the complete separation of space and time, and the concept of the absoluteness of time, are violated by the<br />

Theory of Relativity as discussed <strong>in</strong> chapter 17. However, for most practical applications, relativistic effects<br />

are negligible and Newtonian mechanics is an adequate description at low velocities. Therefore chapters<br />

2 − 16 will assume velocities for which Newton’s laws of motion are applicable.<br />

2.2 Newton’s Laws of motion<br />

Newton def<strong>in</strong>ed a vector quantity called l<strong>in</strong>ear momentum p which is the product of mass and velocity.<br />

p = ṙ (2.1)<br />

S<strong>in</strong>ce the mass is a scalar quantity, then the velocity vector ṙ and the l<strong>in</strong>ear momentum vector p are<br />

col<strong>in</strong>ear.<br />

Newton’s laws, expressed <strong>in</strong> terms of l<strong>in</strong>ear momentum, are:<br />

1 Law of <strong>in</strong>ertia: A body rema<strong>in</strong>s at rest or <strong>in</strong> uniform motion unless acted upon by a force.<br />

2 Equation of motion: A body acted upon by a force moves <strong>in</strong> such a manner that the time rate of change<br />

of momentum equals the force.<br />

F = p<br />

(2.2)<br />

<br />

3 Action and reaction: If two bodies exert forces on each other, these forces are equal <strong>in</strong> magnitude and<br />

opposite <strong>in</strong> direction.<br />

Newton’s second law conta<strong>in</strong>s the essential physics relat<strong>in</strong>g the force F and the rate of change of l<strong>in</strong>ear<br />

momentum p.<br />

Newton’s first law, the law of <strong>in</strong>ertia, is a special case of Newton’s second law <strong>in</strong> that if<br />

F = p =0 (2.3)<br />

<br />

then p is a constant of motion.<br />

Newton’s third law also can be <strong>in</strong>terpreted as a statement of the conservation of momentum, that is, for<br />

a two particle system with no external forces act<strong>in</strong>g,<br />

F 12 = −F 21 (2.4)<br />

9

10 CHAPTER 2. REVIEW OF NEWTONIAN MECHANICS<br />

If the forces act<strong>in</strong>g on two bodies are their mutual action and reaction, then equation 24 simplifies to<br />

F 12 + F 21 = p 1<br />

+ p 2<br />

= (p 1 + p 2 )=0 (2.5)<br />

This implies that the total l<strong>in</strong>ear momentum (P = p 1 + p 2 ) is a constant of motion.<br />

Comb<strong>in</strong><strong>in</strong>g equations 21 and 22 leads to a second-order differential equation<br />

F = p<br />

= r<br />

2 = ¨r (2.6)<br />

2 Note that the force on a body F, and the resultant acceleration a = ¨r are col<strong>in</strong>ear. Appendix 2 gives<br />

explicit expressions for the acceleration a <strong>in</strong> cartesian and curvil<strong>in</strong>ear coord<strong>in</strong>ate systems. The def<strong>in</strong>ition of<br />

forcedependsonthedef<strong>in</strong>ition of the mass . Newton’s laws of motion are obeyed to a high precision for<br />

velocities much less than the velocity of light. For example, recent experiments have shown they are obeyed<br />

with an error <strong>in</strong> the acceleration of ∆ ≤ 5 × 10 −14 2 <br />

2.3 Inertial frames of reference<br />

An <strong>in</strong>ertial frame of reference is one <strong>in</strong> which Newton’s Laws of<br />

motion are valid. It is a non-accelerated frame of reference. An<br />

<strong>in</strong>ertial frame must be homogeneous and isotropic. Physical experiments<br />

can be carried out <strong>in</strong> different <strong>in</strong>ertial reference frames.<br />

The Galilean transformation provides a means of convert<strong>in</strong>g between<br />

two <strong>in</strong>ertial frames of reference mov<strong>in</strong>g at a constant relative<br />

velocity. Consider two reference frames and 0 with 0<br />

mov<strong>in</strong>g with constant velocity V at time Figure 21 shows a<br />

Galilean transformation which can be expressed <strong>in</strong> vector form.<br />

r 0 = r − V (2.7)<br />

0 = <br />

Equation 27 gives the boost, assum<strong>in</strong>g Newton’s hypothesis<br />

that the time is <strong>in</strong>variant to change of <strong>in</strong>ertial frames of reference.<br />

Thetimedifferential of this transformation gives<br />

ṙ 0 = ṙ − V (2.8)<br />

¨r 0 = ¨r<br />

Note that the forces <strong>in</strong> the primed and unprimed <strong>in</strong>ertial frames<br />

are related by<br />

F = p<br />

z<br />

x<br />

O<br />

y<br />

r<br />

Vt<br />

Figure 2.1: Frame 0 mov<strong>in</strong>g with a constant<br />

velocity with respect to frame <br />

at the time .<br />

= ¨r =¨r0 = F 0 (2.9)<br />

Thus Newton’s Laws of motion are <strong>in</strong>variant under a Galilean transformation, that is, the <strong>in</strong>ertial mass is<br />

unchanged under Galilean transformations. If Newton’s laws are valid <strong>in</strong> one <strong>in</strong>ertial frame of reference,<br />

then they are valid <strong>in</strong> any frame of reference <strong>in</strong> uniform motion with respect to the first frame of reference.<br />

This <strong>in</strong>variance is called Galilean <strong>in</strong>variance. There are an <strong>in</strong>f<strong>in</strong>ite number of possible <strong>in</strong>ertial frames all<br />

connected by Galilean transformations.<br />

Galilean <strong>in</strong>variance violates E<strong>in</strong>ste<strong>in</strong>’s Theory of Relativity. In order to satisfy E<strong>in</strong>ste<strong>in</strong>’s postulate<br />

that the laws of physics are the same <strong>in</strong> all <strong>in</strong>ertial frames, as well as satisfy Maxwell’s equations for<br />

electromagnetism, it is necessary to replace the Galilean transformation by the Lorentz transformation. As<br />

will be discussed <strong>in</strong> chapter 17, the Lorentz transformation leads to Lorentz contraction and time dilation both<br />

1−( )<br />

of which are related to the parameter ≡<br />

1<br />

2<br />

where is the velocity of light <strong>in</strong> vacuum. Fortunately,<br />

most situations <strong>in</strong> life <strong>in</strong>volve velocities where ; for example, for a body mov<strong>in</strong>g at 25 000m.p.h.<br />

(11 111 ) which is the escape velocity for a body at the surface of the earth, the factor differs from<br />

unity by about 6810 −10 which is negligible. Relativistic effects are significant only <strong>in</strong> nuclear and particle<br />

physics as well as some exotic conditions <strong>in</strong> astrophysics. Thus, for the purpose of classical mechanics,<br />

usually it is reasonable to assume that the Galilean transformation is valid and is well obeyed under most<br />

practical conditions.<br />

z’<br />

x’<br />

O’<br />

r’<br />

P<br />

y’

2.4. FIRST-ORDER INTEGRALS IN NEWTONIAN MECHANICS 11<br />

2.4 First-order <strong>in</strong>tegrals <strong>in</strong> Newtonian mechanics<br />

A fundamental goal of mechanics is to determ<strong>in</strong>e the equations of motion for an −body system, where<br />

the force F acts on the <strong>in</strong>dividual mass where 1 ≤ ≤ . Newton’s second-order equation of motion,<br />

equation 26 must be solved to calculate the <strong>in</strong>stantaneous spatial locations, velocities, and accelerations for<br />

each mass of an -body system. Both F and ¨r are vectors, each hav<strong>in</strong>g three orthogonal components.<br />

The solution of equation 26 <strong>in</strong>volves <strong>in</strong>tegrat<strong>in</strong>g second-order equations of motion subject to a set of <strong>in</strong>itial<br />

conditions. Although this task appears simple <strong>in</strong> pr<strong>in</strong>ciple, it can be exceed<strong>in</strong>gly complicated for many-body<br />

systems. Fortunately, solution of the motion often can be simplified by exploit<strong>in</strong>g three first-order <strong>in</strong>tegrals<br />

of Newton’s equations of motion, that are related directly to conservation of either the l<strong>in</strong>ear momentum,<br />

angular momentum, or energy of the system. In addition, for the special case of these three first-order<br />

<strong>in</strong>tegrals, the <strong>in</strong>ternal motion of any many-body system can be factored out by a simple transformations <strong>in</strong>to<br />

the center of mass of the system. As a consequence, the follow<strong>in</strong>g three first-order <strong>in</strong>tegrals are exploited<br />

extensively <strong>in</strong> classical mechanics.<br />

2.4.1 L<strong>in</strong>ear Momentum<br />

Newton’s Laws can be written as the differential and <strong>in</strong>tegral forms of the first-order time <strong>in</strong>tegral which<br />

equals the change <strong>in</strong> l<strong>in</strong>ear momentum. That is<br />

F = p <br />

<br />

Z 2<br />

1<br />

F =<br />

Z 2<br />

1<br />

p <br />

=(p 2 − p 1 ) <br />

(2.10)<br />

This allows Newton’s law of motion to be expressed directly <strong>in</strong> terms of the l<strong>in</strong>ear momentum p = ṙ of<br />

each of the 1 bodies <strong>in</strong> the system This first-order time <strong>in</strong>tegral features prom<strong>in</strong>ently <strong>in</strong> classical<br />

mechanics s<strong>in</strong>ce it connects to the important concept of l<strong>in</strong>ear momentum p. Thisfirst-order time <strong>in</strong>tegral<br />

gives that the total l<strong>in</strong>ear momentum is a constant of motion when the sum of the external forces is zero.<br />

2.4.2 Angular momentum<br />

The angular momentum L of a particle with l<strong>in</strong>ear momentum p with respect to an orig<strong>in</strong> from which<br />

the position vector r is measured, is def<strong>in</strong>ed by<br />

L ≡ r × p (2.11)<br />

The torque, or moment of the force N with respect to the same orig<strong>in</strong> is def<strong>in</strong>ed to be<br />

N ≡ r × F (2.12)<br />

where r is the position vector from the orig<strong>in</strong> to the po<strong>in</strong>t where the force F is applied. Note that the<br />

torque N can be written as<br />

N = r × p <br />

(2.13)<br />

<br />

Consider the time differential of the angular momentum, L <br />

<br />

L <br />

<br />

= (r × p )= r <br />

× p + r × p <br />

<br />

(2.14)<br />

However,<br />

r <br />

× p = r <br />

× r <br />

=0 (2.15)<br />

<br />

Equations 213 − 215 canbeusedtowritethefirst-order time <strong>in</strong>tegral for angular momentum <strong>in</strong> either<br />

differential or <strong>in</strong>tegral form as<br />

L <br />

<br />

= r × p <br />

= N <br />

Z 2<br />

1<br />

N =<br />

Z 2<br />

1<br />

L <br />

=(L 2 − L 1 ) <br />

(2.16)<br />

Newton’s Law relates torque and angular momentum about the same axis. When the torque about any axis<br />

is zero then angular momentum about that axis is a constant of motion. If the total torque is zero then the<br />

total angular momentum, as well as the components about three orthogonal axes, all are constants.

12 CHAPTER 2. REVIEW OF NEWTONIAN MECHANICS<br />

2.4.3 K<strong>in</strong>etic energy<br />

The third first-order <strong>in</strong>tegral, that can be used for solv<strong>in</strong>g the equations of motion, is the first-order spatial<br />

<strong>in</strong>tegral R 2<br />

1 F · r . Note that this spatial <strong>in</strong>tegral is a scalar <strong>in</strong> contrast to the first-order time <strong>in</strong>tegrals for<br />

l<strong>in</strong>ear and angular momenta which are vectors. The work done on a mass by a force F <strong>in</strong> transform<strong>in</strong>g<br />

from condition 1 to 2 is def<strong>in</strong>ed to be<br />

Z 2<br />

[ 12 ] <br />

≡ F · r (2.17)<br />

If F is the net resultant force act<strong>in</strong>g on a particle then the <strong>in</strong>tegrand can be written as<br />

F · r = p <br />

· r v <br />

= <br />

· r <br />

= v <br />

<br />

· v = µ <br />

<br />

1<br />

2 (v · v ) = <br />

2 2 = [ ] <br />

(2.18)<br />

where the k<strong>in</strong>etic energy of a particle is def<strong>in</strong>ed as<br />

1<br />

[ ] <br />

≡ 1 2 2 (2.19)<br />

Thus the work done on the particle , thatis,[ 12 ] <br />

equalsthechange<strong>in</strong>k<strong>in</strong>eticenergyoftheparticleif<br />

there is no change <strong>in</strong> other contributions to the total energy such as potential energy, heat dissipation, etc.<br />

That is<br />

∙ 1<br />

[ 12 ] <br />

=<br />

2 2 2 − 1 2<br />

1¸<br />

2 =[ 2 − 1 ] <br />

(2.20)<br />

Thus the differential, and correspond<strong>in</strong>g first <strong>in</strong>tegral, forms of the k<strong>in</strong>etic energy can be written as<br />

F = Z 2<br />

<br />

F · r =( 2 − 1 ) (2.21)<br />

r 1<br />

If the work done on the particle is positive, then the f<strong>in</strong>al k<strong>in</strong>etic energy 2 1 Especially noteworthy is that<br />

the k<strong>in</strong>etic energy [ ] <br />

is a scalar quantity which makes it simple to use. This first-order spatial <strong>in</strong>tegral is the<br />

foundation of the analytic formulation of mechanics that underlies Lagrangian and Hamiltonian mechanics.<br />

2.5 Conservation laws <strong>in</strong> classical mechanics<br />

Elucidat<strong>in</strong>g the dynamics <strong>in</strong> classical mechanics is greatly simplified when conservation laws are applicable.<br />

In nature, isolated many-body systems frequently conserve one or more of the first-order <strong>in</strong>tegrals for l<strong>in</strong>ear<br />

momentum, angular momentum, and mass/energy. Note that mass and energy are coupled <strong>in</strong> the Theory<br />

of Relativity, but for non-relativistic mechanics the conservation of mass and energy are decoupled. Other<br />

observables such as lepton and baryon numbers are conserved, but these conservation laws usually can be<br />

subsumed under conservation of mass for most problems <strong>in</strong> non-relativistic classical mechanics. The power<br />

of conservation laws <strong>in</strong> calculat<strong>in</strong>g classical dynamics makes it useful to comb<strong>in</strong>e the conservation laws<br />

with the first <strong>in</strong>tegrals for l<strong>in</strong>ear momentum, angular momentum, and work-energy, when solv<strong>in</strong>g problems<br />

<strong>in</strong>volv<strong>in</strong>g Newtonian mechanics. These three conservation laws will be derived assum<strong>in</strong>g Newton’s laws of<br />

motion, however, these conservation laws are fundamental laws of nature that apply well beyond the doma<strong>in</strong><br />

of applicability of Newtonian mechanics.<br />

2.6 Motion of f<strong>in</strong>ite-sized and many-body systems<br />

Elementary presentations <strong>in</strong> classical mechanics discuss motion and forces <strong>in</strong>volv<strong>in</strong>g s<strong>in</strong>gle po<strong>in</strong>t particles.<br />

However, <strong>in</strong> real life, s<strong>in</strong>gle bodies have a f<strong>in</strong>ite size <strong>in</strong>troduc<strong>in</strong>g new degrees of freedom such as rotation and<br />

vibration, and frequently many f<strong>in</strong>ite-sized bodies are <strong>in</strong>volved. A f<strong>in</strong>ite-sized body can be thought of as a<br />

system of <strong>in</strong>teract<strong>in</strong>g particles such as the <strong>in</strong>dividual atoms of the body. The <strong>in</strong>teractions between the parts<br />

of the body can be strong which leads to rigid body motion where the positions of the particles are held<br />

fixed with respect to each other, and the body can translate and rotate. When the <strong>in</strong>teraction between the<br />

bodies is weaker, such as for a diatomic molecule, additional vibrational degrees of relative motion between<br />

the <strong>in</strong>dividual atoms are important. Newton’s third law of motion becomes especially important for such<br />

many-body systems.

2.7. CENTER OF MASS OF A MANY-BODY SYSTEM 13<br />

2.7 Centerofmassofamany-bodysystem<br />

r’ i<br />

A f<strong>in</strong>ite sized body needs a reference po<strong>in</strong>t with respect<br />

to which the motion can be described. For example,<br />

there are 8 corners of a cube that could server as reference<br />

po<strong>in</strong>ts, but the motion of each corner is complicated<br />

if the cube is both translat<strong>in</strong>g and rotat<strong>in</strong>g. The<br />

CM<br />

treatment of the behavior of f<strong>in</strong>ite-sized bodies, or manybody<br />

systems, is greatly simplified us<strong>in</strong>g the concept of<br />

center of mass. Thecenterofmassisaparticularfixed<br />

R<br />

po<strong>in</strong>t <strong>in</strong> the body that has an especially valuable property;<br />

that is, the translational motion of a f<strong>in</strong>ite sized<br />

m<br />

i<br />

body can be treated like that of a po<strong>in</strong>t mass located at<br />

r i<br />

the center of mass. In addition the translational motion<br />

is separable from the rotational-vibrational motion of a<br />

many-body system when the motion is described with<br />

respect to the center of mass. Thus it is convenient at<br />

this juncture to <strong>in</strong>troduce the concept of center of mass<br />

of a many-body system.<br />

For a many-body system, the position vector r ,def<strong>in</strong>ed<br />

relative to the laboratory system, is related to the Figure 2.2: Position vector with respect to the<br />

position vector r 0 with respect to the center of mass, and center of mass.<br />

the center-of-mass location R relative to the laboratory<br />

system. That is, as shown <strong>in</strong> figure 22<br />

r = R + r 0 (2.22)<br />

This vector relation def<strong>in</strong>es the transformation between the laboratory and center of mass systems. For<br />

discrete and cont<strong>in</strong>uous systems respectively, the location of the center of mass is uniquely def<strong>in</strong>ed as be<strong>in</strong>g<br />

where<br />

X<br />

Z<br />

r 0 = r 0 =0<br />

(Center of mass def<strong>in</strong>ition)<br />

Def<strong>in</strong>e the total mass as<br />

<br />

=<br />

X<br />

Z<br />

=<br />

<br />

<br />

<br />

(Total mass)<br />

The average location of the system corresponds to the location of the center of mass s<strong>in</strong>ce 1 P<br />

r 0 =0<br />

that is<br />

1 X<br />

r = R + 1 X<br />

r 0 = R (2.23)<br />

<br />

<br />

<br />

The vector R which describes the location of the center of mass, depends on the orig<strong>in</strong> and coord<strong>in</strong>ate<br />

system chosen. For a cont<strong>in</strong>uous mass distribution the location vector of the center of mass is given by<br />

R = 1 X<br />

r = 1 Z<br />

r (2.24)<br />

<br />

<br />

<br />

The center of mass can be evaluated by calculat<strong>in</strong>g the <strong>in</strong>dividual components along three orthogonal axes.<br />

The center-of-mass frame of reference is def<strong>in</strong>ed as the frame for which the center of mass is stationary.<br />

This frame of reference is especially valuable for elucidat<strong>in</strong>g the underly<strong>in</strong>g physics which <strong>in</strong>volves only the<br />

relative motion of the many bodies. That is, the trivial translational motion of the center of mass frame,<br />

which has no <strong>in</strong>fluence on the relative motion of the bodies, is factored out and can be ignored. For example,<br />

a tennis ball (006) approach<strong>in</strong>g the earth (6 × 10 24 ) with velocity could be treated <strong>in</strong> three frames,<br />

(a) assume the earth is stationary, (b) assume the tennis ball is stationary, or (c) the center-of-mass frame.<br />

The latter frame ignores the center of mass motion which has no <strong>in</strong>fluence on the relative motion of the<br />

tennis ball and the earth. The center of l<strong>in</strong>ear momentum and center of mass coord<strong>in</strong>ate frames are identical<br />

<strong>in</strong> Newtonian mechanics but not <strong>in</strong> relativistic mechanics as described <strong>in</strong> chapter 1743.

14 CHAPTER 2. REVIEW OF NEWTONIAN MECHANICS<br />

2.8 Total l<strong>in</strong>ear momentum of a many-body system<br />

2.8.1 Center-of-mass decomposition<br />

The total l<strong>in</strong>ear momentum P for a system of particlesisgivenby<br />

P =<br />

X<br />

p = <br />

<br />

X<br />

r (2.25)<br />

<br />

It is convenient to describe a many-body system by a position vector r 0 with respect to the center of mass.<br />

That is,<br />

P =<br />

X<br />

p = <br />

<br />

X<br />

r = R + <br />

<br />

s<strong>in</strong>ce P <br />

r 0 =0as given by the def<strong>in</strong>ition of the center of mass. That is;<br />

r = R + r 0 (2.26)<br />

X<br />

<br />

r 0 = R +0=Ṙ (2.27)<br />

<br />

P = Ṙ (2.28)<br />

Thus the total l<strong>in</strong>ear momentum for a system is the same as the momentum of a s<strong>in</strong>gle particle of mass<br />

= P <br />

located at the center of mass of the system.<br />

2.8.2 Equations of motion<br />

The force act<strong>in</strong>g on particle <strong>in</strong> an -particle many-body system, can be separated <strong>in</strong>to an external force<br />

plus <strong>in</strong>ternal forces f between the particles of the system<br />

F <br />

<br />

F = F +<br />

X<br />

f (2.29)<br />

The orig<strong>in</strong> of the external force is from outside of the system while the <strong>in</strong>ternal force is due to the mutual<br />

<strong>in</strong>teraction between the particles <strong>in</strong> the system. Newton’s Law tells us that<br />

Thustherateofchangeoftotalmomentumis<br />

<br />

6=<br />

ṗ = F = F +<br />

X<br />

f (2.30)<br />

<br />

6=<br />

Ṗ =<br />

X<br />

ṗ =<br />

<br />

X<br />

F +<br />

<br />

X<br />

<br />

X<br />

f (2.31)<br />

<br />

6=<br />

Note that s<strong>in</strong>ce the <strong>in</strong>dices are dummy then<br />

X<br />

<br />

X<br />

f = X <br />

<br />

6=<br />

X<br />

f (2.32)<br />

<br />

6=<br />

Substitut<strong>in</strong>g Newton’s third law f = −f <strong>in</strong>to equation 232 implies that<br />

X<br />

<br />

X<br />

f = X <br />

<br />

6=<br />

X<br />

f = −<br />

<br />

6=<br />

X<br />

<br />

X<br />

f =0 (2.33)<br />

<br />

6=

2.8. TOTAL LINEAR MOMENTUM OF A MANY-BODY SYSTEM 15<br />

which is satisfied only for the case where the summations equal zero. That is, for every <strong>in</strong>ternal force, there<br />

is an equal and opposite reaction force that cancels that <strong>in</strong>ternal force.<br />

Therefore the first-order <strong>in</strong>tegral for l<strong>in</strong>ear momentum can be written <strong>in</strong> differential and <strong>in</strong>tegral forms<br />

as<br />

X<br />

Z 2 X<br />

Ṗ =<br />

F = P 2 − P 1 (2.34)<br />

<br />

F <br />

The reaction of a body to an external force is equivalent to a s<strong>in</strong>gle particle of mass located at the center<br />

of mass assum<strong>in</strong>g that the <strong>in</strong>ternal forces cancel due to Newton’s third law.<br />

Note that the total l<strong>in</strong>ear momentum P is conserved if the net external force F is zero, that is<br />

F = P =0 (2.35)<br />

<br />

Therefore the P of the center of mass is a constant. Moreover, if the component of the force along any<br />

direction be is zero, that is,<br />

F P · be<br />

· be = =0 (2.36)<br />

<br />

then P · be is a constant. This fact is used frequently to solve problems <strong>in</strong>volv<strong>in</strong>g motion <strong>in</strong> a constant force<br />

field. For example, <strong>in</strong> the earth’s gravitational field,themomentumofanobjectmov<strong>in</strong>g<strong>in</strong>vacuum<strong>in</strong>the<br />

vertical direction is time dependent because of the gravitational force, whereas the horizontal component of<br />

momentum is constant if no forces act <strong>in</strong> the horizontal direction.<br />

2.1 Example: Explod<strong>in</strong>g cannon shell<br />

Consider a cannon shell of mass moves along a parabolic trajectory <strong>in</strong> the earths gravitational field.<br />

An <strong>in</strong>ternal explosion, generat<strong>in</strong>g an amount of mechanical energy, blows the shell <strong>in</strong>to two parts. One<br />

part of mass where 1 cont<strong>in</strong>ues mov<strong>in</strong>g along the same trajectory with velocity 0 while the other<br />

part is reduced to rest. F<strong>in</strong>d the velocity of the mass immediately after the explosion.<br />

It is important to remember that the energy release is given <strong>in</strong><br />

the center of mass. If the velocity of the shell immediately before the<br />

explosion is and 0 is the velocity of the part immediately after the<br />

M<br />

explosion, then energy conservation gives that 1 2 2 + = 1 2 02 <br />

The conservation of l<strong>in</strong>ear momentum gives = 0 .Elim<strong>in</strong>at<strong>in</strong>g<br />

v<br />

from these equations gives<br />

s<br />

(1-k)M<br />

0 2<br />

=<br />

[(1 − )]<br />

2.2 Example: Billiard-ball collisions<br />

1<br />

<br />

kM<br />

Explod<strong>in</strong>g cannon shell<br />

A billiard ball with mass and <strong>in</strong>cident velocity collides with an identical stationary ball. Assume that<br />

the balls bounce off each other elastically <strong>in</strong> such a way that the <strong>in</strong>cident ball is deflected at a scatter<strong>in</strong>g angle<br />

to the <strong>in</strong>cident direction. Calculate the f<strong>in</strong>al velocities and of the two balls and the scatter<strong>in</strong>g angle<br />

of the target ball. The conservation of l<strong>in</strong>ear momentum <strong>in</strong> the <strong>in</strong>cident direction , and the perpendicular<br />

direction give<br />

= cos + cos <br />

0= s<strong>in</strong> − s<strong>in</strong> <br />

Energy conservation gives .<br />

<br />

2 2 = 2 2 + 2 <br />

2<br />

Solv<strong>in</strong>gthesethreeequationsgives =90 0 − that is, the balls bounce off perpendicular to each other <strong>in</strong><br />

the laboratory frame. The f<strong>in</strong>al velocities are<br />

v’<br />

= cos <br />

= s<strong>in</strong>

16 CHAPTER 2. REVIEW OF NEWTONIAN MECHANICS<br />

2.9 Angular momentum of a many-body system<br />

2.9.1 Center-of-mass decomposition<br />

As was the case for l<strong>in</strong>ear momentum, for a many-body system it is possible to separate the angular momentum<br />

<strong>in</strong>to two components. One component is the angular momentum about the center of mass and the<br />

other component is the angular motion of the center of mass about the orig<strong>in</strong> of the coord<strong>in</strong>ate system. This<br />

separation is done by describ<strong>in</strong>g the angular momentum of a many-body system us<strong>in</strong>g a position vector r 0 <br />

with respect to the center of mass plus the vector location R of the center of mass.<br />

The total angular momentum<br />

L =<br />

=<br />

=<br />

X<br />

L =<br />

<br />

r = R + r 0 (2.37)<br />

X<br />

r × p <br />

<br />

X<br />

(R + r 0 ) × <br />

³Ṙ + ṙ<br />

0<br />

´<br />

<br />

X<br />

<br />

i<br />

<br />

hr 0 × ṙ 0 + r 0 × Ṙ + R × ṙ 0 + R × Ṙ<br />

(2.38)<br />

Note that if the position vectors are with respect to the center of mass, then P <br />

r 0 =0result<strong>in</strong>g <strong>in</strong> the<br />

middle two terms <strong>in</strong> the bracket be<strong>in</strong>g zero, that is;<br />

L =<br />

X<br />

r 0 × p 0 + R × P (2.39)<br />

<br />

The total angular momentum separates <strong>in</strong>to two terms, the angular momentum about the center of mass,<br />

plus the angular momentum of the center of mass about the orig<strong>in</strong> of the axis system. This factor<strong>in</strong>g of the<br />

angular momentum only applies for the center of mass. This is called Samuel König’s first theorem.<br />

2.9.2 Equations of motion<br />

The time derivative of the angular momentum<br />

˙L = r × p = ṙ × p + r × ṗ (2.40)<br />

But<br />

ṙ × p = ṙ × ṙ =0 (2.41)<br />

Thus the torque act<strong>in</strong>g on mass is given by<br />

N = ˙L = r × ṗ = r × F (2.42)<br />

Consider that the resultant force act<strong>in</strong>g on particle <strong>in</strong> this -particle system can be separated <strong>in</strong>to an<br />

external force F <br />

plus <strong>in</strong>ternal forces between the particles of the system<br />

X<br />

F = F + f (2.43)<br />

The orig<strong>in</strong> of the external force is from outside of the system while the <strong>in</strong>ternal force is due to the <strong>in</strong>teraction<br />

with the other − 1 particles <strong>in</strong> the system. Newton’s Law tells us that<br />

<br />

6=<br />

ṗ = F = F +<br />

X<br />

f (2.44)<br />

<br />

6=

2.9. ANGULAR MOMENTUM OF A MANY-BODY SYSTEM 17<br />

The rate of change of total angular momentum is<br />

˙L = X ˙L = X r × ṗ = X<br />

<br />

<br />

<br />

r × F <br />

+ X <br />

X<br />

r × f (2.45)<br />

<br />

6=<br />

S<strong>in</strong>ce f = −f the last expression can be written as<br />

X X<br />

r × f = X<br />

<br />

<br />

<br />

6=<br />

X<br />

(r − r ) × f (2.46)<br />

<br />

<br />

Note that (r − r ) is the vector r connect<strong>in</strong>g to . For central forces the force vector f = cr thus<br />

X X<br />

(r − r ) × f = X X<br />

r × cr =0 (2.47)<br />

<br />

<br />

<br />

<br />

That is, for central <strong>in</strong>ternal forces the total <strong>in</strong>ternal torque on a system of particles is zero, and the rate of<br />

change of total angular momentum for central <strong>in</strong>ternal forces becomes<br />

<br />

<br />

˙L = X <br />

r × F <br />

= X <br />

N = N (2.48)<br />

where N is the net external torque act<strong>in</strong>g on the system. Equation 248 leads to the differential and <strong>in</strong>tegral<br />

forms of the first <strong>in</strong>tegral relat<strong>in</strong>g the total angular momentum to total external torque.<br />

Z2<br />

˙L = N <br />

1<br />

N = L 2 − L 1 (2.49)<br />

Angular momentum conservation occurs <strong>in</strong> many problems <strong>in</strong>volv<strong>in</strong>g zero external torques N =0 plus<br />

two-body central forces F =()ˆr s<strong>in</strong>ce the torque on the particle about the center of the force is zero<br />

N = r × F =()[r × ˆr] =0 (2.50)<br />

Examples are, the central gravitational force for stellar or planetary systems <strong>in</strong> astrophysics, and the central<br />

electrostatic force manifest for motion of electrons <strong>in</strong> the atom. In addition, the component of angular<br />

momentum about any axis Lê is conserved if the net external torque about that axis Nê =0.<br />

2.3 Example: Bolas thrown by gaucho<br />

Consider the bolas thrown by a gaucho to catch cattle. This is a<br />

system with conserved l<strong>in</strong>ear and angular momentum about certa<strong>in</strong><br />

axes. When the bolas leaves the gaucho’s hand the center of mass<br />

has a l<strong>in</strong>ear velocity V plus an angular momentum about the center<br />

of mass of L If no external torques act, then the center of mass of<br />

the bolas will follow a typical ballistic trajectory <strong>in</strong> the earth’s gravitational<br />

field while the angular momentum vector L is conserved,<br />

that is, both <strong>in</strong> magnitude and direction. The tension <strong>in</strong> the ropes<br />

connect<strong>in</strong>g the three balls does not impact the motion of the system<br />

as long as the ropes do not snap due to centrifugal forces.<br />

CM<br />

V 0<br />

m 1<br />

Bolas thrown by a gaucho

18 CHAPTER 2. REVIEW OF NEWTONIAN MECHANICS<br />

2.10 Work and k<strong>in</strong>etic energy for a many-body system<br />

2.10.1 Center-of-mass k<strong>in</strong>etic energy<br />

For a many-body system the position vector r 0 with respect to the center of mass is given by.<br />

r = R + r 0 (2.51)<br />

The location of the center of mass is uniquely def<strong>in</strong>ed as be<strong>in</strong>g at the location where R r 0 =0 The<br />

velocity of the particle can be expressed <strong>in</strong> terms of the velocity of the center of mass Ṙ plus the velocity<br />

of the particle with respect to the center of mass ṙ 0 .Thatis,<br />

The total k<strong>in</strong>etic energy is<br />

=<br />

X<br />

<br />

1<br />

2 2 =<br />

X<br />

<br />

1<br />

2 ṙ · ṙ =<br />

X<br />

<br />

ṙ = Ṙ + ṙ 0 (2.52)<br />

Ã<br />

1<br />

2 ṙ 0 · ṙ 0 <br />

+<br />

<br />

!<br />

X<br />

ṙ 0 · Ṙ + X<br />

<br />

<br />

1<br />

2 Ṙ · Ṙ (2.53)<br />

P<br />

For the special case of the center of mass, the middle term is zero s<strong>in</strong>ce, by def<strong>in</strong>ition of the center of mass,<br />

ṙ 0 =0 Therefore X 1<br />

=<br />

2 02 + 1 2 2 (2.54)<br />

<br />

Thus the total k<strong>in</strong>etic energy of the system is equal to the sum of the k<strong>in</strong>etic energy of a mass mov<strong>in</strong>g<br />

with the center of mass velocity plus the k<strong>in</strong>etic energy of motion of the <strong>in</strong>dividual particles relative to the<br />

center of mass. This is called Samuel König’s second theorem.<br />

Note that for a fixed center-of-mass energy, the total k<strong>in</strong>etic energy has a m<strong>in</strong>imum value of P <br />

1 2 <br />

02<br />

when the velocity of the center of mass =0. For a given <strong>in</strong>ternal excitation energy, the m<strong>in</strong>imum energy<br />

required to accelerate collid<strong>in</strong>g bodies occurs when the collid<strong>in</strong>g bodies have identical, but opposite, l<strong>in</strong>ear<br />

momenta. That is, when the center-of-mass velocity =0.<br />

2.10.2 Conservative forces and potential energy<br />

In general, the l<strong>in</strong>e <strong>in</strong>tegral of a force field F, thatis, R 2<br />

F·r is both path and time dependent. However,<br />

1<br />

an important class of forces, called conservative forces, exist for which the follow<strong>in</strong>g two facts are obeyed.<br />

1) Time <strong>in</strong>dependence:<br />

The force depends only on the particle position r, that is, it does not depend on velocity or time.<br />

2) Path <strong>in</strong>dependence:<br />

For any two po<strong>in</strong>ts 1 and 2 , the work done by F is <strong>in</strong>dependent of the path taken between 1 and 2 .<br />

If forces are path <strong>in</strong>dependent, then it is possible to def<strong>in</strong>e a scalar field, called potential energy, denoted<br />

by (r) that is only a function of position. The path <strong>in</strong>dependence can be expressed by not<strong>in</strong>g that the<br />

<strong>in</strong>tegral around a closed loop is zero. That is<br />

I<br />

F · r =0 (2.55)<br />

Apply<strong>in</strong>g Stokes theorem for a path-<strong>in</strong>dependent force leads to the alternate statement that the curl is zero.<br />

See appendix 33<br />

∇ × F =0 (2.56)<br />

Note that the vector product of two del operators ∇ act<strong>in</strong>g on a scalar field equals<br />

∇ × ∇ =0 (2.57)<br />

Thus it is possible to express a path-<strong>in</strong>dependent force field as the gradient of a scalar field, , thatis<br />

F = −∇ (2.58)

2.10. WORK AND KINETIC ENERGY FOR A MANY-BODY SYSTEM 19<br />

Then the spatial <strong>in</strong>tegral<br />

Z 2<br />

1<br />

F · r = −<br />

Z 2<br />

1<br />

(∇) · r = 1 − 2 (2.59)<br />

Thus for a path-<strong>in</strong>dependent force, the work done on the particle is given by the change <strong>in</strong> potential energy<br />

if there is no change <strong>in</strong> k<strong>in</strong>etic energy. For example, if an object is lifted aga<strong>in</strong>st the gravitational field, then<br />

work is done on the particle and the f<strong>in</strong>al potential energy 2 exceeds the <strong>in</strong>itial potential energy, 1 .<br />

2.10.3 Total mechanical energy<br />

The total mechanical energy of a particle is def<strong>in</strong>ed as the sum of the k<strong>in</strong>etic and potential energies.<br />

= + (2.60)<br />

Note that the potential energy is def<strong>in</strong>ed only to with<strong>in</strong> an additive constant s<strong>in</strong>ce the force F = −∇<br />

depends only on difference <strong>in</strong> potential energy. Similarly, the k<strong>in</strong>etic energy is not absolute s<strong>in</strong>ce any <strong>in</strong>ertial<br />

frame of reference can be used to describe the motion and the velocity of a particle depends on the relative<br />

velocities of <strong>in</strong>ertial frames. Thus the total mechanical energy = + is not absolute.<br />

If a s<strong>in</strong>gle particle is subject to several path-<strong>in</strong>dependent forces, such as gravity, l<strong>in</strong>ear restor<strong>in</strong>g forces,<br />

etc., then a potential energy canbeascribedtoeachofthe forces where for each force F = −∇ .In<br />

X<br />

contrast to the forces, which add vectorially, these scalar potential energies are additive, = .Thus<br />

the total mechanical energy for potential energies equals<br />

<br />

= + (r) = +<br />

X<br />

(r) (2.61)<br />

<br />

Thetimederivativeofthetotalmechanicalenergy = + equals<br />

<br />

= <br />

+ <br />

<br />

Equation 218 gave that = F · r. Thus,thefirst term <strong>in</strong> equation 262 equals<br />

(2.62)<br />

<br />

= F · r<br />

(2.63)<br />

<br />

The potential energy can be a function of both position and time. Thus the time difference <strong>in</strong> potential<br />

energy due to change <strong>in</strong> both time and position is given as<br />

<br />

= X <br />

<br />

+ <br />

<br />

r<br />

=(∇) ·<br />

+ <br />

<br />

(2.64)<br />

The time derivative of the total mechanical energy is given us<strong>in</strong>g equations 263 264 <strong>in</strong> equation 262<br />

<br />

= <br />

+ <br />

= F · r r<br />

+(∇) ·<br />

+ <br />

<br />

r<br />

=[F +(∇)] ·<br />

+ <br />

<br />

Note that if the field is path <strong>in</strong>dependent, that is ∇ × F =0 then the force and potential are related by<br />

(2.65)<br />

F = −∇ (2.66)<br />

Therefore, for path <strong>in</strong>dependent forces, the first term <strong>in</strong> the time derivative of the total energy <strong>in</strong> equation<br />

265 is zero. That is,<br />

<br />

= <br />

(2.67)<br />

<br />

In addition, when the potential energy is not an explicit function of time, then <br />

<br />

=0and thus the total<br />

energy is conserved. That is, for the comb<strong>in</strong>ation of (a) path <strong>in</strong>dependence plus (b) time <strong>in</strong>dependence, then<br />

the total energy of a conservative field is conserved.

20 CHAPTER 2. REVIEW OF NEWTONIAN MECHANICS<br />

Note that there are cases where the concept of potential still is useful even when it is time dependent.<br />

That is, if path <strong>in</strong>dependence applies, i.e. F = −∇ at any <strong>in</strong>stant. For example, a Coulomb field problem<br />

where charges are slowly chang<strong>in</strong>g due to leakage etc., or dur<strong>in</strong>g a peripheral collision between two charged<br />

bodies such as nuclei.<br />

2.4 Example: Central force<br />

A particle of mass moves along a trajectory given by = 0 cos 1 and = 0 s<strong>in</strong> 2 .<br />

a) F<strong>in</strong>d the and components of the force and determ<strong>in</strong>e the condition for which the force is a central<br />

force.<br />

Differentiat<strong>in</strong>g with respect to time gives<br />

˙ = − 0 1 s<strong>in</strong> ( 1 ) ¨ = − 0 2 1 cos ( 1 )<br />

˙ = − 0 2 cos ( 2 ) ¨ = − 0 2 2 s<strong>in</strong> ( 2 )<br />

Newton’s second law gives<br />

F= (¨ˆ+¨ˆ) =− £ 0 2 1 cos ( 1 )ˆ + 0 2 2 s<strong>in</strong> ( 2 ) ˆ ¤ = − £ 2 1ˆ + 2 2ˆ ¤<br />

Note that if 1 = 2 = then<br />

F ==− 2 [ˆ + ˆ] =− 2 r<br />

That is, it is a central force if 1 = 2 = <br />

b) F<strong>in</strong>d the potential energy as a function of and .<br />

S<strong>in</strong>ce<br />

∙ ˆ¸<br />

F = −∇ = − +<br />

ˆ<br />

<br />

then<br />

= 1 2 ¡ 2 1 2 + 2 2 2¢<br />

assum<strong>in</strong>g that =0at the orig<strong>in</strong>.<br />

c) Determ<strong>in</strong>e the k<strong>in</strong>etic energy of the particle and show that it is conserved.<br />

The total energy<br />

= + = 1 2 ¡ ˙ 2 ¢ 1<br />

+ ˙ 2 +<br />

2 ¡ 2 1 2 + 2 2 2¢ = 1 2 ¡ 2 0 2 1 + 0 2 2 ¢<br />

2<br />

s<strong>in</strong>ce cos 2 +s<strong>in</strong> 2 =1. Thus the total energy is a constant and is conserved.<br />

2.10.4 Total mechanical energy for conservative systems<br />

Equation 220 showed that, us<strong>in</strong>g Newton’s second law, F = p<br />

<br />

the first-order spatial <strong>in</strong>tegral gives that<br />

the work done, 12 is related to the change <strong>in</strong> the k<strong>in</strong>etic energy. That is,<br />

12 ≡<br />

Z 2<br />

1<br />

F · r = 1 2 2 2 − 1 2 2 1 = 2 − 1 (2.68)<br />

The work done 12 also can be evaluated <strong>in</strong> terms of the known forces F <strong>in</strong> the spatial <strong>in</strong>tegral.<br />

Consider that the resultant force act<strong>in</strong>g on particle <strong>in</strong> this -particle system can be separated <strong>in</strong>to an<br />

external force F <br />

plus <strong>in</strong>ternal forces between the particles of the system<br />

X<br />

F = F + f (2.69)<br />

The orig<strong>in</strong> of the external force is from outside of the system while the <strong>in</strong>ternal force is due to the <strong>in</strong>teraction<br />

with the other − 1 particles <strong>in</strong> the system. Newton’s Law tells us that<br />

X<br />

ṗ = F = F + f (2.70)<br />

<br />

6=<br />

<br />

6=

2.10. WORK AND KINETIC ENERGY FOR A MANY-BODY SYSTEM 21<br />

The work done on the system by a force mov<strong>in</strong>g from configuration 1 → 2 is given by<br />

1→2 =<br />

X<br />

Z 2<br />

1<br />

F · r +<br />

X<br />

<br />

X<br />

<br />

6=<br />

Z 2<br />

1<br />

f · r (2.71)<br />

S<strong>in</strong>ce f = −f then<br />

1→2 =<br />

X<br />

<br />

Z 2<br />

1<br />

F · r +<br />

X<br />

<br />

X<br />

<br />

<br />

Z 2<br />

1<br />

f · (r − r ) (2.72)<br />

where r − r = r is the vector from to <br />

Assume that both the external and <strong>in</strong>ternal forces are conservative, and thus can be derived from time<br />

<strong>in</strong>dependent potentials, that is<br />

F = −∇ <br />

(2.73)<br />

f = −∇ <br />

(2.74)<br />

Then<br />

1→2 = −<br />

=<br />

X<br />

<br />

X<br />

<br />

Z 2<br />

1<br />

∇ <br />

<br />

<br />

(1) −<br />

Def<strong>in</strong>e the total external potential energy,<br />

and the total <strong>in</strong>ternal energy<br />

X<br />

<br />

· r −<br />

X<br />

<br />

<br />

(2) +<br />

X<br />

<br />

<br />

X<br />

<br />

Z 2<br />

1<br />

∇ <br />

<br />

<br />

(1) −<br />

· r <br />

X<br />

<br />

<br />

(2)<br />

= (1) − (2) + (1) − (2) (2.75)<br />

=<br />

=<br />

Equat<strong>in</strong>g the two equivalent equations for 1→2 ,thatis268 and 275gives that<br />

Regroup these terms <strong>in</strong> equation 278 gives<br />

X<br />

<br />

X<br />

<br />

<br />

(2.76)<br />

<br />

(2.77)<br />

1→2 = 2 − 1 = (1) − (2) + (1) − (2) (2.78)<br />

1 + (1) + (1) = 2 + (2) + (2)<br />

This shows that, for conservative forces, the total energy is conserved and is given by<br />

= + + (2.79)<br />

The three first-order <strong>in</strong>tegrals for l<strong>in</strong>ear momentum, angular momentum, and energy provide powerful<br />

approaches for solv<strong>in</strong>g the motion of Newtonian systems due to the applicability of conservation laws for the<br />

correspond<strong>in</strong>g l<strong>in</strong>ear and angular momentum plus energy conservation for conservative forces. In addition,<br />

the important concept of center-of-mass motion naturally separates out for these three first-order <strong>in</strong>tegrals.<br />

Although these conservation laws were derived assum<strong>in</strong>g Newton’s Laws of motion, these conservation laws<br />

are more generally applicable, and these conservation laws surpass the range of validity of Newton’s Laws of<br />

motion. For example, <strong>in</strong> 1930 Pauli and Fermi postulated the existence of the neutr<strong>in</strong>o <strong>in</strong> order to account for<br />

non-conservation of energy and momentum <strong>in</strong> -decay because they did not wish to rel<strong>in</strong>quish the concepts<br />

of energy and momentum conservation. The neutr<strong>in</strong>o was first detected <strong>in</strong> 1956 confirm<strong>in</strong>g the correctness<br />

of this hypothesis.

22 CHAPTER 2. REVIEW OF NEWTONIAN MECHANICS<br />

2.11 Virial Theorem<br />

The Virial theorem is an important theorem for a system of mov<strong>in</strong>g particles both <strong>in</strong> classical physics and<br />

quantum physics. The Virial Theorem is useful when consider<strong>in</strong>g a collection of many particles and has a<br />

special importance to central-force motion. For a general system of mass po<strong>in</strong>ts with position vectors r and<br />

applied forces F , consider the scalar product <br />

≡ X <br />

p · r (2.80)<br />

where sums over all particles. The time derivative of is<br />

<br />

= X <br />

p · ṙ + X <br />

ṗ · r (2.81)<br />

However,<br />

X<br />

p · ṙ = X<br />

<br />

<br />

ṙ · ṙ = X <br />

2 =2 (2.82)<br />

Also, s<strong>in</strong>ce ṗ = F <br />

X<br />

<br />

ṗ · r = X <br />

F · r (2.83)<br />

Thus<br />

<br />

=2 + X <br />

F · r (2.84)<br />

Thetimeaverageoveraperiod is<br />

Z<br />

1 <br />

0<br />

() − (0)<br />

= = h2 i +<br />

<br />

* X<br />

<br />

F · r <br />

+<br />

(2.85)<br />

where the hi brackets refer to the time average. Note that if the motion is periodic and the chosen time <br />

equals a multiple of the period, then ()−(0)<br />

<br />

=0. Even if the motion is not periodic, if the constra<strong>in</strong>ts and<br />

velocities of all the particles rema<strong>in</strong> f<strong>in</strong>ite, then there is an upper bound to This implies that choos<strong>in</strong>g<br />

→∞means that ()−(0)<br />

<br />

→ 0 In both cases the left-hand side of the equation tends to zero giv<strong>in</strong>g the<br />

Virial theorem<br />

* +<br />

h i = − 1 X<br />

F · r (2.86)<br />

2<br />

<br />

The right-hand side of this equation is called the Virial of the system. For a s<strong>in</strong>gle particle subject to a<br />

conservative central force F = −∇ the Virial theorem equals<br />

h i = 1 2 h∇ · ri = 1 ¿<br />

À<br />

(2.87)<br />

2 <br />

If the potential is of the form = +1 that is, = −( +1) ,then <br />

<br />

=( +1). Thus for a s<strong>in</strong>gle<br />

particle <strong>in</strong> a central potential = +1 the Virial theorem reduces to<br />

h i = +1 hi (2.88)<br />

2<br />

The follow<strong>in</strong>g two special cases are of considerable importance <strong>in</strong> physics.<br />

Hooke’s Law: Notethatforal<strong>in</strong>earrestor<strong>in</strong>gforce =1then<br />

h i =+hi ( =1)<br />

You may be familiar with this fact for simple harmonic motion where the average k<strong>in</strong>etic and potential<br />

energies are the same and both equal half of the total energy.

2.11. VIRIAL THEOREM 23<br />

Inverse-square law:<br />

Theother<strong>in</strong>terest<strong>in</strong>gcaseisforthe<strong>in</strong>versesquarelaw = −2 where<br />

h i = − 1 hi ( = −2)<br />

2<br />

The Virial theorem is useful for solv<strong>in</strong>g problems <strong>in</strong> that know<strong>in</strong>g the exponent of the field makes it<br />

possible to write down directly the average total energy <strong>in</strong> the field. For example, for = −2<br />

hi = h i + hi = − 1 2 hi + hi = 1 hi (2.89)<br />

2<br />

This occurs for the Bohr model of the hydrogen atom where the k<strong>in</strong>etic energy of the bound electron is half<br />

of the potential energy. The same result occurs for planetary motion <strong>in</strong> the solar system.<br />

2.5 Example: The ideal gas law<br />

The Virial theorem deals with average properties and has applications to statistical mechanics. Consider<br />

an ideal gas. Accord<strong>in</strong>g to the Equipartition theorem the average k<strong>in</strong>etic energy per atom <strong>in</strong> an ideal gas is<br />

3<br />

2 where is the absolute temperature and is the Boltzmann constant. Thus the average total k<strong>in</strong>etic<br />

energy for atoms is hi = 3 2 . The right-hand side of the Virial theorem conta<strong>in</strong>s the force .For<br />

an ideal gas it is assumed that there are no <strong>in</strong>teraction forces between atoms, that is the only force is the<br />

force of constra<strong>in</strong>t of the walls of the pressure vessel. The pressure is force per unit area and thus the<br />

<strong>in</strong>stantaneous force on an area of wall is F = −ˆn where ˆ designates the unit vector normal to<br />

the surface. Thus the right-hand side of the Virial theorem is<br />

* +<br />

− 1 X<br />

F · r = Z<br />

ˆn · r <br />

2<br />

2<br />

<br />

Use of the divergence theorem thus gives that R ˆn·r = R ∇ · r =3 R =3 Thus the Virial theorem<br />

leads to the ideal gas law, that is<br />

= <br />

2.6 Example: The mass of galaxies<br />

The Virial theorem can be used to make a crude estimate of the mass of a cluster of galaxies. Assum<strong>in</strong>g a<br />

spherically-symmetric cluster of galaxies,eachofmass then the total mass of the cluster is = .<br />

A crude estimate of the cluster potential energy is<br />

hi ≈ 2<br />

<br />

()<br />

where is the radius of a cluster. The average k<strong>in</strong>etic energy per galaxy is 1 2 hi2 where hi 2 is the average<br />

square of the galaxy velocities with respect to the center of mass of the cluster. Thus the total k<strong>in</strong>etic energy<br />

of the cluster is<br />

hi ≈ hi2 = hi2<br />

()<br />

2 2<br />

The Virial theorem tells us that a central force hav<strong>in</strong>g a radial dependence of the form ∝ gives hi =<br />

+1<br />

2<br />

hi. For the <strong>in</strong>verse-square gravitational force then<br />

hi = − 1 2 hi<br />

()<br />

Thus equations and give an estimate of the total mass of the cluster to be<br />

≈ hi2<br />

<br />

This estimate is larger than the value estimated from the lum<strong>in</strong>osity of the cluster imply<strong>in</strong>g a large amount<br />

of “dark matter” must exist <strong>in</strong> galaxies which rema<strong>in</strong>s an open question <strong>in</strong> physics.

24 CHAPTER 2. REVIEW OF NEWTONIAN MECHANICS<br />

2.12 Applications of Newton’s equations of motion<br />

Newton’s equation of motion can be written <strong>in</strong> the form<br />

F = p<br />

= v = r<br />

2 2 (2.90)<br />

A description of the motion of a particle requires a solution of this second-order differential equation of<br />

motion. This equation of motion may be <strong>in</strong>tegrated to f<strong>in</strong>d r() and v() if the <strong>in</strong>itial conditions and<br />

the force field F() are known. Solution of the equation of motion can be complicated for many practical<br />

examples, but there are various approaches to simplify the solution. It is of value to learn efficient approaches<br />

to solv<strong>in</strong>g problems.<br />

The follow<strong>in</strong>g sequence is recommended<br />

a) Make a vector diagram of the problem <strong>in</strong>dicat<strong>in</strong>g forces, velocities, etc.<br />

b) Write down the known quantities.<br />

c) Before try<strong>in</strong>g to solve the equation of motion directly, look to see if a basic conservation law applies.<br />

That is, check if any of the three first-order <strong>in</strong>tegrals, can be used to simplify the solution. The use of<br />

conservation of energy or conservation of momentum can greatly simplify solv<strong>in</strong>g problems.<br />

The follow<strong>in</strong>g examples show the solution of typical types of problem encountered us<strong>in</strong>g Newtonian<br />

mechanics.<br />

2.12.1 Constant force problems<br />

Problems hav<strong>in</strong>g a constant force imply constant acceleration. The classic example is a block slid<strong>in</strong>g on an<br />

<strong>in</strong>cl<strong>in</strong>ed plane, where the block of mass is acted upon by both gravity and friction. The net force F is<br />

given by the vector sum of the gravitational force F ,normalforceN and frictional force f .<br />

Tak<strong>in</strong>g components perpendicular to the <strong>in</strong>cl<strong>in</strong>ed plane <strong>in</strong> the direction<br />

That is, s<strong>in</strong>ce = <br />

F = F + N + f = a (2.91)<br />

− cos + =0 (2.92)<br />

= cos (2.93)<br />

Similarly, tak<strong>in</strong>g components along the <strong>in</strong>cl<strong>in</strong>ed plane <strong>in</strong> the direction<br />

s<strong>in</strong> − = 2 <br />

2 (2.94)<br />

Us<strong>in</strong>g the concept of coefficient of friction <br />

f f<br />

N<br />

y<br />

Thus the equation of motion can be written as<br />

= (2.95)<br />

(s<strong>in</strong> − cos ) = 2 <br />

2 (2.96)<br />

The block accelerates if s<strong>in</strong> cos that is, tan The<br />

acceleration is constant if and are constant, that is<br />

Fg<br />

Figure 2.3: Block on an <strong>in</strong>cl<strong>in</strong>ed plane<br />

2 <br />

= (s<strong>in</strong> − cos ) (2.97)<br />

2 Remember that if the block is stationary, the friction coefficient balances such that (s<strong>in</strong> − cos ) =0<br />

that is, tan = . However, there is a maximum static friction coefficient beyond which the block starts<br />

slid<strong>in</strong>g. The k<strong>in</strong>etic coefficient of friction is applicable for slid<strong>in</strong>g friction and usually <br />

Another example of constant force and acceleration is motion of objects free fall<strong>in</strong>g <strong>in</strong> a uniform gravitational<br />

field when air drag is neglected. Then one obta<strong>in</strong>s the simple relations such as = + , etc.<br />

x

2.12. APPLICATIONS OF NEWTON’S EQUATIONS OF MOTION 25<br />

2.12.2 L<strong>in</strong>ear Restor<strong>in</strong>g Force<br />

An important class of problems <strong>in</strong>volve a l<strong>in</strong>ear restor<strong>in</strong>g force, that is, they obey Hooke’s law. The equation<br />

of motion for this case is<br />

() =− = ¨ (2.98)<br />

It is usual to def<strong>in</strong>e<br />

2 0 ≡ <br />

(2.99)<br />

Then the equation of motion then can be written as<br />

¨ + 2 0 =0 (2.100)<br />

which is the equation of the harmonic oscillator. Examples are small oscillations of a mass on a spr<strong>in</strong>g,<br />

vibrations of a stretched piano str<strong>in</strong>g, etc.<br />

The solution of this second order equation is<br />

() = s<strong>in</strong> ( 0 − ) (2.101)<br />

This is the well known s<strong>in</strong>usoidal behavior of the displacement for the simple harmonic oscillator.<br />

angular frequency 0 is<br />

The<br />

r<br />

<br />

0 =<br />

<br />

(2.102)<br />

Note that this l<strong>in</strong>ear system has no dissipative forces, thus the total energy is a constant of motion as<br />

discussed previously. That is, it is a conservative system with a total energy given by<br />

1<br />

2 ˙2 + 1 2 2 = (2.103)<br />

The first term is the k<strong>in</strong>etic energy and the second term is the potential energy. The Virial theorem gives<br />

that for the l<strong>in</strong>ear restor<strong>in</strong>g force the average k<strong>in</strong>etic energy equals the average potential energy.<br />

2.12.3 Position-dependent conservative forces<br />

The l<strong>in</strong>ear restor<strong>in</strong>g force is an example of a conservative field. The total energy is conserved, and if the<br />

field is time <strong>in</strong>dependent, then the conservative forces are a function only of position. The easiest way to<br />

solve such problems is to use the concept of potential energy illustrated <strong>in</strong> Figure 24.<br />

2 − 1 = −<br />

Z 2<br />

1<br />

F · r (2.104)<br />

Consider a conservative force <strong>in</strong> one dimension. S<strong>in</strong>ce it was shown that the total energy = + is<br />

conserved for a conservative field, then<br />

= + = 1 2 2 + () (2.105)<br />

Therefore:<br />

Integration of this gives<br />

= <br />

r<br />

2<br />

= ± [ − ()] (2.106)<br />

<br />

− 0 =<br />

Z <br />

0<br />

±<br />

q<br />

(2.107)<br />

2<br />

[ − ()]<br />

where = 0 when = 0 Know<strong>in</strong>g () it is possible to solve this equation as a function of time.

26 CHAPTER 2. REVIEW OF NEWTONIAN MECHANICS<br />

It is possible to understand the general features of the<br />

solution just from <strong>in</strong>spection of the function () For example,<br />

as shown <strong>in</strong> figure 24 the motion for energy 1<br />

is periodic between the turn<strong>in</strong>g po<strong>in</strong>ts and S<strong>in</strong>ce<br />

the potential energy curve is approximately parabolic between<br />

these limits the motion will exhibit simple harmonic<br />

motion. For 0 the turn<strong>in</strong>g po<strong>in</strong>t coalesce to 0 that is<br />

U(x)<br />

E 4<br />

E 3<br />

E 2<br />

E 1<br />

E 0<br />

x x x x<br />

g<br />

x<br />

a o b<br />

x<br />

c d e f<br />

Figure 2.4: One-dimensional potential ()<br />

there is no motion. For total energy 2 the motion is<br />

periodic <strong>in</strong> two <strong>in</strong>dependent regimes, ≤ ≤ and<br />

≤ ≤ <strong>Classical</strong>ly the particle cannot jump from<br />

one pocket to the other. The motion for the particle with<br />

total energy 3 is that it moves freely from <strong>in</strong>f<strong>in</strong>ity, stops<br />

and rebounds at = andthenreturnsto<strong>in</strong>f<strong>in</strong>ity. That<br />

is the particle bounces off the potential at For energy<br />

4 the particle moves freely and is unbounded. For all<br />

these cases, the actual velocity is given by the above relation<br />

for () Thus the k<strong>in</strong>etic energy is largest where<br />

the potential is deepest. An example would be motion of<br />

a roller coaster car.<br />

Position-dependent forces are encountered extensively<br />

<strong>in</strong> classical mechanics. Examples are the many manifestations<br />

of motion <strong>in</strong> gravitational fields, such as <strong>in</strong>terplanetary<br />

probes, a roller coaster, and automobile suspension systems. The l<strong>in</strong>ear restor<strong>in</strong>g force is an especially<br />

simple example of a position-dependent force while the most frequently encountered conservative potentials<br />

are <strong>in</strong> electrostatics and gravitation for which the potentials are;<br />

() = 1 1 2<br />

4 0 12<br />

2 (Electrostatic potential energy)<br />

() =− 1 2<br />

12<br />

2 (Gravitational potential energy)<br />

Know<strong>in</strong>g () it is possible to solve the equation of motion as a function of time.<br />

2.7 Example: Diatomic molecule<br />

An example of a conservative field is a vibrat<strong>in</strong>g diatomic molecule which has a potential energy dependence<br />

with separation distance that is described approximately by the Morse function<br />

h<br />

i<br />

() = 0 1 − − (− 0 ) 2<br />

− 0<br />

where 0 0 and are parameters chosen to best describe the particular pair of atoms. The restor<strong>in</strong>g force<br />

is given by<br />

() =− () =2 i<br />

0<br />

h1 − − (− 0 )<br />

<br />

ih − (− 0 )<br />

<br />

<br />

This has a m<strong>in</strong>imum value of ( 0 )= 0 at = 0 <br />

Note that for small amplitude oscillations, where<br />

( − 0 ) <br />

the exponential term <strong>in</strong> the potential function can be expanded<br />

to give<br />

∙<br />

() ≈ 0 1 − (1 −− ( − ¸2<br />

0)<br />

) − 0 ≈ 0<br />

<br />

2 (− 0) 2 − 0<br />

This gives a restor<strong>in</strong>g force<br />

() =− () = −2 0<br />

( − 0)<br />

That is, for small amplitudes the restor<strong>in</strong>g force is l<strong>in</strong>ear.<br />

U(x)<br />

U o<br />

1.0<br />

0.8<br />

0.6<br />

0.4<br />

0.2<br />

0.0<br />

-0.2<br />

-0.4<br />

-0.6<br />

-0.8<br />

-1.0<br />

x<br />

1 2 3 4 5<br />

Potential energy function () 0 versus <br />

for the diatomic molecule.<br />

x<br />

x<br />

x

2.12. APPLICATIONS OF NEWTON’S EQUATIONS OF MOTION 27<br />

2.12.4 Constra<strong>in</strong>ed motion<br />

A frequently encountered problem <strong>in</strong>volv<strong>in</strong>g position dependent forces, is when the motion is constra<strong>in</strong>ed to<br />

follow a certa<strong>in</strong> trajectory. Forces of constra<strong>in</strong>t must exist to constra<strong>in</strong> the motion to a specific trajectory.<br />

Examples are, the roller coaster, a roll<strong>in</strong>g ball on an undulat<strong>in</strong>g surface, or a downhill skier, where the<br />

motion is constra<strong>in</strong>ed to follow the surface or track contours. The potential energy can be evaluated at all<br />

positions along the constra<strong>in</strong>ed trajectory for conservative forces such as gravity. However, the additional<br />

forces of constra<strong>in</strong>t that must exist to constra<strong>in</strong> the motion, can be complicated and depend on the motion.<br />

For example, the roller coaster must always balance the gravitational and centripetal forces. Fortunately<br />

forces of constra<strong>in</strong>t F often are normal to the direction of motion and thus do not contribute to the total<br />

mechanical energy s<strong>in</strong>ce then the work done F · l is zero. Magnetic forces F =v × B exhibit this feature<br />

of hav<strong>in</strong>g the force normal to the motion.<br />

Solution of constra<strong>in</strong>ed problems is greatly simplifiediftheotherforcesareconservativeandtheforces<br />

of constra<strong>in</strong>t are normal to the motion, s<strong>in</strong>ce then energy conservation can be used.<br />

2.8 Example: Roller coaster<br />

Consider motion of a roller coaster shown <strong>in</strong> the<br />

adjacent figure. This system is conservative if the friction<br />

and air drag are neglected and then the forces of<br />

constra<strong>in</strong>t are normal to the direction of motion.<br />

The k<strong>in</strong>etic energy at any position is just given by<br />

energy conservation and the fact that<br />

= + <br />

where depends on the height of the track at any the<br />

given location. The k<strong>in</strong>etic energy is greatest when the<br />

potential energy is lowest. The forces of constra<strong>in</strong>t<br />

can be deduced if the velocity of motion on the track<br />

is known. Assum<strong>in</strong>g that the motion is conf<strong>in</strong>edtoa<br />

vertical plane, then one has a centripetal force of constra<strong>in</strong>t<br />

2<br />

<br />

normal to the track <strong>in</strong>wards towards the<br />

center of the radius of curvature , plus the gravitation<br />

force downwards of <br />

The constra<strong>in</strong>t force is 2 <br />

<br />

− upwards at the<br />

top of the loop, while it is 2 <br />

<br />

+ downwards at<br />

the bottom of the loop. To ensure that the car and<br />

occupants do not leave the required trajectory, the force<br />

upwards at the top of the loop has to be positive, that<br />

is, 2 ≥ . The velocity at the bottom of the loop<br />

is given by 1 2 2 = 1 2 2 +2 assum<strong>in</strong>g that the<br />

track has a constant radius of curvature . That is;<br />

at a m<strong>in</strong>imum 2 = +4 =5 Therefore the<br />

occupants now will feel an acceleration downwards of<br />

Roller coaster (CCO Public Doma<strong>in</strong>)<br />

at least 2 <br />

<br />

+ =6 at the bottom of the loop The<br />

first roller coaster was built with such a constant radius of curvature but an acceleration of 6 was too much<br />

for the average passenger. Therefore roller coasters are designed such that the radius of curvature is much<br />

larger at the bottom of the loop, as illustrated, <strong>in</strong> order to ma<strong>in</strong>ta<strong>in</strong> sufficiently low loads and also ensure<br />

that the required constra<strong>in</strong>t forces exist.<br />

Note that the m<strong>in</strong>imum velocity at the top of the loop, , implies that if the cart starts from rest it must<br />

startataheight > 2<br />

above the top of the loop if friction is negligible. Note that the solution for the roll<strong>in</strong>g<br />

ball on such a roller coaster differs from that for a slid<strong>in</strong>g object s<strong>in</strong>ce one must <strong>in</strong>clude the rotational energy<br />

of the ball as well as the l<strong>in</strong>ear velocity.<br />

Loop<strong>in</strong>g the loop <strong>in</strong> a sailplane <strong>in</strong>volves the same physics mak<strong>in</strong>g it necessary to vary the elevator control<br />

to vary the radius of curvature throughout the loop to m<strong>in</strong>imize the maximum load.

28 CHAPTER 2. REVIEW OF NEWTONIAN MECHANICS<br />

2.12.5 Velocity Dependent Forces<br />

Velocity dependent forces are encountered frequently <strong>in</strong> practical problems. For example, motion of an<br />

object <strong>in</strong> a fluid, such as air, where viscous forces retard the motion. In general the retard<strong>in</strong>g force has a<br />

complicated dependence on velocity. A quadrative-velocity drag force <strong>in</strong> air often can be expressed <strong>in</strong> the<br />

form,<br />

F () =− 1 2 2 bv (2.108)<br />

where is a dimensionless drag coefficient, is the density of air, is the cross sectional area perpendicular<br />

to the direction of motion, and is the velocity. Modern automobiles have drag coefficients as low as 03. As<br />

described<strong>in</strong>chapter16, the drag coefficient depends on the Reynold’s number which relates the <strong>in</strong>ertial to<br />

viscous drag forces. Small sized objects at low velocity, such as light ra<strong>in</strong>drops, have low Reynold’s numbers<br />

for which is roughly proportional to −1 lead<strong>in</strong>g to a l<strong>in</strong>ear dependence of the drag force on velocity, i.e.<br />

() ∝ . Larger objects mov<strong>in</strong>g at higher velocities, suchasacarorsky-diver,havehigherReynold’s<br />

numbers for which is roughly <strong>in</strong>dependent of velocity lead<strong>in</strong>g to a drag force () ∝ 2 .Thisdragforce<br />

always po<strong>in</strong>ts <strong>in</strong> the opposite direction to the unit velocity vector. Approximately for air<br />

F () =− ¡ 1 + 2 2¢ bv (2.109)<br />

where for spherical objects of diameter , 1 ≈ 155×10 −4 and 2 ≈ 022 2 <strong>in</strong> MKS units. Fortunately, the<br />

equation of motion usually can be <strong>in</strong>tegrated when the retard<strong>in</strong>g force has a simple power law dependence.<br />

As an example, consider free fall <strong>in</strong> the Earth’s gravitational field.<br />

2.9 Example: Vertical fall <strong>in</strong> the earth’s gravitational field.<br />

L<strong>in</strong>ear regime<br />

1 2 <br />

For small objects at low-velocity, i.e. low Reynold’s number, the drag approximately has a l<strong>in</strong>ear dependence<br />

on velocity. Then the equation of motion is<br />

Separate the variables and <strong>in</strong>tegrate<br />

That is<br />

=<br />

Z <br />

0<br />

− − 1 = <br />

<br />

µ <br />

<br />

+<br />

− − 1 = − 1 <br />

ln<br />

1 + 1 0<br />

= − µ <br />

+ + 0 − 1<br />

<br />

1 1<br />

Note that for À 1<br />

the velocity approaches a term<strong>in</strong>al velocity of ∞ = − <br />

1<br />

Thecharacteristictime<br />

constant is = 1<br />

= ∞ Note that if 0 =0 then<br />

´<br />

= ∞<br />

³1 − − <br />

For the case of small ra<strong>in</strong>drops with =05 then ∞ =8 (18) and time constant =08sec<br />

Note that <strong>in</strong> the absence of air drag, these ra<strong>in</strong> drops fall<strong>in</strong>g from 2000 would atta<strong>in</strong> a velocity of over<br />

400 m.p.h. It is fortunate that the drag reduces the speed of ra<strong>in</strong> drops to non-damag<strong>in</strong>g values. Note that<br />

the above relation would predict high velocities for hail. Fortunately, the drag <strong>in</strong>creases quadratically at the<br />

higher velocities atta<strong>in</strong>ed by large ra<strong>in</strong> drops or hail, and this limits the term<strong>in</strong>al velocity to moderate values.<br />

For the United States these velocities still are sufficienttodoconsiderablecropdamage<strong>in</strong>themid-west.<br />

Quadratic regime 2 1<br />

For larger objects at higher velocities, i.e. high Reynold’s number, the drag depends on the square of the<br />

velocity mak<strong>in</strong>g it necessary to differentiate between objects ris<strong>in</strong>g and fall<strong>in</strong>g. The equation of motion is<br />

− ± 2 2 =

2.12. APPLICATIONS OF NEWTON’S EQUATIONS OF MOTION 29<br />

where the positive sign is for fall<strong>in</strong>g objects and negative sign for ris<strong>in</strong>g objects. Integrat<strong>in</strong>g the equation of<br />

motion for fall<strong>in</strong>g gives<br />

Z <br />

µ<br />

<br />

<br />

=<br />

− + 2 2 = tanh −1 0<br />

− tanh −1 <br />

∞ ∞<br />

q<br />

where = <br />

and 2 ∞ =<br />

velocity gives<br />

0<br />

q<br />

<br />

2<br />

That is, = ∞<br />

<br />

For the case of a fall<strong>in</strong>g object with 0 =0 solv<strong>in</strong>g for<br />

= ∞ tanh <br />

As an example, a 06 basket ball with =025 will have ∞ =20 ( 43 m.p.h.) and =21.<br />

Consider President George H.W. Bush skydiv<strong>in</strong>g. Assume his mass is 70kg and assume an equivalent<br />

spherical shape of the former President to have a diameter of =1. This gives that ∞ =56<br />

( 120) and =56. When Bush senior opens his 8 diameter parachute his term<strong>in</strong>al velocity is<br />

estimated to decrease to 7 ( 15 ) which is close to the value for a typical ( 8) diameter emergency<br />

parachute which has a measured term<strong>in</strong>al velocity of 11 <strong>in</strong> spite of air leakage through the central vent<br />

needed to stabilize the parachute motion.<br />

2.10 Example: Projectile motion <strong>in</strong> air<br />

Consider a projectile <strong>in</strong>itially at = =0at =0,thatisfired at an <strong>in</strong>itial velocity v 0 at an angle<br />

to the horizontal. In order to understand the general features of the solution, assume that the drag is<br />

proportional to velocity. This is <strong>in</strong>correct for typical projectile velocities, but simplifies the mathematics. The<br />

equations of motion can be expressed as<br />

¨ = − ˙<br />

¨ = − ˙ − <br />

where is the coefficient for air drag. Take the <strong>in</strong>itial conditions at =0to be = =0 ˙ = cos <br />

˙ = s<strong>in</strong> <br />

Solv<strong>in</strong>g <strong>in</strong> the x coord<strong>in</strong>ate,<br />

˙<br />

= − ˙<br />

<br />

Therefore<br />

˙ = cos −<br />

That is, the velocity decays to zero with a time constant = 1 .<br />

Integration of the velocity equation gives<br />

= <br />

<br />

¡<br />

1 − <br />

− ¢<br />

Note that this implies that the body approaches a value of = <br />

<br />

as →∞<br />

The trajectory of an object is distorted from the parabolic shape, that occurs for =0 due to the rapid<br />

drop <strong>in</strong> range as the drag coefficient <strong>in</strong>creases. For realistic cases it is necessary to use a computer to solve<br />

this numerically.<br />

2.12.6 Systems with Variable Mass<br />

Classic examples of systems with variable mass are the rocket, a fall<strong>in</strong>g cha<strong>in</strong>, and nuclear fission. Consider<br />

the problem of vertical rocket motion <strong>in</strong> a gravitational field us<strong>in</strong>g Newtonian mechanics. When there is a<br />

vertical gravitational external field, the vertical momentum is not conserved due to both gravity and the<br />

ejection of rocket propellant. In a time the rocket ejects propellant vertically with exhaust velocity<br />

relative to the rocket of . Thus the momentum imparted to this propellant is<br />

= − (2.110)<br />

Therefore the rocket is given an equal and opposite <strong>in</strong>crease <strong>in</strong> momentum <br />

=+ (2.111)

30 CHAPTER 2. REVIEW OF NEWTONIAN MECHANICS<br />

In the time <strong>in</strong>terval the net change <strong>in</strong> the l<strong>in</strong>ear momentum of the rocket plus fuel system is given by<br />

=( − )( + )+ ( − ) − = − (2.112)<br />

Therateofchangeofthel<strong>in</strong>earmomentumthusequals<br />

= <br />

= − <br />

<br />

(2.113)<br />

Consider the problem for the special case of vertical ascent of the rocket aga<strong>in</strong>st the external gravitational<br />

force = −. Then<br />

This can be rewritten as<br />

− + <br />

<br />

= <br />

<br />

− + ˙ = ˙<br />

The second term comes from the variable rocket mass where<br />

the loss of mass of the rocket equals the mass of the ejected<br />

propellant. Assum<strong>in</strong>g a constant fuel burn ˙ = then<br />

˙ = − ˙ = − (2.115)<br />

y<br />

v<br />

(2.114)<br />

where 0 Then the equation becomes<br />

=<br />

³− + ´<br />

(2.116)<br />

<br />

m<br />

g<br />

S<strong>in</strong>ce<br />

then<br />

<br />

<br />

− <br />

= − (2.117)<br />

= (2.118)<br />

Earth<br />

dm’<br />

u<br />

Insert<strong>in</strong>g this <strong>in</strong> the above equation gives<br />

³ <br />

=<br />

´<br />

− (2.119)<br />

Figure 2.5: Vertical motion of a rocket <strong>in</strong> a<br />

gravitational field<br />

Integration gives<br />

But the change <strong>in</strong> mass is given by<br />

That is<br />

= − ( 0 − )+ ln<br />

Z <br />

0<br />

= −<br />

Z <br />

0<br />

³ 0<br />

´<br />

<br />

(2.120)<br />

(2.121)<br />

0 − = (2.122)<br />

Thus<br />

³ 0<br />

´<br />

= − + ln<br />

(2.123)<br />

<br />

Note that once the propellant is exhausted the rocket will cont<strong>in</strong>ue to fly upwards as it decelerates <strong>in</strong><br />

the gravitational field. You can easily calculate the maximum height. Note that this formula assumes that<br />

the acceleration due to gravity is constant whereas for large heights above the Earth it is necessary to use<br />

the true gravitational force − <br />

<br />

where isthedistancefromthecenteroftheearth. Inrealsituations<br />

2<br />

it is necessary to <strong>in</strong>clude air drag which requires a computer to numerically solve the equations of motion.<br />

The highest rocket velocity is atta<strong>in</strong>ed by maximiz<strong>in</strong>g the exhaust velocity and the ratio of <strong>in</strong>itial to f<strong>in</strong>al<br />

mass. Because the term<strong>in</strong>al velocity is limited by the mass ratio, eng<strong>in</strong>eers construct multistage rockets that<br />

jettison the spent fuel conta<strong>in</strong>ers and rockets. The variational-pr<strong>in</strong>ciple approach applied to variable mass<br />

problems is discussed <strong>in</strong> chapter 87

2.12. APPLICATIONS OF NEWTON’S EQUATIONS OF MOTION 31<br />

2.12.7 Rigid-body rotation about a body-fixedrotationaxis<br />

The most general case of rigid-body rotation <strong>in</strong>volves rotation about some body-fixed po<strong>in</strong>t with the orientation<br />

of the rotation axis undef<strong>in</strong>ed. For example, an object sp<strong>in</strong>n<strong>in</strong>g <strong>in</strong> space will rotate about the center<br />

of mass with the rotation axis hav<strong>in</strong>g any orientation. Another example is a child’s sp<strong>in</strong>n<strong>in</strong>g top which sp<strong>in</strong>s<br />

with arbitrary orientation of the axis of rotation about the po<strong>in</strong>ted end which touches the ground about a<br />

static location. Such rotation about a body-fixed po<strong>in</strong>t is complicated and will be discussed <strong>in</strong> chapter 13.<br />

Rigid-body rotation is easier to handle if the orientation of the axis of rotation is fixed with respect to the<br />

rigid body. An example of such motion is a h<strong>in</strong>ged door.<br />

For a rigid body rotat<strong>in</strong>g with angular velocity the total angular momentum L is given by<br />

L =<br />

X<br />

L =<br />

<br />

X<br />

r × p (2.124)<br />

<br />

For rotation equation appendix 29 gives<br />

thus the angular momentum can be written as<br />

v = ω × r <br />

(294)<br />

L =<br />

X<br />

r × p =<br />

<br />

X<br />

r × ω × r (2.125)<br />

<br />

The vector triple product can be simplified us<strong>in</strong>g the vector identity equation 24 giv<strong>in</strong>g<br />

L =<br />

X £¡<br />

2 ¢ ¤<br />

ω − (r · ω) r <br />

<br />

(2.126)<br />

Rigid-body rotation about a body-fixed symmetry axis<br />

The simplest case for rigid-body rotation is when the body has a symmetry axis with the angular velocity ω<br />

parallel to this body-fixed symmetry axis. For this case then r can be taken perpendicular to ω for which<br />

the second term <strong>in</strong> equation 2126, i.e. (r · ω) =0,thus<br />

L =<br />

X<br />

<br />

¡<br />

<br />

2 ¢<br />

ω (r perpendicular to ω)<br />

The moment of <strong>in</strong>ertia about the symmetry axis is def<strong>in</strong>ed as<br />

=<br />

X<br />

2 (2.127)<br />

<br />

where is the perpendicular distance from the axis of rotation to the body, For a cont<strong>in</strong>uous body the<br />

moment of <strong>in</strong>ertia can be generalized to an <strong>in</strong>tegral over the mass density of the body<br />

Z<br />

= 2 (2.128)<br />

where is perpendicular to the rotation axis. The def<strong>in</strong>ition of the moment of <strong>in</strong>ertia allows rewrit<strong>in</strong>g the<br />

angular momentum about a symmetry axis L <strong>in</strong> the form<br />

L = ω (2.129)<br />

where the moment of <strong>in</strong>ertia is taken about the symmetry axis and assum<strong>in</strong>g that the angular velocity<br />

of rotation vector is parallel to the symmetry axis.

32 CHAPTER 2. REVIEW OF NEWTONIAN MECHANICS<br />

Rigid-body rotation about a non-symmetric body-fixed axis<br />

In general the fixed axis of rotation is not aligned with a symmetry axis of the body, or the body does not<br />

have a symmetry axis, both of which complicate the problem.<br />

For illustration consider that the rigid body comprises a system of masses located at positions r <br />

with the rigid body rotat<strong>in</strong>g about the axis with angular velocity ω That is,<br />

ω = ẑ (2.130)<br />

In cartesian coord<strong>in</strong>ates the fixed-frame vector for particle is<br />

r =( ) (2.131)<br />

us<strong>in</strong>g these <strong>in</strong> the cross product (294) gives<br />

⎛<br />

v = ω × r = ⎝ − <br />

<br />

0<br />

⎞<br />

⎠ (2.132)<br />

which is written as a column vector for clarity. Insert<strong>in</strong>g v <strong>in</strong> the cross-product r ×v gives the components<br />

of the angular momentum to be<br />

⎛<br />

X<br />

X<br />

L = r × v = <br />

⎝<br />

− ⎞<br />

<br />

− <br />

⎠<br />

<br />

2 + 2 <br />

That is, the components of the angular momentum are<br />

Ã <br />

!<br />

X<br />

= − ≡ (2.133)<br />

<br />

Ã <br />

!<br />

X<br />

= − ≡ <br />

=<br />

<br />

Ã<br />

X <br />

£<br />

<br />

2<br />

+ 2 ¤ ! ≡ <br />

<br />

Note that the perpendicular distance from the axis<br />

<strong>in</strong> cyl<strong>in</strong>drical coord<strong>in</strong>ates is = p 2 + 2 <br />

thus the angular<br />

momentum about the axis can be written<br />

as<br />

Ã <br />

!<br />

X<br />

= 2 = (2.134)<br />

<br />

where (2134) gives the elementary formula for the moment<br />

of <strong>in</strong>ertia = about the axis given earlier<br />

<strong>in</strong> (2129).<br />

The surpris<strong>in</strong>g result is that and are non-zero<br />

imply<strong>in</strong>g that the total angular momentum vector L is<br />

<strong>in</strong> general not parallel with ω This can be understood<br />

by consider<strong>in</strong>g the s<strong>in</strong>gle body shown <strong>in</strong> figure 26.<br />

When the body is <strong>in</strong> the plane then = 0 and<br />

=0 Thus the angular momentum vector L has a<br />

component along the − direction as shown which is<br />

not parallel with ω and, s<strong>in</strong>ce the vectors ω L r are<br />

L<br />

x<br />

Figure 2.6: A rigid rotat<strong>in</strong>g body compris<strong>in</strong>g a s<strong>in</strong>gle<br />

mass attached by a massless rod at a fixed<br />

angle shown at the <strong>in</strong>stant when happens to<br />

lie <strong>in</strong> the plane. As the body rotates about<br />

the − axis the mass has a velocity and momentum<br />

<strong>in</strong>to the page (the negative direction).<br />

Therefore the angular momentum L = r × p is <strong>in</strong><br />

the direction shown which is not parallel to the<br />

angular velocity <br />

coplanar, then L must sweep around the rotation axis ω to rema<strong>in</strong> coplanar with the body as it rotates<br />

about the axis. Instantaneously the velocity of the body v is <strong>in</strong>to the plane of the paper and, s<strong>in</strong>ce<br />

z<br />

O<br />

r<br />

y

2.12. APPLICATIONS OF NEWTON’S EQUATIONS OF MOTION 33<br />

L = r × v then L is at an angle (90 ◦ − ) to the axis. This implies that a torque must be applied<br />

to rotate the angular momentum vector. This expla<strong>in</strong>s why your automobile shakes if the rotation axis and<br />

symmetry axis are not parallel for one wheel.<br />

The first two moments <strong>in</strong> (2133) are called products of <strong>in</strong>ertia of the body designated by the pair of<br />

axes <strong>in</strong>volved. Therefore, to avoid confusion, it is necessary to def<strong>in</strong>e the diagonal moment, which is called<br />

the moment of <strong>in</strong>ertia, by two subscripts as Thus <strong>in</strong> general, a body can have three moments of <strong>in</strong>ertia<br />

about the three axes plus three products of <strong>in</strong>ertia. This group of moments comprise the <strong>in</strong>ertia tensor<br />

which will be discussed further <strong>in</strong> chapter 13. Ifabodyhasanaxisofsymmetryalongthe axis then the<br />

summations will give = =0while will be unchanged. That is, for rotation about a symmetry<br />

axis the angular momentum and rotation axes are parallel. For any axis along which the angular momentum<br />

and angular velocity co<strong>in</strong>cide is called a pr<strong>in</strong>cipal axis of the body.<br />

2.11 Example: Moment of <strong>in</strong>ertia of a th<strong>in</strong> door<br />

Consider that the door has width and height and assume the door thickness is negligible with areal<br />

density 2. Assume that the door is h<strong>in</strong>ged about the axis. The mass of a surface element of<br />

dimension at a distance from the rotation axis is = thus the mass of the complete door<br />

is = The moment of <strong>in</strong>ertia about the axis is given by<br />

=<br />

Z <br />

=0<br />

Z <br />

=0<br />

2 = 1 3 3 = 1 3 2<br />

2.12 Example: Merry-go-round<br />

A child of mass jumps onto the outside edge of a circular merry-go-round of moment of <strong>in</strong>ertia , and<br />

radius and <strong>in</strong>itial angular velocity 0 What is the f<strong>in</strong>al angular velocity ?<br />

If the <strong>in</strong>itial angular momentum is 0 and, assum<strong>in</strong>g the child jumps with zero angular velocity, then the<br />

conservation of angular momentum implies that<br />

That is<br />

0 = <br />

0 = + <br />

0<br />

= <br />

( + 2 )<br />

<br />

= <br />

=<br />

0 0 + 2<br />

Note that this is true <strong>in</strong>dependent of the details of the acceleration of the <strong>in</strong>itially stationary child.<br />

2.13 Example: Cue pushes a billiard ball<br />

Cue push<strong>in</strong>g a billiard ball horizontally at the height<br />

of the centre of rotation of the ball.<br />

Consider a billiard ball of mass and radius <br />

is pushed by a cue <strong>in</strong> a direction that passes through<br />

the center of gravity such that the ball atta<strong>in</strong>s a velocity<br />

0 . The friction coefficient between the table and<br />

the ball is . How far does the ball move before the<br />

<strong>in</strong>itial slipp<strong>in</strong>g motion changes to pure roll<strong>in</strong>g motion?<br />

S<strong>in</strong>ce the direction of the cue force passes through<br />

the center of mass of the ball, it contributes zero<br />

torque to the ball. Thus the <strong>in</strong>itial angular momentum<br />

is zero at =0. The friction force po<strong>in</strong>ts opposite to the direction of motion and causes a torque <br />

about the center of mass <strong>in</strong> the direction ˆ.<br />

N = f · R =<br />

0

34 CHAPTER 2. REVIEW OF NEWTONIAN MECHANICS<br />

S<strong>in</strong>ce the moment of <strong>in</strong>ertia about the center of a uniform sphere is = 2 5 2 then the angular acceleration<br />

of the ball is<br />

˙ = = <br />

<br />

2<br />

= 5 <br />

()<br />

5 2 2 <br />

Moreover the frictional force causes a deceleration of the l<strong>in</strong>ear velocity of the center of mass of<br />

= − = −<br />

()<br />

Integrat<strong>in</strong>g from time zero to gives<br />

Z <br />

= ˙ = 5 <br />

0 2 <br />

The l<strong>in</strong>ear velocity of the center of mass at time is given by <strong>in</strong>tegration of equation <br />

=<br />

Z <br />

0<br />

= 0 − <br />

The billiard ball stops slid<strong>in</strong>g and only rolls when = , thatis,when<br />

5 <br />

2 = 0 − <br />

That is, when<br />

Thus the ball slips for a distance<br />

0<br />

= 2 7 <br />

=<br />

Z <br />

0<br />

= 0 − 2 <br />

2<br />

= 12 0<br />

2<br />

49 <br />

Note that if the ball is pushed at a distance above the center of mass, besides the l<strong>in</strong>ear velocity there<br />

is an <strong>in</strong>itial angular momentum of<br />

= 0<br />

2<br />

= 5 0 <br />

5 2 2 2<br />

For the special case = 2 5 the ball immediately assumes a pure non-slipp<strong>in</strong>g roll. For 2 5 one has<br />

0<br />

<br />

while 2 5 corresponds to 0<br />

<br />

. In the latter case the frictional force po<strong>in</strong>ts forward.<br />

2.12.8 Time dependent forces<br />

Many problems <strong>in</strong>volve action <strong>in</strong> the presence of a time dependent force. There are two extreme cases that<br />

are often encountered. One case is an impulsive force that acts for a very short time, for example, strik<strong>in</strong>g<br />

a ball with a bat, or the collision of two cars. The second case <strong>in</strong>volves an oscillatory time dependent force.<br />

The response to impulsive forces is discussed below whereas the response to oscillatory time-dependent forces<br />

is discussed <strong>in</strong> chapter 3.<br />

Translational impulsive forces<br />

An impulsive force acts for a very short time relative to the response time of the mechanical system be<strong>in</strong>g<br />

discussed. In pr<strong>in</strong>ciple the equation of motion can be solved if the complicated time dependence of the force,<br />

() is known. However, often it is possible to use the much simpler approach employ<strong>in</strong>g the concept of an<br />

impulse and the pr<strong>in</strong>ciple of the conservation of l<strong>in</strong>ear momentum.<br />

Def<strong>in</strong>e the l<strong>in</strong>ear impulse P to be the first-order time <strong>in</strong>tegral of the time-dependent force.<br />

Z<br />

P ≡ F() (2.135)

2.12. APPLICATIONS OF NEWTON’S EQUATIONS OF MOTION 35<br />

S<strong>in</strong>ce F() = p<br />

<br />

then equation 2135 gives that<br />

Z <br />

Z <br />

p<br />

P =<br />

0 0 0 = p = p() − p 0 = ∆p (2.136)<br />

0<br />

Thus the impulse P is an unambiguous quantity that equals the change <strong>in</strong> l<strong>in</strong>ear momentum of the object<br />

that has been struck which is <strong>in</strong>dependent of the details of the time dependence of the impulsive force.<br />

Computation of the spatial motion still requires knowledge of () s<strong>in</strong>ce the 2136 can be written as<br />

Z <br />

v() = 1 F( 0 ) 0 + v 0 (2.137)<br />

0<br />

Integration gives<br />

Z "<br />

Z #<br />

1 <br />

”<br />

r() − r 0 = v 0 +<br />

F( 0 ) 0 ” (2.138)<br />

0 0<br />

In general this is complicated. However, for the case of a constant force F() =F 0 this simplifies to the<br />

constant acceleration equation<br />

r() − r 0 = v 0 + 1 F 0<br />

2 2 (2.139)<br />

where the constant acceleration a = F 0<br />

.<br />

Angular impulsive torques<br />

Note that the pr<strong>in</strong>ciple of impulse also applies to angular motion. Def<strong>in</strong>e an impulsive torque T as the<br />

first-order time <strong>in</strong>tegral of the time-dependent torque.<br />

Z<br />

T ≡ N() (2.140)<br />

S<strong>in</strong>ce torque is related to the rate of change of angular momentum<br />

N() = L<br />

(2.141)<br />

<br />

then<br />

Z Z<br />

L<br />

<br />

T =<br />

0 0 0 = L = L() − L 0 = ∆L (2.142)<br />

0<br />

Thus the impulsive torque T equals the change <strong>in</strong> angular momentum ∆L of the struck body.<br />

2.14 Example: Center of percussion of a baseball bat<br />

When an impulsive force strikes a bat of mass at a distance<br />

s from the center of mass, then both the l<strong>in</strong>ear momentum<br />

of the center of mass, and angular momenta about the center<br />

of mass, of the bat are changed. Assume that the ball strikes<br />

the bat with an impulsive force = ∆ perpendicular to the<br />

symmetry axis of the bat at the strike po<strong>in</strong>t which is a distance<br />

from the center of mass of the bat. The translational impulse<br />

given to the bat equals the change <strong>in</strong> l<strong>in</strong>ear momentum of the<br />

ball as given by equation 2136 coupledwiththeconservationof<br />

l<strong>in</strong>ear momentum<br />

y<br />

O<br />

C<br />

P = ∆p <br />

= ∆v<br />

<br />

s<br />

M<br />

y<br />

Similarly equation 2142 gives that the angular impulse equals<br />

the change <strong>in</strong> angular momentum about the center of mass to be<br />

0<br />

S<br />

x<br />

T= s × P = ∆L = ∆ω

36 CHAPTER 2. REVIEW OF NEWTONIAN MECHANICS<br />

The above equations give that<br />

∆v = P <br />

∆ω <br />

= s × P<br />

<br />

Assume that the bat was stationary prior to the strike, then after the strike the net translational velocity<br />

of a po<strong>in</strong>t along the body-fixed symmetry axis of the bat at a distance from the center of mass, is given<br />

by<br />

v () =∆v + ∆ω × y = P + 1 ((s × P) × y) = P + 1 [(s · y) P− (s · P) y]<br />

<br />

It is assumed that and are perpendicular and thus (s · P) =0which simplifies the above equation to<br />

v () =∆v + ∆ω × y = P µ<br />

<br />

(s · y)<br />

1+<br />

<br />

Note that the translational velocity of the location along the bat symmetry axis at a distance from the<br />

center of mass, is zero if the bracket equals zero, that is, if<br />

s · y = − <br />

= −2 <br />

where is called the radius of gyration of the body about the center of mass. Note that when the scalar<br />

product · = − <br />

<br />

= −2 then there will be no translational motion at the po<strong>in</strong>t . This po<strong>in</strong>t on the<br />

axis lies on the opposite side of the center of mass from the strike po<strong>in</strong>t , and is called the center of<br />

percussion correspond<strong>in</strong>g to the impulse at the po<strong>in</strong>t . The center of percussion often is referred to as the<br />

“sweet spot” for an object correspond<strong>in</strong>g to the impulse at the po<strong>in</strong>t . For a baseball bat the batter holds<br />

the bat at the center of percussion so that they do not feel an impulse <strong>in</strong> their hands when the ball is struck<br />

at the po<strong>in</strong>t . This pr<strong>in</strong>ciple is used extensively to design bats for all sports <strong>in</strong>volv<strong>in</strong>g strik<strong>in</strong>g a ball with<br />

a bat, such as, cricket, squash, tennis, etc. as well as weapons such of swords and axes used to decapitate<br />

opponents.<br />

2.15 Example: Energy transfer <strong>in</strong> charged-particle scatter<strong>in</strong>g<br />

Consider a particle of charge + 1 mov<strong>in</strong>g with very high<br />

velocity 0 along a straight l<strong>in</strong>e that passes a distance <br />

from another charge + 2 and mass . F<strong>in</strong>d the energy <br />

transferredtothemass dur<strong>in</strong>g the encounter assum<strong>in</strong>g<br />

the force is given by Coulomb’s law electrostatics. S<strong>in</strong>ce the<br />

charged particle 1 moves at very high speed it is assumed<br />

that charge 2 does not change position dur<strong>in</strong>g the encounter.<br />

Assume that charge 1 moves along the − axis through the<br />

orig<strong>in</strong> while charge 2 is located on the axis at = .<br />

Let us consider the impulse given to charge 2 dur<strong>in</strong>g the<br />

encounter. By symmetry the componentmustcancelwhile<br />

the component is given by<br />

But<br />

= = − 1 2<br />

4 0 2 cos = − 1 2 <br />

cos <br />

4 0 2 <br />

ṗ<br />

O<br />

y<br />

+e 1<br />

V 0<br />

m<br />

Charged-particle scatter<strong>in</strong>g<br />

+e 2<br />

x<br />

˙ = − 0 cos <br />

where<br />

<br />

=cos( − ) =− cos

2.13. SOLUTION OF MANY-BODY EQUATIONS OF MOTION 37<br />

Thus<br />

Integrate from 2 3 2<br />

= − 1 2<br />

4 0 0<br />

cos <br />

gives that the total momentum imparted to 2 is<br />

= − Z 3<br />

1 2 2<br />

4 0 0 <br />

2<br />

Thus the recoil energy of charge 2 is given by<br />

2 = 2 <br />

2 = 1<br />

2<br />

cos = 1 2<br />

2 0 0<br />

µ<br />

1 2<br />

2 0 0<br />

2<br />

2.13 Solution of many-body equations of motion<br />

The follow<strong>in</strong>g are general methods used to solve Newton’s many-body equations of motion for practical<br />

problems.<br />

2.13.1 Analytic solution<br />

In practical problems one has to solve a set of equations of motion s<strong>in</strong>ce the forces depend on the location<br />

of every body <strong>in</strong>volved. For example one may be deal<strong>in</strong>g with a set of coupled oscillators such as the<br />

many components that comprise the suspension system of an automobile. Often the coupled equations of<br />

motion comprise a set of coupled second-order differential equations. The first approach to solve such a<br />

system is to try an analytic solution compris<strong>in</strong>g a general solution of the <strong>in</strong>homogeneous equation plus one<br />

particular solution of the <strong>in</strong>homogeneous equation. Another approach is to employ numeric <strong>in</strong>tegration us<strong>in</strong>g<br />

acomputer.<br />

2.13.2 Successive approximation<br />

When the system of coupled differential equations of motion is too complicated to solve analytically, one<br />

can use the method of successive approximation. The differential equations are transformed to <strong>in</strong>tegral<br />

equations. Then one starts with some <strong>in</strong>itial conditions to make a first order estimate of the functions. The<br />

functions determ<strong>in</strong>ed by this first order estimate then are used <strong>in</strong> a second iteration and this is repeated<br />

until the solution converges. An example of this approach is when mak<strong>in</strong>g Hartree-Foch calculations of the<br />

electron distributions <strong>in</strong> an atom. The first order calculation uses the electron distributions predicted by<br />

the one-electron model of the atom. This result then is used to compute the <strong>in</strong>fluence of the electron charge<br />

distribution around the nucleus on the charge distribution of the atom for a second iteration etc.<br />

2.13.3 Perturbation method<br />

The perturbation technique can be applied if the force separates <strong>in</strong>to two parts = 1 + 2 where 1 2<br />

and the solution is known for the dom<strong>in</strong>ant 1 part of the force. Then the correction to this solution due<br />

to addition of the perturbation 2 usually is easier to evaluate. As an example, consider that one of the<br />

Space Shuttle thrusters fires. In pr<strong>in</strong>ciple one has all the gravitational forces act<strong>in</strong>g plus the thrust force<br />

of the thruster. The perturbation approach is to assume that the trajectory of the Space Shuttle <strong>in</strong> the<br />

earth’s gravitational field is known. Then the perturbation to this motion due to the very small thrust,<br />

produced by the thruster, is evaluated as a small correction to the motion <strong>in</strong> the Earth’s gravitational field.<br />

This perturbation technique is used extensively <strong>in</strong> physics, especially <strong>in</strong> quantum physics. An example<br />

from my own research is scatter<strong>in</strong>g of a 1 208 ion<strong>in</strong>theCoulombfield of a 197 nucleus The<br />

trajectory for elastic scatter<strong>in</strong>g is simple to calculate s<strong>in</strong>ce neither nucleus is excited and the total energy and<br />

momenta are conserved. However, usually one of these nuclei will be <strong>in</strong>ternally excited by the electromagnetic<br />

<strong>in</strong>teraction. This is called Coulomb excitation. The effect of the Coulomb excitation usually can be treated as<br />

a perturbation by assum<strong>in</strong>g that the trajectory is given by the elastic scatter<strong>in</strong>g solution and then calculate<br />

the excitation probability assum<strong>in</strong>g the Coulomb excitation of the nucleus is a small perturbation to the<br />

trajectory.

38 CHAPTER 2. REVIEW OF NEWTONIAN MECHANICS<br />

2.14 Newton’s Law of Gravitation<br />

Gravitation plays a fundamental role <strong>in</strong> classical mechanics<br />

as well as be<strong>in</strong>g an important example of a conservative<br />

central ¡ ¢<br />

1 2<br />

force. Although you may not be familiar with<br />

use of vector calculus for the gravitational field g, itisassumed<br />

that you have met the identical approach for studies<br />

of the electric field E <strong>in</strong> electrostatics. The primary difference<br />

is that mass replaces charge and gravitational<br />

field g replaces the electric field E. This chapter reviews the<br />

concepts of vector calculus as used for study of conservative<br />

<strong>in</strong>verse-square law central fields.<br />

In 1666 Newton formulated the Theory of Gravitation<br />

which he eventually published <strong>in</strong> the Pr<strong>in</strong>cipia <strong>in</strong> 1687 Newton’s<br />

Law of Gravitation states that each mass particle attracts<br />

every other particle <strong>in</strong> the universe with a force that<br />

varies directly as the product of the mass and <strong>in</strong>versely as<br />

the square of the distance between them. That is, the force<br />

on a gravitational po<strong>in</strong>t mass produced by a mass <br />

F = − <br />

br<br />

2 (2.143)<br />

where br is the unit vector po<strong>in</strong>t<strong>in</strong>g from the gravitational<br />

x<br />

z<br />

r’<br />

dx’dy’dz’<br />

Figure 2.7: Gravitational force on mass m due<br />

to an <strong>in</strong>f<strong>in</strong>itessimal volume element of the mass<br />

density distribution.<br />

mass to the gravitational mass as shown <strong>in</strong> figure 27. Note that the force is attractive, that<br />

is, it po<strong>in</strong>ts toward the other mass. This is <strong>in</strong> contrast to the repulsive electrostatic force between two<br />

similar charges. Newton’s law was verified by Cavendish us<strong>in</strong>g a torsion balance. The experimental value of<br />

=(66726 ± 00008) × 10 −11 · 2 2 <br />

The gravitational force between po<strong>in</strong>t particles can be extended to f<strong>in</strong>ite-sized bodies us<strong>in</strong>g the fact that<br />

the gravitational force field satisfies the superposition pr<strong>in</strong>ciple, that is, the net force is the vector sum of the<br />

<strong>in</strong>dividual forces between the component po<strong>in</strong>t particles. Thus the force summed over the mass distribution<br />

is<br />

F (r) <br />

= − <br />

X<br />

=1<br />

<br />

2 <br />

r<br />

r-r’<br />

br (2.144)<br />

where r is the vector from the gravitational mass to the gravitational mass at the position r.<br />

For a cont<strong>in</strong>uous gravitational mass distribution (r 0 ), the net force on the gravitational mass at<br />

the location r can be written as<br />

Z (r 0 )<br />

³br − br 0´<br />

F (r) =− <br />

(r − r 0 ) 2 0 (2.145)<br />

where 0 is the volume element at the po<strong>in</strong>t r 0 as illustrated <strong>in</strong> figure 27.<br />

2.14.1 Gravitational and <strong>in</strong>ertial mass<br />

Newton’s Laws use the concept of <strong>in</strong>ertial mass ≡ <strong>in</strong> relat<strong>in</strong>g the force F to acceleration a<br />

F = a (2.146)<br />

and momentum p to velocity v<br />

p = v (2.147)<br />

That is, <strong>in</strong>ertial mass is the constant of proportionality relat<strong>in</strong>g the acceleration to the applied force.<br />

The concept of gravitational mass is the constant of proportionality between the gravitational force<br />

and the amount of matter. That is, on the surface of the earth, the gravitational force is assumed to be<br />

F = <br />

"−<br />

X<br />

=1<br />

<br />

2 <br />

br <br />

#<br />

= g (2.148)<br />

m<br />

y

2.14. NEWTON’S LAW OF GRAVITATION 39<br />

where g is the gravitational field which is a position-dependent force per unit gravitational mass po<strong>in</strong>t<strong>in</strong>g<br />

towards the center of the Earth. The gravitational mass is measured when an object is weighed.<br />

Newton’s Law of Gravitation leads to the relation for the gravitational field g (r) at the location r due<br />

to a gravitational mass distribution at the location r 0 as given by the <strong>in</strong>tegral over the gravitational mass<br />

density <br />

Z<br />

g (r) =−<br />

<br />

(r 0 )<br />

³br − br 0´<br />

(r − r 0 ) 2 0 (2.149)<br />

The acceleration of matter <strong>in</strong> a gravitational field relates the gravitational and <strong>in</strong>ertial masses<br />

F = g = a (2.150)<br />

Thus<br />

a = <br />

g (2.151)<br />

<br />

That is, the acceleration of a body depends on the gravitational strength and the ratio of the gravitational<br />

and <strong>in</strong>ertial masses. It has been shown experimentally that all matter is subject to the same acceleration<br />

<strong>in</strong> vacuum at a given location <strong>in</strong> a gravitational field. That is, <br />

<br />

is a constant common to all materials.<br />

Galileo first showed this when he dropped objects from the Tower of Pisa. Modern experiments have shown<br />

that this is true to 5 parts <strong>in</strong> 10 13 .<br />

The exact equivalence of gravitational mass and <strong>in</strong>ertial mass is called the weak pr<strong>in</strong>ciple of equivalence<br />

which underlies the General Theory of Relativity as discussed <strong>in</strong> chapter 17. Itisconvenienttouse<br />

the same unit for the gravitational and <strong>in</strong>ertial masses and thus they both can be written <strong>in</strong> terms of the<br />

common mass symbol .<br />

= = (2.152)<br />

Therefore the subscripts and can be omitted <strong>in</strong> equations 2150 and 2152. Also the local acceleration<br />

due to gravity a can be written as<br />

a = g (2.153)<br />

The gravitational field g ≡ F has units of <strong>in</strong> the MKS system while the acceleration a has units 2 .<br />

2.14.2 Gravitational potential energy <br />

Chapter 2102 showed that a conservative field can be expressed <strong>in</strong><br />

termsoftheconceptofapotentialenergy(r) which depends on<br />

position. The potential energy difference ∆ → between two po<strong>in</strong>ts<br />

r and r , is the work done mov<strong>in</strong>g from to aga<strong>in</strong>st a force F. That<br />

is:<br />

Z <br />

∆ → = (r ) − (r )=− F · l (2.154)<br />

<br />

In general, this l<strong>in</strong>e <strong>in</strong>tegral depends on the path taken.<br />

Consider the gravitational field produced by a s<strong>in</strong>gle po<strong>in</strong>t mass<br />

1 The work done mov<strong>in</strong>g a mass 0 from to <strong>in</strong> this gravitational<br />

field can be calculated along an arbitrary path shown <strong>in</strong> figure<br />

28 by assum<strong>in</strong>g Newton’s law of gravitation. Then the force on 0<br />

due to po<strong>in</strong>t mass 1 is;<br />

mg<br />

F<br />

dl<br />

F = − 1 0<br />

br<br />

2 (2.155)<br />

Express<strong>in</strong>g l <strong>in</strong> spherical coord<strong>in</strong>ates l =ˆr+ˆθ+ s<strong>in</strong> ˆφ gives<br />

the path <strong>in</strong>tegral (2154) from ( ) to ( ) is<br />

Z <br />

Z <br />

h<br />

∆ → = − F · l =<br />

<br />

<br />

∙ 1<br />

= − 1 0 − 1 ¸<br />

<br />

1 0<br />

2<br />

(ˆr·br + ˆr · ˆθ + s<strong>in</strong> ˆr · ˆφ)<br />

i<br />

= <br />

Figure 2.8: Work done aga<strong>in</strong>st a<br />

force field mov<strong>in</strong>g from a to b.<br />

Z <br />

<br />

1 0<br />

br<br />

2 · br<br />

(2.156)

40 CHAPTER 2. REVIEW OF NEWTONIAN MECHANICS<br />

s<strong>in</strong>ce the scalar product of the unit vectors br · br =1 Note that the second two terms also cancel s<strong>in</strong>ce<br />

br · ˆθ = ˆr · ˆφ =0s<strong>in</strong>ce the unit vectors are mutually orthogonal. Thusthel<strong>in</strong>e<strong>in</strong>tegraljustdependsonlyon<br />

the start<strong>in</strong>g and end<strong>in</strong>g radii and is <strong>in</strong>dependent of the angular coord<strong>in</strong>ates or the detailed path taken between<br />

( ) and ( ) <br />

Consider the Pr<strong>in</strong>ciple of Superposition for a gravitational field produced by a set of po<strong>in</strong>t masses. The<br />

l<strong>in</strong>e <strong>in</strong>tegral then can be written as:<br />

Z <br />

X<br />

Z <br />

X<br />

∆→ = − F · l = − F · l = ∆→ (2.157)<br />

<br />

=1 =1<br />

Thus the net potential energy difference is the sum of the contributions from each po<strong>in</strong>t mass produc<strong>in</strong>g the<br />

gravitational force field. S<strong>in</strong>ce each component is conservative, then the total potential energy difference also<br />

must be conservative. For a conservative force, this l<strong>in</strong>e <strong>in</strong>tegral is <strong>in</strong>dependent of the path taken, itdepends<br />

only on the start<strong>in</strong>g and end<strong>in</strong>g positions, r and r . That is, the potential energy is a local function<br />

dependent only on position. The usefulness of gravitational potential energy is that, s<strong>in</strong>ce the gravitational<br />

force is a conservative force, it is possible to solve many problems <strong>in</strong> classical mechanics us<strong>in</strong>g the fact<br />

that the sum of the k<strong>in</strong>etic energy and potential energy is a constant. Note that the gravitational field is<br />

conservative, s<strong>in</strong>ce the potential energy difference ∆→ is <strong>in</strong>dependent of the path taken. It is conservative<br />

becausetheforceisradial and time <strong>in</strong>dependent, it is not due to the 1 <br />

dependence of the field.<br />

2<br />

2.14.3 Gravitational potential <br />

Us<strong>in</strong>g F = 0 g gives that the change <strong>in</strong> potential energy due to mov<strong>in</strong>g a mass 0 from to <strong>in</strong> a<br />

gravitational field g is:<br />

∆ <br />

→ = − 0<br />

Z <br />

<br />

g · l (2.158)<br />

Note that the probe mass 0 factors out from the <strong>in</strong>tegral. It is convenient to def<strong>in</strong>e a new quantity called<br />

gravitational potential where<br />

∆ <br />

→ =<br />

<br />

Z<br />

∆<br />

<br />

→<br />

= − g · l (2.159)<br />

0<br />

<br />

That is; gravitational potential differenceistheworkthatmustbedone,perunitmass,tomovefroma to<br />

b with no change <strong>in</strong> k<strong>in</strong>etic energy. Be careful not to confuse the gravitational potential energy difference<br />

∆ → and gravitational potential difference ∆ → ,thatis,∆ has units of energy, ,while∆ has<br />

units of .<br />

The gravitational potential is a property of the gravitational force field; it is given as m<strong>in</strong>us the l<strong>in</strong>e<br />

<strong>in</strong>tegral of the gravitational field from to . The change <strong>in</strong> gravitational potential energy for mov<strong>in</strong>g a<br />

mass 0 from to is given <strong>in</strong> terms of gravitational potential by:<br />

Superposition and potential<br />

∆ <br />

→ = 0 ∆ <br />

→ (2.160)<br />

Previously it was shown that the gravitational force is conservative for the superposition of many masses.<br />

To recap, if the gravitational field<br />

g = g 1 + g 2 + g 3 (2.161)<br />

then<br />

Z <br />

Z <br />

Z <br />

Z <br />

<br />

→ = − g · l = − g 1 · l − g 2 · l − g 3 · l = Σ → (2.162)<br />

<br />

<br />

<br />

<br />

Thus gravitational potential is a simple additive scalar field because the Pr<strong>in</strong>ciple of Superposition applies.<br />

The gravitational potential, between two po<strong>in</strong>ts differ<strong>in</strong>g by <strong>in</strong> height, is . Clearly, the greater or ,<br />

the greater the energy released by the gravitational field when dropp<strong>in</strong>g a body through the height . The<br />

unit of gravitational potential is the

2.14. NEWTON’S LAW OF GRAVITATION 41<br />

2.14.4 Potential theory<br />

The gravitational force and electrostatic force both obey the <strong>in</strong>verse square law, for which the field and<br />

correspond<strong>in</strong>g potential are related by:<br />

Z <br />

∆ → = − g · l (2.163)<br />

<br />

For an arbitrary <strong>in</strong>f<strong>in</strong>itessimal element distance l the change <strong>in</strong> gravitational potential is<br />

= −g · l (2.164)<br />

Us<strong>in</strong>g cartesian coord<strong>in</strong>ates both g and l can be written as<br />

g = b i + b j + k b l = b i + b j + k b (2.165)<br />

Tak<strong>in</strong>g the scalar product gives:<br />

= −g · l = − − − (2.166)<br />

Differential calculus expresses the change <strong>in</strong> potential <strong>in</strong> terms of partial derivatives by:<br />

By association, 2166 and 2167 imply that<br />

= <br />

<br />

+<br />

<br />

<br />

<br />

+ (2.167)<br />

<br />

= − <br />

<br />

= − <br />

<br />

= − <br />

<br />

(2.168)<br />

Thus on each axis, the gravitational field can be written as m<strong>in</strong>us the gradient of the gravitational potential.<br />

In three dimensions, the gravitational field is m<strong>in</strong>us the total gradient of potential and the gradient of the<br />

scalar function canbewrittenas:<br />

g = −∇ (2.169)<br />

In cartesian coord<strong>in</strong>ates this equals<br />

∙<br />

g = − b<br />

i<br />

+b j <br />

+ k b ¸<br />

(2.170)<br />

<br />

Thus the gravitational field is just the gradient of the gravitational potential, which always is perpendicular<br />

to the equipotentials. Skiers are familiar with the concept of gravitational equipotentials and the fact that<br />

the l<strong>in</strong>e of steepest descent, and thus maximum acceleration, is perpendicular to gravitational equipotentials<br />

of constant height. The advantage of us<strong>in</strong>g potential theory for <strong>in</strong>verse-square law forces is that scalar<br />

potentials replace the more complicated vector forces, which greatly simplifies calculation. Potential theory<br />

plays a crucial role for handl<strong>in</strong>g both gravitational and electrostatic forces.<br />

2.14.5 Curl of the gravitational field<br />

It has been shown that the gravitational field is conservative, that is<br />

∆ → is <strong>in</strong>dependent of the path taken between and . Therefore,<br />

equation 2159 gives that the gravitational potential is <strong>in</strong>dependent of<br />

the path taken between two po<strong>in</strong>ts and . Consider two possible paths<br />

between and as shown <strong>in</strong> figure 29. The l<strong>in</strong>e <strong>in</strong>tegral from to via<br />

route 1 is equal and opposite to the l<strong>in</strong>e <strong>in</strong>tegral back from to via<br />

route 2 if the gravitational field is conservative as shown earlier.<br />

A better way of express<strong>in</strong>g this is that the l<strong>in</strong>e <strong>in</strong>tegral of the gravitational<br />

field is zero around any closed path. Thus the l<strong>in</strong>e <strong>in</strong>tegral between<br />

and , viapath1, and return<strong>in</strong>g back to , viapath2, areequaland<br />

opposite. That is, the net l<strong>in</strong>e <strong>in</strong>tegral for a closed loop is zero<br />

1<br />

2<br />

Figure 2.9: Circulation of the<br />

gravitational field.

42 CHAPTER 2. REVIEW OF NEWTONIAN MECHANICS<br />

I<br />

g · l =0 (2.171)<br />

which is a measure of the circulation of the gravitational field. The fact that the circulation equals zero<br />

corresponds to the statement that the gravitational field is radial for a po<strong>in</strong>t mass.<br />

Stokes Theorem, discussed <strong>in</strong> appendix 3 states that<br />

I Z<br />

<br />

F · l =<br />

<br />

<br />

<br />

<br />

(∇ × F) · S (2.172)<br />

Thus the zero circulation of the gravitational field can be rewritten as<br />

I Z<br />

g · l = (∇ × g) · S =0 (2.173)<br />

<br />

<br />

<br />

<br />

<br />

S<strong>in</strong>ce this is <strong>in</strong>dependent of the shape of the perimeter , therefore<br />

∇ × g =0 (2.174)<br />

That is, the gravitational field is a curl-free field.<br />

A property of any curl-free field is that it can be expressed as the gradient of a scalar potential s<strong>in</strong>ce<br />

∇ × ∇ =0 (2.175)<br />

Therefore, the curl-free gravitational field can be related to a scalar potential as<br />

g = −∇ (2.176)<br />

Thus is consistent with the above def<strong>in</strong>ition of gravitational potential <strong>in</strong> that the scalar product<br />

Z Z <br />

Z X<br />

Z<br />

<br />

<br />

∆ → = − g · l = (∇) · l =<br />

= (2.177)<br />

<br />

<br />

<br />

An identical relation between the electric field and electric potential applies for the <strong>in</strong>verse-square law<br />

electrostatic field.<br />

Reference potentials:<br />

Note that only differences <strong>in</strong> potential energy, , and gravitational potential, , are mean<strong>in</strong>gful, the absolute<br />

values depend on some arbitrarily chosen reference. However, often it is useful to measure gravitational<br />

potential with respect to a particular arbitrarily chosen reference po<strong>in</strong>t such as to sea level. Aircraft<br />

pilots are required to set their altimeters to read with respect to sea level rather than their departure<br />

airport. This ensures that aircraft leav<strong>in</strong>g from say both Rochester, 559 0 and Denver 5000 0 , have<br />

their altimeters set to a common reference to ensure that they do not collide. The gravitational force is the<br />

gradient of the gravitational field which only depends on differences <strong>in</strong> potential, and thus is <strong>in</strong>dependent of<br />

any constant reference.<br />

Gravitational potential due to cont<strong>in</strong>uous distributions of charge Suppose mass is distributed<br />

over a volume with a density at any po<strong>in</strong>t with<strong>in</strong> the volume. The gravitational potential at any field<br />

po<strong>in</strong>t due to an element of mass = at the po<strong>in</strong>t 0 is given by:<br />

Z<br />

( 0 ) 0<br />

∆ ∞→ = −<br />

(2.178)<br />

0 <br />

This <strong>in</strong>tegral is over a scalar quantity. S<strong>in</strong>ce gravitational potential is a scalar quantity, it is easier to<br />

compute than is the vector gravitational field g . If the scalar potential field is known, then the gravitational<br />

field is derived by tak<strong>in</strong>g the gradient of the gravitational potential.

2.14. NEWTON’S LAW OF GRAVITATION 43<br />

2.14.6 Gauss’s Law for Gravitation<br />

The flux Φ of the gravitational field g through a surface<br />

, asshown<strong>in</strong>figure 210, isdef<strong>in</strong>ed as<br />

Z<br />

Φ ≡ g · S (2.179)<br />

<br />

Note that there are two possible perpendicular directions<br />

that could be chosen for the surface vector S Us<strong>in</strong>g<br />

g<br />

Newton’s law of gravitation for a po<strong>in</strong>t mass the flux<br />

through the surface is<br />

Z<br />

br · S<br />

Φ = −<br />

2 (2.180)<br />

Note that the solid angle subtended by the surface <br />

at an angle to the normal from the po<strong>in</strong>t mass is given<br />

by<br />

cos br · S<br />

Ω =<br />

2 =<br />

2 (2.181)<br />

Thus the net gravitational flux equals<br />

Z<br />

Figure 2.10: Flux of the gravitational field through<br />

an <strong>in</strong>f<strong>in</strong>itessimal surface element dS.<br />

Φ = − Ω (2.182)<br />

<br />

Consider a closed surface where the direction of the surface vector S is def<strong>in</strong>ed as outwards. The net<br />

flux out of this closed surface is given by<br />

I<br />

I<br />

br · S<br />

Φ = −<br />

2 = − Ω = −4 (2.183)<br />

<br />

This is <strong>in</strong>dependent of where the po<strong>in</strong>t mass lies with<strong>in</strong> the closed surface or on the shape of the closed<br />

surface. Note that the solid angle subtended is zero if the po<strong>in</strong>t mass lies outside the closed surface. Thus<br />

the flux is as given by equation 2183 if the mass is enclosed by the closed surface, while it is zero if the mass<br />

is outside of the closed surface.<br />

S<strong>in</strong>ce the flux for a po<strong>in</strong>t mass is <strong>in</strong>dependent of the location of the mass with<strong>in</strong> the volume enclosed by<br />

the closed surface, and us<strong>in</strong>g the pr<strong>in</strong>ciple of superposition for the gravitational field, then for enclosed<br />

po<strong>in</strong>t masses the net flux is<br />

Z<br />

X<br />

Φ ≡ g · S = −4 (2.184)<br />

<br />

This can be extended to cont<strong>in</strong>uous mass distributions, with local mass density giv<strong>in</strong>g that the net flux<br />

Z<br />

Z<br />

Φ ≡ g · S = −4 (2.185)<br />

<br />

<br />

<br />

<br />

Gauss’s Divergence Theorem was given <strong>in</strong> appendix 2 as<br />

I Z<br />

Φ = F · S = ∇ · F (2.186)<br />

<br />

<br />

<br />

Apply<strong>in</strong>g the Divergence Theorem to Gauss’s law gives that<br />

I Z<br />

Z<br />

Φ = g · S = ∇ · g = −4<br />

or<br />

<br />

Z<br />

<br />

<br />

<br />

<br />

<br />

<br />

<br />

[∇ · g +4] =0 (2.187)<br />

dS

44 CHAPTER 2. REVIEW OF NEWTONIAN MECHANICS<br />

This is true <strong>in</strong>dependent of the shape of the surface, thus the divergence of the gravitational field<br />

∇ · g = −4 (2.188)<br />

This is a statement that the gravitational field of a po<strong>in</strong>t mass has a 1<br />

<br />

dependence.<br />

2<br />

Us<strong>in</strong>g the fact that the gravitational field is conservative, this can be expressed as the gradient of the<br />

gravitational potential <br />

g = −∇ (2.189)<br />

and Gauss’s law, then becomes<br />

which also can be written as Poisson’s equation<br />

∇ · ∇ =4 (2.190)<br />

∇ 2 =4 (2.191)<br />

Know<strong>in</strong>g the mass distribution allows determ<strong>in</strong>ation of the potential by solv<strong>in</strong>g Poisson’s equation.<br />

A special case that often is encountered is when the mass distribution is zero <strong>in</strong> a given region. Then the<br />

potential for this region can be determ<strong>in</strong>ed by solv<strong>in</strong>g Laplace’s equation with known boundary conditions.<br />

∇ 2 =0 (2.192)<br />

For example, Laplace’s equation applies <strong>in</strong> the free space between the masses. It is used extensively <strong>in</strong> electrostatics<br />

to compute the electric potential between charged conductors which themselves are equipotentials.<br />

2.14.7 Condensed forms of Newton’s Law of Gravitation<br />

The above discussion has resulted <strong>in</strong> several alternative expressions of Newton’s Law of Gravitation that will<br />

be summarized here. The most direct statement of Newton’s law is<br />

Z<br />

g (r) =−<br />

<br />

(r 0 )<br />

³br − br 0´<br />

(r − r 0 ) 2 0 (2.193)<br />

An elegant way to express Newton’s Law of Gravitation is <strong>in</strong> terms of the flux and circulation of the<br />

gravitational field. That is,<br />

Flux:<br />

Z<br />

Circulation:<br />

Φ ≡<br />

<br />

Z<br />

g · S = −4<br />

I<br />

<br />

<br />

(2.194)<br />

g · l =0 (2.195)<br />

The flux and circulation are better expressed <strong>in</strong> terms of the vector differential concepts of divergence<br />

and curl.<br />

Divergence:<br />

∇ · g = −4 (2.196)<br />

Curl:<br />

∇ × g =0 (2.197)<br />

Remember that the flux and divergence of the gravitational field are statements that the field between<br />

po<strong>in</strong>t masses has a 1<br />

<br />

dependence. The circulation and curl are statements that the field between po<strong>in</strong>t<br />

2<br />

masses is radial.<br />

Because the gravitational field is conservative it is possible to use the concept of the scalar potential<br />

field This concept is especially useful for solv<strong>in</strong>g some problems s<strong>in</strong>ce the gravitational potential can be<br />

evaluated us<strong>in</strong>g the scalar <strong>in</strong>tegral<br />

Z<br />

( 0 ) 0<br />

∆ ∞→ = −<br />

(2.198)<br />

0

2.14. NEWTON’S LAW OF GRAVITATION 45<br />

An alternate approach is to solve Poisson’s equation if the boundary values and mass distributions are known<br />

where Poisson’s equation is:<br />

∇ 2 =4 (2.199)<br />

These alternate expressions of Newton’s law of gravitation can be exploited to solve problems.<br />

method of solution is identical to that used <strong>in</strong> electrostatics.<br />

The<br />

2.16 Example: Field of a uniform sphere<br />

Consider the simple case of the gravitational field due to a uniform sphere of matter of radius and<br />

mass . Then the volume mass density<br />

=<br />

3<br />

4 3<br />

The gravitational field and potential for this uniform sphere of matter can be derived three ways;<br />

a) The field can be evaluated by directly <strong>in</strong>tegrat<strong>in</strong>g over the volume<br />

Z (r 0 )<br />

³br − br 0´<br />

g (r) =−<br />

(r − r 0 ) 2 0<br />

b) The potential can be evaluated directly by <strong>in</strong>tegration of<br />

Z<br />

( 0 ) 0<br />

∆ ∞→ = −<br />

0 <br />

and then<br />

g = −∇<br />

c) The obvious spherical symmetry can be used <strong>in</strong> conjunction<br />

with Gauss’s law to easily solve this problem.<br />

Z<br />

Z<br />

g · S = −4 <br />

<br />

<br />

<br />

g<br />

0<br />

-GM r<br />

-GM<br />

r²<br />

4 2 () =−4<br />

(rR)<br />

That is: for <br />

g = − 2 br<br />

(rR)<br />

-GM | 3R²-r² |<br />

-GM<br />

r<br />

Similarly, for <br />

4 2 () = 4 3 3 <br />

(rR)<br />

Gravitational field g and gravitational<br />

potential Φ of a uniformly-dense<br />

spherical mass distribution of radius .<br />

That is:<br />

g = − 3 r<br />

(rR)<br />

The field <strong>in</strong>side the Earth is radial and is proportional to the distance from the center of the Earth. This<br />

is Hooke’s Law, and thus ignor<strong>in</strong>g air drag, any body dropped down a qhole through the center of the Earth<br />

<br />

will undergo harmonic oscillations with an angular frequency of 0 =<br />

<br />

= p <br />

3 This gives a period of<br />

oscillation of 14 hours, which is about the length of a 235 lecture <strong>in</strong> classical mechanics, which may seem<br />

like a long time.<br />

Clearly method (c) is much simpler to solve for this case. In general, look for a symmetry that allows<br />

identification of a surface upon which the magnitude and direction of the fieldisconstant. Forsuchcases<br />

use Gauss’s law. Otherwise use methods (a) or (b) whichever one is easiest to apply. Further examples will<br />

not be given here s<strong>in</strong>ce they are essentially identical to those discussed extensively <strong>in</strong> electrostatics.

46 CHAPTER 2. REVIEW OF NEWTONIAN MECHANICS<br />

2.15 Summary<br />

Newton’s Laws of Motion:<br />

A cursory review of Newtonian mechanics has been presented. The concept of <strong>in</strong>ertial frames of reference<br />

was <strong>in</strong>troduced s<strong>in</strong>ce Newton’s laws of motion apply only to <strong>in</strong>ertial frames of reference.<br />

Newton’s Law of motion<br />

F = p<br />

(26)<br />

<br />

leads to second-order equations of motion which can be difficult to handle for many-body systems.<br />

Solution of Newton’s second-order equations of motion can be simplified us<strong>in</strong>g the three first-order <strong>in</strong>tegrals<br />

coupled with correspond<strong>in</strong>g conservation laws. The first-order time <strong>in</strong>tegral for l<strong>in</strong>ear momentum<br />

is Z 2<br />

F =<br />

1<br />

Z 2<br />

The first-order time <strong>in</strong>tegral for angular momentum is<br />

L <br />

<br />

= r × p <br />

= N <br />

1<br />

p <br />

=(p 2 − p 1 ) <br />

Z 2<br />

1<br />

N =<br />

Z 2<br />

1<br />

L <br />

=(L 2 − L 1 ) <br />

The first-order spatial <strong>in</strong>tegral is related to k<strong>in</strong>etic energy and the concept of work. That is<br />

(210)<br />

(216)<br />

F = Z 2<br />

<br />

F · r =( 2 − 1 ) <br />

r <br />

<br />

1<br />

(221)<br />

The conditions that lead to conservation of l<strong>in</strong>ear and angular momentum and total mechanical energy<br />

were discussed for many-body systems. The important class of conservative forces was shown to apply if<br />

the position-dependent force do not depend on time or velocity, and if the work done by a force R 2<br />

1 F · r <br />

is <strong>in</strong>dependent of the path taken between the <strong>in</strong>itial and f<strong>in</strong>al locations. The total mechanical energy is a<br />

constant of motion when the forces are conservative.<br />

It was shown that the concept of center of mass of a many-body or f<strong>in</strong>ite sized body separates naturally<br />

for all three first-order <strong>in</strong>tegrals. The center of mass is that po<strong>in</strong>t about which<br />

X<br />

Z<br />

r 0 = r 0 =0<br />

(Centre of mass def<strong>in</strong>ition)<br />

where r 0 is the vector def<strong>in</strong><strong>in</strong>g the location of mass with respect to the center of mass. The concept of<br />

center of mass greatly simplifies the description of the motion of f<strong>in</strong>ite-sized bodies and many-body systems<br />

by separat<strong>in</strong>g out the important <strong>in</strong>ternal <strong>in</strong>teractions and correspond<strong>in</strong>g underly<strong>in</strong>g physics, from the trivial<br />

overall translational motion of a many-body system..<br />

The Virial theorem states that the time-averaged properties are related by<br />

* +<br />

h i = − 1 X<br />

F · r (286)<br />

2<br />

<br />

It was shown that the Virial theorem is useful for relat<strong>in</strong>g the time-averaged k<strong>in</strong>etic and potential energies,<br />

especially for cases <strong>in</strong>volv<strong>in</strong>g either l<strong>in</strong>ear or <strong>in</strong>verse-square forces.<br />

Typical examples were presented of application of Newton’s equations of motion to solv<strong>in</strong>g systems<br />

<strong>in</strong>volv<strong>in</strong>g constant, l<strong>in</strong>ear, position-dependent, velocity-dependent, and time-dependent forces, to constra<strong>in</strong>ed<br />

and unconstra<strong>in</strong>ed systems, as well as systems with variable mass. Rigid-body rotation about a body-fixed<br />

rotation axis also was discussed.<br />

It is important to be cognizant of the follow<strong>in</strong>g limitations that apply to Newton’s laws of motion:<br />

1) Newtonian mechanics assumes that all observables are measured to unlimited precision, that is <br />

p r are known exactly. Quantum physics <strong>in</strong>troduces limits to measurement due to wave-particle duality.<br />

2) The Newtonian view is that time and position are absolute concepts. The Theory of Relativity shows<br />

that this is not true. Fortunately for most problems and thus Newtonian mechanics is an excellent<br />

approximation.

2.15. SUMMARY 47<br />

3) Another limitation, to be discussed later, is that it is impractical to solve the equations of motion for<br />

many <strong>in</strong>teract<strong>in</strong>g bodies such as all the molecules <strong>in</strong> a gas. Then it is necessary to resort to us<strong>in</strong>g statistical<br />

averages, this approach is called statistical mechanics.<br />

Newton’s work constitutes a theory of motion <strong>in</strong> the universe that <strong>in</strong>troduces the concept of causality.<br />

Causality is that there is a one-to-one correspondence between cause of effect. Each force causes a known<br />

effect that can be calculated. Thus the causal universe is pictured by philosophers to be a giant mach<strong>in</strong>e<br />

whose parts move like clockwork <strong>in</strong> a predictable and predeterm<strong>in</strong>ed way accord<strong>in</strong>g to the laws of nature. This<br />

is a determ<strong>in</strong>istic view of nature. There are philosophical problems <strong>in</strong> that such a determ<strong>in</strong>istic viewpo<strong>in</strong>t<br />

appears to be contrary to free will. That is, taken to the extreme it implies that you were predest<strong>in</strong>ed to<br />

read this book because it is a natural consequence of this mechanical universe!<br />

Newton’s Laws of Gravitation<br />

Newton’s Laws of Gravitation and the Laws of Electrostatics are essentially identical s<strong>in</strong>ce they both<br />

<strong>in</strong>volve a central <strong>in</strong>verse square-law dependence of the forces. The important difference is that the gravitational<br />

force is attractive whereas the electrostatic force between identical charges is repulsive. That is,<br />

the gravitational constant is replaced by − 1<br />

4 0<br />

,andthemassdensity becomes the charge density for<br />

the case of electrostatics. As a consequence it is unnecessary to make a detailed study of Newton’s law of<br />

gravitation s<strong>in</strong>ce it is identical to what has already been studied <strong>in</strong> your accompany<strong>in</strong>g electrostatic courses.<br />

Table 21 summarizes and compares the laws of gravitation and electrostatics. For both gravitation and<br />

electrostatics the field is central and conservative and depends as 1<br />

2ˆr<br />

The laws of gravitation and electrostatics can be expressed <strong>in</strong> a more useful form <strong>in</strong> terms of the flux and<br />

circulation of the gravitational field as given either <strong>in</strong> the vector <strong>in</strong>tegral or vector differential forms. The<br />

radial <strong>in</strong>dependence of the flux, and correspond<strong>in</strong>g divergence, is a statement that the fields are radial and<br />

have a 1<br />

2ˆr dependence. The statement that the circulation, and correspond<strong>in</strong>g curl, are zero is a statement<br />

that the fields are radial and conservative.<br />

Table 21; Comparison of Newton’s law of gravitation and electrostatics.<br />

Gravitation<br />

Electrostatics<br />

Force field g ≡ F <br />

E ≡ F<br />

<br />

Density Mass density (r 0 ) Charge density (r 0 )<br />

Conservative central field g (r) =− R (r 0 )(r− r 0 )<br />

0 E (r) = 1 (r<br />

(r−r 0 ) 2 4 0<br />

R<br />

)(r− r 0 )<br />

0<br />

(r−r 0 ) 2<br />

Flux<br />

Φ ≡ R g · S = −4 R Φ ≡ R E · S = 1 <br />

<br />

0<br />

R <br />

I<br />

I<br />

<br />

Circulation<br />

g · l =0<br />

E · l =0<br />

Divergence ∇ · g = −4 ∇ · E = 1 0<br />

<br />

Curl ∇ × g =0 ∇ × E =0<br />

Potential<br />

∆ ∞→ = − R ( 0 ) 0<br />

<br />

∆<br />

0 ∞→ = 1<br />

4 0<br />

R<br />

Poisson’s equation ∇ 2 =4 ∇ 2 = − 1 0<br />

<br />

( 0 ) 0<br />

0 <br />

Both the gravitational and electrostatic central fields are conservative mak<strong>in</strong>g it possible to use the<br />

concept of the scalar potential field This concept is especially useful for solv<strong>in</strong>g some problems s<strong>in</strong>ce the<br />

potential can be evaluated us<strong>in</strong>g a scalar <strong>in</strong>tegral. An alternate approach is to solve Poisson’s equation if the<br />

boundary values and mass distributions are known. The methods of solution of Newton’s law of gravitation<br />

are identical to those used <strong>in</strong> electrostatics and are readily accessible <strong>in</strong> the literature.

48 CHAPTER 2. REVIEW OF NEWTONIAN MECHANICS<br />

Problems<br />

1. Two particles are projected from the same po<strong>in</strong>t with velocities 1 and 2 ,atelevations 1 and 2 , respectively<br />

( 1 2 ). Show that if they are to collide <strong>in</strong> mid-air the <strong>in</strong>terval between the fir<strong>in</strong>gs must be<br />

2 1 2 s<strong>in</strong>( 1 − 2 )<br />

( 1 cos 1 + 2 cos 2 ) <br />

2. The teeter totter comprises two identical weights which hang on droop<strong>in</strong>g arms attached to a peg as shown.<br />

The arrangement is unexpectedly stable and can be spun and rocked with little danger of toppl<strong>in</strong>g over.<br />

m<br />

l<br />

L<br />

l<br />

m<br />

(a) F<strong>in</strong>d an expression for the potential energy of the teeter toy as a function of when the teeter toy is<br />

cocked at an angle about the pivot po<strong>in</strong>t. For simplicity, consider only rock<strong>in</strong>g motion <strong>in</strong> the vertical<br />

plane.<br />

(b) Determ<strong>in</strong>e the equilibrium values(s) of .<br />

(c) Determ<strong>in</strong>e whether the equilibrium is stable, unstable, or neutral for the value(s) of found <strong>in</strong> part (b).<br />

(d) How could you determ<strong>in</strong>e the answers to parts (b) and (c) from a graph of the potential energy versus ?<br />

(e) Expand the expression for the potential energy about = 0 and determ<strong>in</strong>e the frequency of small<br />

oscillations.<br />

3. A particle of mass is constra<strong>in</strong>ed to move on the frictionless <strong>in</strong>ner surface of a cone of half-angle .<br />

(a) F<strong>in</strong>d the restrictions on the <strong>in</strong>itial conditions such that the particle moves <strong>in</strong> a circular orbit about the<br />

vertical axis.<br />

(b) Determ<strong>in</strong>e whether this k<strong>in</strong>d of orbit is stable.<br />

4. Consider a th<strong>in</strong> rod of length and mass .<br />

(a) Draw gravitational field l<strong>in</strong>es and equipotential l<strong>in</strong>es for the rod. What can you say about the equipotential<br />

surfaces of the rod?<br />

(b) Calculate the gravitational potential at a po<strong>in</strong>t that is a distance from one end of the rod and <strong>in</strong> a<br />

direction perpendicular to the rod.<br />

(c) Calculate the gravitational field at by direct <strong>in</strong>tegration.<br />

(d) Could you have used Gauss’s law to f<strong>in</strong>d the gravitational field at ?Whyorwhynot?<br />

5. Consider a s<strong>in</strong>gle particle of mass <br />

(a) Determ<strong>in</strong>e the position and velocity of a particle <strong>in</strong> spherical coord<strong>in</strong>ates.<br />

(b) Determ<strong>in</strong>e the total mechanical energy of the particle <strong>in</strong> potential .<br />

(c) Assume the force is conservative. Show that = −∇ .ShowthatitagreeswithStoke’stheorem.<br />

(d) Show that the angular momentum = × of the particle is conserved.<br />

× dB<br />

d + dA<br />

d × .<br />

H<strong>in</strong>t:<br />

d<br />

d<br />

( × ) =

2.15. SUMMARY 49<br />

6. Consider a fluidwithdensity and velocity <strong>in</strong> some volume . The mass current = determ<strong>in</strong>es the<br />

amount of mass exit<strong>in</strong>g the surface per unit time by the <strong>in</strong>tegral · <br />

(a) Us<strong>in</strong>g the divergence theorem, prove the cont<strong>in</strong>uity equation, ∇ · + <br />

=0<br />

7. A rocket of <strong>in</strong>itial mass burns fuel at constant rate (kilograms per second), produc<strong>in</strong>g a constant force .<br />

The total mass of available fuel is . Assume the rocket starts from rest and moves <strong>in</strong> a fixed direction with<br />

no external forces act<strong>in</strong>g on it.<br />

(a) Determ<strong>in</strong>e the equation of motion of the rocket.<br />

(b) Determ<strong>in</strong>e the f<strong>in</strong>al velocity of the rocket.<br />

(c) Determ<strong>in</strong>e the displacement of the rocket <strong>in</strong> time.<br />

8. Consider a solid hemisphere of radius . Compute the coord<strong>in</strong>ates of the center of mass relative to the center<br />

of the spherical surface used to def<strong>in</strong>e the hemisphere.<br />

9. A 2000kg Ford was travell<strong>in</strong>g south on Mt. Hope Avenue when it collided with your 1000kg sports car travell<strong>in</strong>g<br />

west on Elmwood Avenue. The two badly-damaged cars became entangled <strong>in</strong> the collision and leave a skid mark<br />

that is 20 meters long <strong>in</strong> a direction 14 ◦ to the west of the orig<strong>in</strong>al direction of travel of the Excursion. The<br />

wealthy Excursion driver hires a high-powered lawyer who accuses you of speed<strong>in</strong>g through the <strong>in</strong>tersection.<br />

Use your P235 knowledge, plus the police officer’s report of the recoil direction, the skid length, and knowledge<br />

that the coefficient of slid<strong>in</strong>g friction between the tires and road is =06, to deduce the orig<strong>in</strong>al velocities of<br />

both cars. Were either of the cars exceed<strong>in</strong>g the 30mph speed limit?<br />

10. A particle of mass mov<strong>in</strong>g <strong>in</strong> one dimension has potential energy () = 0 [2( )2 − ( )4 ] where 0 and <br />

are positive constants.<br />

a) F<strong>in</strong>d the force () that acts on the particle.<br />

b) Sketch (). F<strong>in</strong>d the positions of stable and unstable equilibrium.<br />

c) What is the angular frequency of oscillations about the po<strong>in</strong>t of stable equilibrium?<br />

d) What is the m<strong>in</strong>imum speed the particle must have at the orig<strong>in</strong> to escape to <strong>in</strong>f<strong>in</strong>ity?<br />

e) At =0the particle is at the orig<strong>in</strong> and its velocity is positive and equal to the escape velocity. F<strong>in</strong>d ()<br />

andsketchtheresult.<br />

11. a) Consider a s<strong>in</strong>gle-stage rocket travell<strong>in</strong>g <strong>in</strong> a straight l<strong>in</strong>e subject to an external force act<strong>in</strong>g along the<br />

same l<strong>in</strong>e where is the exhaust velocity of the ejected fuel relative to the rocket. Show that the equation of<br />

motion is<br />

˙ = − ˙ + <br />

b) Specialize to the case of a rocket tak<strong>in</strong>g off vertically from rest <strong>in</strong> a uniform gravitational field Assume<br />

that the rocket ejects mass at a constant rate of ˙ = − where is a positive constant. Solve the equation of<br />

motion to derive the dependence of velocity on time.<br />

c) The first couple of m<strong>in</strong>utes of the launch of the Space Shuttle can be described roughly by; <strong>in</strong>itial mass<br />

= 2 × 10 6 kg, mass after 2 m<strong>in</strong>utes = 1 × 10 6 kg, exhaust speed = 3000 and <strong>in</strong>itial velocity is zero.<br />

Estimate the velocity of the Space Shuttle after two m<strong>in</strong>utes of flight.<br />

d) Describe what would happen to a rocket where ˙ <br />

12. A time <strong>in</strong>dependent field is conservative if ∇ × =0. Use this fact to test if the follow<strong>in</strong>g fields are<br />

conservative, and derive the correspond<strong>in</strong>g potential .<br />

a) = + + = + = + <br />

b) = − − =ln = − +

50 CHAPTER 2. REVIEW OF NEWTONIAN MECHANICS<br />

13. Consider a solid cyl<strong>in</strong>der of mass and radius slid<strong>in</strong>g without roll<strong>in</strong>g down the smooth <strong>in</strong>cl<strong>in</strong>ed face of a<br />

wedge of mass that is free to slide without friction on a horizontal plane floor. Use the coord<strong>in</strong>ates shown<br />

<strong>in</strong> the figure.<br />

a) How far has the wedge moved by the time the cyl<strong>in</strong>der has descended from rest a vertical distance ?<br />

b) Now suppose that the cyl<strong>in</strong>der is free to roll down the wedge without slipp<strong>in</strong>g. How far does the wedge<br />

move <strong>in</strong> this case if the cyl<strong>in</strong>der rolls down a vertical distance ?<br />

c) In which case does the cyl<strong>in</strong>der reach the bottom faster? How does this depend on the radius of the cyl<strong>in</strong>der?<br />

y<br />

.<br />

x<br />

y<br />

x<br />

14. If the gravitational field vector is <strong>in</strong>dependent of the radial distance with<strong>in</strong> a sphere, f<strong>in</strong>d the function describ<strong>in</strong>g<br />

the mass density () of the sphere.

Chapter 3<br />

L<strong>in</strong>ear oscillators<br />

3.1 Introduction<br />

Oscillations are a ubiquitous feature <strong>in</strong> nature. Examples are periodic motion of planets, the rise and fall<br />

of the tides, water waves, pendulum <strong>in</strong> a clock, musical <strong>in</strong>struments, sound waves, electromagnetic waves,<br />

and wave-particle duality <strong>in</strong> quantal physics. Oscillatory systems all have the same basic mathematical form<br />

although the names of the variables and parameters are different. The classical l<strong>in</strong>ear theory of oscillations<br />

will be assumed <strong>in</strong> this chapter s<strong>in</strong>ce: (1) The l<strong>in</strong>ear approximation is well obeyed when the amplitudes of<br />

oscillation are small, that is, the restor<strong>in</strong>g force obeys Hooke’s Law. (2) The Pr<strong>in</strong>ciple of Superposition<br />

applies. (3) The l<strong>in</strong>ear theory allows most problems to be solved explicitly <strong>in</strong> closed form. This is <strong>in</strong> contrast<br />

to non-l<strong>in</strong>ear system where the motion can be complicated and even chaotic as discussed <strong>in</strong> chapter 4.<br />

3.2 L<strong>in</strong>ear restor<strong>in</strong>g forces<br />

An oscillatory system requires that there be a stable equilibrium about<br />

which the oscillations occur. Consider a conservative system with potential<br />

energy for which the force is given by<br />

F = −∇ (3.1)<br />

Figure 31 illustrates a conservative system that has three locations at<br />

which the restor<strong>in</strong>g force is zero, that is, where the gradient of the potential<br />

is zero. Stable oscillations occur only around locations 1 and 3 whereas<br />

the system is unstable at the zero gradient location 2. Po<strong>in</strong>t 2 is called a<br />

separatrix <strong>in</strong> that an <strong>in</strong>f<strong>in</strong>itessimal displacement of the particle from this<br />

separatrix will cause the particle to diverge towards either m<strong>in</strong>imum 1 or<br />

3 depend<strong>in</strong>g on which side of the separatrix the particle is displaced.<br />

The requirements for stable oscillations about any po<strong>in</strong>t 0 are that<br />

the potential energy must have the follow<strong>in</strong>g properties.<br />

Figure 3.1: Stability for a onedimensional<br />

potential U(x).<br />

Stability requirements<br />

1) The potential has a stable position for which the restor<strong>in</strong>g force is zero, i.e. ¡ ¢<br />

<br />

= 0<br />

=0<br />

³<br />

<br />

2) The potential must be positive and an even function of displacement − 0 That is.<br />

<br />

<br />

0<br />

´ 0<br />

where is even.<br />

The requirement for the restor<strong>in</strong>g force to be l<strong>in</strong>ear is that the restor<strong>in</strong>g force for perturbation about a<br />

stable equilibrium at 0 is of the form<br />

F = −(− 0 )=¨ (3.2)<br />

The potential energy function for a l<strong>in</strong>ear oscillator has a pure parabolic shape about the m<strong>in</strong>imum location,<br />

that is,<br />

= 1 2 ( − 0) 2 (3.3)<br />

51

52 CHAPTER 3. LINEAR OSCILLATORS<br />

where 0 is the location of the m<strong>in</strong>imum.<br />

Fortunately, oscillatory systems <strong>in</strong>volve small amplitude oscillations about a stable m<strong>in</strong>imum. For weak<br />

non-l<strong>in</strong>ear systems, where the amplitude of oscillation ∆ about the m<strong>in</strong>imum is small, it is useful to make<br />

a Taylor expansion of the potential energy about the m<strong>in</strong>imum. That is<br />

(∆) =( 0 )+∆ ( 0)<br />

<br />

By def<strong>in</strong>ition, at the m<strong>in</strong>imum (0)<br />

<br />

∆ = (∆) − ( 0 )= ∆2<br />

2!<br />

+ ∆2<br />

2!<br />

2 ( 0 )<br />

2<br />

+ ∆3<br />

3!<br />

3 ( 0 )<br />

3<br />

+ ∆4<br />

4!<br />

=0 and thus equation 33 can be written as<br />

2 ( 0 )<br />

2<br />

+ ∆3<br />

3!<br />

3 ( 0 )<br />

3<br />

+ ∆4<br />

4!<br />

4 ( 0 )<br />

4 + (3.4)<br />

4 ( 0 )<br />

4 + (3.5)<br />

2 ( 0)<br />

2<br />

For small amplitude oscillations, the system is l<strong>in</strong>ear if the second-order ∆2<br />

2!<br />

term <strong>in</strong> equation 32 is<br />

dom<strong>in</strong>ant.<br />

The l<strong>in</strong>earity for small amplitude oscillations greatly simplifies description of the oscillatory motion and<br />

complicated chaotic motion is avoided. Most physical systems are approximately l<strong>in</strong>ear for small amplitude<br />

oscillations, and thus the motion close to equilibrium approximates a l<strong>in</strong>ear harmonic oscillator.<br />

3.3 L<strong>in</strong>earity and superposition<br />

An important aspect of l<strong>in</strong>ear systems is that the solutions obey the Pr<strong>in</strong>ciple of Superposition, thatis,for<br />

the superposition of different oscillatory modes, the amplitudes add l<strong>in</strong>early. The l<strong>in</strong>early-damped l<strong>in</strong>ear<br />

oscillator is an example of a l<strong>in</strong>ear system <strong>in</strong> that it <strong>in</strong>volves only l<strong>in</strong>ear operators, that is, it can be written<br />

<strong>in</strong> the operator form (appendix 2)<br />

µ <br />

2<br />

2 + Γ <br />

+ 2 () = cos (3.6)<br />

The quantity <strong>in</strong> the brackets on the left hand side is a l<strong>in</strong>ear operator that can be designated by L where<br />

L() = () (3.7)<br />

An important feature of l<strong>in</strong>ear operators is that they obey the pr<strong>in</strong>ciple of superposition. This property<br />

results from the fact that l<strong>in</strong>ear operators are distributive, that is<br />

L( 1 + 2 )=L ( 1 )+L ( 2 ) (3.8)<br />

Therefore if there are two solutions 1 () and 2 () for two different forc<strong>in</strong>g functions 1 () and 2 ()<br />

L 1 () = 1 () (3.9)<br />

L 2 () = 2 ()<br />

then the addition of these two solutions, with arbitrary constants, also is a solution for l<strong>in</strong>ear operators.<br />

In general then<br />

L( 1 1 + 2 2 )= 1 1 ()+ 2 2 () (3.10)<br />

Ã <br />

! Ã<br />

X<br />

<br />

!<br />

X<br />

L () = ()<br />

=1<br />

=1<br />

(3.11)<br />

The left hand bracket can be identified as the l<strong>in</strong>ear comb<strong>in</strong>ation of solutions<br />

X<br />

() = () (3.12)<br />

=1<br />

while the driv<strong>in</strong>g force is a l<strong>in</strong>ear superposition of harmonic forces<br />

() =<br />

X<br />

() (3.13)<br />

=1

3.4. GEOMETRICAL REPRESENTATIONS OF DYNAMICAL MOTION 53<br />

Thus these l<strong>in</strong>ear comb<strong>in</strong>ations also satisfy the general l<strong>in</strong>ear equation<br />

L() = () (3.14)<br />

Applicability of the Pr<strong>in</strong>ciple of Superposition to a system provides a tremendous advantage for handl<strong>in</strong>g<br />

and solv<strong>in</strong>g the equations of motion of oscillatory systems.<br />

3.4 Geometrical representations of dynamical motion<br />

The powerful pattern-recognition capabilities of the human bra<strong>in</strong>, coupled with geometrical representations<br />

of the motion of dynamical systems, provide a sensitive probe of periodic motion. The geometry of the<br />

motion often can provide more <strong>in</strong>sight <strong>in</strong>to the dynamics than <strong>in</strong>spection of mathematical functions. A<br />

system with degrees of freedom is characterized by locations , velocities ˙ and momenta <strong>in</strong> addition<br />

to the time and <strong>in</strong>stantaneous energy (). Geometrical representations of the dynamical correlations are<br />

illustrated by the configuration space and phase space representations of these 2 +2variables.<br />

3.4.1 Configuration space ( )<br />

Aconfiguration space plot shows the correlated motion of two spatial coord<strong>in</strong>ates and averaged over<br />

time. An example is the two-dimensional l<strong>in</strong>ear oscillator with two equations of motion and solutions<br />

q<br />

<br />

¨ + =0 ¨ + =0 (3.15)<br />

() = cos ( ) () = cos ( − ) (3.16)<br />

where =<br />

. For unequal restor<strong>in</strong>g force constants, 6= the trajectory executes complicated<br />

Lissajous figures that depend on the angular frequencies and the phase factor . When the ratio of<br />

the angular frequencies along the two axes is rational, that is <br />

<br />

is a rational fraction, then the curve will<br />

repeat at regular <strong>in</strong>tervals as shown <strong>in</strong> figure 32 and this shape depends on the phase difference. Otherwise<br />

the trajectory uniformly traverses the whole rectangle.<br />

Figure 3.2: Configuration plots of ( ) where =cos(4) and =cos(5 − ) at four different phase values<br />

. The curves are called Lissajous figures

54 CHAPTER 3. LINEAR OSCILLATORS<br />

3.4.2 State space, ( ˙ )<br />

Visualization of a trajectory is enhanced by correlation of configuration and it’s correspond<strong>in</strong>g velocity<br />

˙ which specifies the direction of the motion. The state space representation 1 is especially valuable when<br />

discuss<strong>in</strong>g Lagrangian mechanics which is based on the Lagrangian (q ˙q).<br />

The free undamped harmonic oscillator provides a simple illustration of state space. Consider a mass <br />

attached to a spr<strong>in</strong>g with l<strong>in</strong>ear spr<strong>in</strong>g constant for which the equation of motion is<br />

− = ¨ = ˙ ˙<br />

(3.17)<br />

<br />

By <strong>in</strong>tegration this gives<br />

1<br />

2 ˙2 + 1 2 2 = (3.18)<br />

The first term <strong>in</strong> equation 318 is the k<strong>in</strong>etic energy, the second term is the potential energy, and is the<br />

total energy which is conserved for this system. This equation can be expressed <strong>in</strong> terms of the state space<br />

coord<strong>in</strong>ates as<br />

˙ 2<br />

¡ 2<br />

<br />

¢ + ¡ 2<br />

2<br />

¢ =1 (3.19)<br />

<br />

This corresponds to the equation of an ellipse for a state-space plot of ˙ versus as shown <strong>in</strong> figure 33.<br />

The elliptical paths shown correspond to contours of constant total energy which is partitioned between<br />

k<strong>in</strong>etic and potential energy. For the coord<strong>in</strong>ate axis shown, the motion of a representative po<strong>in</strong>t will be <strong>in</strong><br />

a clockwise direction as the total oscillator energy is redistributed between potential to k<strong>in</strong>etic energy. The<br />

area of the ellipse is proportional to the total energy .<br />

3.4.3 Phase space, ( )<br />

Phase space, which was <strong>in</strong>troduced by J.W. Gibbs for the field of statistical<br />

mechanics, provides a fundamental graphical representation <strong>in</strong><br />

classical mechanics. The phase space coord<strong>in</strong>ates are the conjugate<br />

coord<strong>in</strong>ates (q p) and are fundamental to Hamiltonian mechanics<br />

which is based on the Hamiltonian (q p). For a conservative system,<br />

only one phase-space curve passes through any po<strong>in</strong>t <strong>in</strong> phase space<br />

like the flow of an <strong>in</strong>compressible fluid. This makes phase space more<br />

useful than state space where many curves pass through any location.<br />

Lanczos [La49] def<strong>in</strong>ed an extended phase space us<strong>in</strong>g four-dimensional<br />

relativistic space-time as discussed <strong>in</strong> chapter 17.<br />

S<strong>in</strong>ce = ˙ for the non-relativistic, one-dimensional, l<strong>in</strong>ear oscillator,<br />

then equation 319 can be rewritten <strong>in</strong> the form<br />

2 <br />

2 + ¡ 2<br />

2<br />

¢ =1 (3.20)<br />

<br />

This is the equation of an ellipse <strong>in</strong> the phase space diagram shown <strong>in</strong><br />

Fig.33- which looks identical to Fig 33- where the ord<strong>in</strong>ate<br />

variable = ˙. That is, the only difference is the phase-space coord<strong>in</strong>ates<br />

( ) replace the state-space coord<strong>in</strong>ates ( ˙). State space<br />

plots are used extensively <strong>in</strong> this chapter to describe oscillatory motion.<br />

Although phase space is more fundamental, both state space and<br />

phase space plots provide useful representations for characteriz<strong>in</strong>g and<br />

elucidat<strong>in</strong>g a wide variety of motion <strong>in</strong> classical mechanics. The follow<strong>in</strong>g<br />

discussion of the undamped simple pendulum illustrates the general<br />

features of state space.<br />

Figure 3.3: State space (upper),<br />

and phase space (lower) diagrams,<br />

for the l<strong>in</strong>ear harmonic oscillator.<br />

1 A universal name for the (q ˙q) representation has not been adopted <strong>in</strong> the literature. Therefore this book has adopted<br />

the name "state space" <strong>in</strong> common with reference [Ta05]. Lanczos [La49] uses the term "state space" to refer to the extended<br />

phase space (q p) discussed <strong>in</strong> chapter 17

3.4. GEOMETRICAL REPRESENTATIONS OF DYNAMICAL MOTION 55<br />

3.4.4 Plane pendulum<br />

Consider a simple plane pendulum of mass attachedtoastr<strong>in</strong>goflength <strong>in</strong> a uniform gravitational field<br />

. There is only one generalized coord<strong>in</strong>ate, S<strong>in</strong>ce the moment of <strong>in</strong>ertia of the simple plane-pendulum is<br />

= 2 then the k<strong>in</strong>etic energy is<br />

= 1 2 2 ˙2<br />

(3.21)<br />

and the potential energy relative to the bottom dead center is<br />

= (1 − cos ) (3.22)<br />

Thus the total energy equals<br />

= 1 2 2 ˙2 + (1 − cos ) = + (1 − cos ) (3.23)<br />

22 where is a constant of motion. Note that the angular momentum is not a constant of motion s<strong>in</strong>ce the<br />

angular acceleration ˙ explicitly depends on .<br />

³<br />

It is <strong>in</strong>terest<strong>in</strong>g to look at the solutions for the equation of motion for a plane pendulum on a ˙<br />

´<br />

state space diagram shown <strong>in</strong> figure 34. The curves shown are equally-spaced contours of constant total<br />

energy. Note that the trajectories are ellipses only at very small angles where 1−cos ≈ 2 , the contours are<br />

non-elliptical for higher amplitude oscillations. When the energy is <strong>in</strong> the range 0 2 the motion<br />

corresponds to oscillations of the pendulum about =0. The center of the ellipse is at (0 0) which is a<br />

stable equilibrium po<strong>in</strong>t for the oscillation. However, when || 2 there is a phase change to rotational<br />

motion about the horizontal axis, that is, the pendulum sw<strong>in</strong>gs around and over top dead center, i.e. it<br />

rotates cont<strong>in</strong>uously <strong>in</strong> one direction about the horizontal axis. The phase change occurs at =2 and<br />

is designated by the separatrix trajectory.<br />

Figure 34 shows two cycles for to better illustrate<br />

the cyclic nature of the phase diagram. The closed loops,<br />

shown as f<strong>in</strong>e solid l<strong>in</strong>es, correspond to pendulum oscillations<br />

about =0or 2 for 2. The dashed<br />

l<strong>in</strong>es show roll<strong>in</strong>g motion for cases where the total energy<br />

2. The broad solid l<strong>in</strong>e is the separatrix<br />

that separates the roll<strong>in</strong>g and oscillatory motion. Note<br />

that at the separatrix, the k<strong>in</strong>etic energy and ˙ are zero<br />

when the pendulum is at top dead center which occurs<br />

when = ±The po<strong>in</strong>t ( 0) is an unstable equilibrium<br />

characterized by phase l<strong>in</strong>es that are hyperbolic<br />

to this unstable equilibrium po<strong>in</strong>t. Note that =+<br />

and − correspond to the same physical po<strong>in</strong>t, that is,<br />

the phase diagram is better presented on a cyl<strong>in</strong>drical<br />

phase space representation s<strong>in</strong>ce is a cyclic variable<br />

that cycles around the cyl<strong>in</strong>der whereas ˙ oscillates<br />

equally about zero hav<strong>in</strong>g both positive and negative values.<br />

The state-space diagram can be wrapped around a lum. The axis is <strong>in</strong> units of radians. Note that<br />

Figure 3.4: State space diagram for a plane pendu-<br />

cyl<strong>in</strong>der, then the unstable and stable equilibrium po<strong>in</strong>ts =+ and − correspond to the same physical<br />

will be at diametrically opposite locations on the surface po<strong>in</strong>t, that is the phase diagram should be rolled<br />

of the cyl<strong>in</strong>der at ˙ =0. For small oscillations about <strong>in</strong>to a cyl<strong>in</strong>der connected at = ±.<br />

equilibrium, also called librations, the correlation between<br />

˙ and is given by the clockwise closed loops wrapped on the cyl<strong>in</strong>drical surface, whereas for energies<br />

|| 2 the positive ˙ corresponds to counterclockwise rotations while the negative ˙ corresponds to<br />

clockwise rotations.<br />

State-space diagrams will be used for describ<strong>in</strong>g oscillatory motion <strong>in</strong> chapters 3 and 4 Phase space is<br />

used <strong>in</strong> statistical mechanics <strong>in</strong> order to handle the equations of motion for ensembles of ∼ 10 23 <strong>in</strong>dependent<br />

particles s<strong>in</strong>ce momentum is more fundamental than velocity. Rather than try to account separately for<br />

the motion of each particle for an ensemble, it is best to specify the region of phase space conta<strong>in</strong><strong>in</strong>g the<br />

ensemble. If the number of particles is conserved, then every po<strong>in</strong>t <strong>in</strong> the <strong>in</strong>itial phase space must transform<br />

to correspond<strong>in</strong>g po<strong>in</strong>ts <strong>in</strong> the f<strong>in</strong>al phase space. This will be discussed <strong>in</strong> chapters 83 and 1527.<br />

2

56 CHAPTER 3. LINEAR OSCILLATORS<br />

3.5 L<strong>in</strong>early-damped free l<strong>in</strong>ear oscillator<br />

3.5.1 General solution<br />

All simple harmonic oscillations are damped to some degree due to energy dissipation via friction, viscous<br />

forces, or electrical resistance etc. The motion of damped systems is not conservative <strong>in</strong> that energy is<br />

dissipated as heat. As was discussed <strong>in</strong> chapter 2 the damp<strong>in</strong>g force can be expressed as<br />

F () =−()bv (3.24)<br />

where the velocity dependent function () can be complicated. Fortunately there is a very large class of<br />

problems <strong>in</strong> electricity and magnetism, classical mechanics, molecular, atomic, and nuclear physics, where<br />

the damp<strong>in</strong>g force depends l<strong>in</strong>early on velocity which greatly simplifies solution of the equations of motion.<br />

This chapter discusses the special case of l<strong>in</strong>ear damp<strong>in</strong>g.<br />

Consider the free simple harmonic oscillator, that is, assum<strong>in</strong>g no oscillatory forc<strong>in</strong>g function, with a<br />

l<strong>in</strong>ear damp<strong>in</strong>g term F () =−v where the parameter is the damp<strong>in</strong>g factor. Then the equation of<br />

motion is<br />

− − ˙ = ¨ (3.25)<br />

This can be rewritten as<br />

¨ + Γ ˙ + 2 0 =0 (3.26)<br />

where the damp<strong>in</strong>g parameter<br />

Γ = <br />

(3.27)<br />

and the characteristic angular frequency<br />

0 =<br />

r<br />

<br />

<br />

(3.28)<br />

The general solution to the l<strong>in</strong>early-damped free oscillator is obta<strong>in</strong>ed by <strong>in</strong>sert<strong>in</strong>g the complex trial<br />

solution = 0 Then<br />

() 2 0 + Γ 0 + 2 0 0 =0 (3.29)<br />

This implies that<br />

2 − Γ − 2 0 =0 (3.30)<br />

The solution is<br />

± = Γ 2 ± s<br />

2 0 − µ Γ<br />

2<br />

2<br />

(3.31)<br />

The two solutions ± are complex conjugates and thus the solutions of the damped free oscillator are<br />

= 1 <br />

Γ 2 + <br />

2 0 −( Γ 2 ) 2 <br />

<br />

+ 2 <br />

Γ 2 − <br />

2 0 −( Γ 2 ) 2 <br />

(3.32)<br />

This can be written as<br />

where<br />

= −( Γ 2 ) £ 1 1 + 2 −1¤ (3.33)<br />

s<br />

µ 2 Γ<br />

1 ≡ 2 −<br />

(3.34)<br />

2

3.5. LINEARLY-DAMPED FREE LINEAR OSCILLATOR 57<br />

Underdamped motion 2 1 ≡ 2 − ¡ Γ<br />

2<br />

¢ 2<br />

0<br />

When 2 1 0 then the square root is real so the solution can be written tak<strong>in</strong>g the real part of which<br />

gives that equation 333 equals<br />

() = −( Γ 2 ) cos ( 1 − ) (3.35)<br />

Where and are adjustable constants fit to the <strong>in</strong>itial conditions. Therefore the velocity is given by<br />

˙() =− − Γ 2<br />

∙ 1 s<strong>in</strong> ( 1 − )+ Γ ¸<br />

2 cos ( 1 − )<br />

(3.36)<br />

This is the damped s<strong>in</strong>usoidal oscillation illustrated <strong>in</strong> figure 35.<br />

characteristics:<br />

a) The oscillation amplitude decreases exponentially with a time constant = 2 Γ <br />

The solution has the follow<strong>in</strong>g<br />

q<br />

b) There is a small reduction <strong>in</strong> the frequency of the oscillation due to the damp<strong>in</strong>g lead<strong>in</strong>g to 1 =<br />

2 − ¡ ¢<br />

Γ 2<br />

2<br />

Figure 3.5: The amplitude-time dependence and state-space diagrams for the free l<strong>in</strong>early-damped harmonic<br />

oscillator. The upper row shows the underdamped system for the case with damp<strong>in</strong>g Γ = 0<br />

5<br />

. The lower<br />

row shows the overdamped ( Γ 2 0) [solid l<strong>in</strong>e] and critically damped ( Γ 2 = 0) [dashed l<strong>in</strong>e] <strong>in</strong> both cases<br />

assum<strong>in</strong>g that <strong>in</strong>itially the system is at rest.

58 CHAPTER 3. LINEAR OSCILLATORS<br />

Figure 3.6: Real and imag<strong>in</strong>ary solutions ± of the damped harmonic oscillator. A phase transition occurs<br />

at Γ =2 0 For Γ 2 0 (dashed) the two solutions are complex conjugates and imag<strong>in</strong>ary. For Γ 2 0 ,<br />

(solid), there are two real solutions + and − with widely different decay constants where + dom<strong>in</strong>ates<br />

the decay at long times.<br />

Overdamped case 2 1 ≡ 2 − ¡ Γ<br />

2<br />

¢ 2<br />

0<br />

¡<br />

In this case the square root of 2 1 is imag<strong>in</strong>ary and can be expressed as 0 1 = q<br />

Γ<br />

¢ 2<br />

2 − <br />

2 Therefore the<br />

solution is obta<strong>in</strong>ed more naturally by us<strong>in</strong>g a real trial solution = 0 <strong>in</strong> equation 333 which leads to<br />

two roots<br />

⎡ s ⎤<br />

µΓ<br />

± = − ⎣− Γ 2<br />

2 ± − <br />

2<br />

2 ⎦<br />

<br />

Thus the exponentially damped decay has two time constants + and − <br />

() = £ 1 − + + 2 − − ¤ (3.37)<br />

1<br />

The time constant<br />

−<br />

1<br />

+<br />

thus the first term 1 − + <strong>in</strong> the bracket decays <strong>in</strong> a shorter time than the<br />

second term 2 −− As illustrated <strong>in</strong> figure 36 the decay rate, which is imag<strong>in</strong>ary when underdamped, i.e.<br />

Γ<br />

2 bifurcates <strong>in</strong>to two real values ± for overdamped, i.e. Γ 2 . At large times the dom<strong>in</strong>ant term<br />

when overdamped is for + which has the smallest decay rate, that is, the longest decay constant + = 1<br />

+<br />

.<br />

There is no oscillatory motion for the overdamped case, it slowly moves monotonically to zero as shown <strong>in</strong><br />

fig 35. The amplitude decays away with a time constant that is longer than 2 Γ <br />

Critically damped 2 1 ≡ 2 − ¡ ¢<br />

Γ 2<br />

2 =0<br />

This is the limit<strong>in</strong>g case where Γ 2 = For this case the solution is of the form<br />

() =( + ) −( Γ 2 )<br />

(3.38)<br />

This motion also is non-s<strong>in</strong>usoidal and evolves monotonically to zero. As shown <strong>in</strong> figure 35 the criticallydamped<br />

solution goes to zero with the shortest time constant, that is, largest . Thus analog electric meters<br />

are built almost critically damped so the needle moves to the new equilibrium value <strong>in</strong> the shortest time<br />

without oscillation.<br />

It is useful to graphically represent the motion of the damped l<strong>in</strong>ear oscillator on either a state space<br />

( ˙ ) diagram or phase space ( ) diagram as discussed <strong>in</strong> chapter 34. The state space plots for the<br />

undamped, overdamped, and critically-damped solutions of the damped harmonic oscillator are shown <strong>in</strong><br />

figure 35 For underdamped motion the state space diagram spirals <strong>in</strong>wards to the orig<strong>in</strong> <strong>in</strong> contrast to<br />

critical or overdamped motion where the state and phase space diagrams move monotonically to zero.

3.5. LINEARLY-DAMPED FREE LINEAR OSCILLATOR 59<br />

3.5.2 Energy dissipation<br />

The <strong>in</strong>stantaneous energy is the sum of the <strong>in</strong>stantaneous k<strong>in</strong>etic and potential energies<br />

where and ˙ are given by the solution of the equation of motion.<br />

Consider the total energy of the underdamped system<br />

= 1 2 ˙2 + 1 2 2 (3.39)<br />

= 1 2 2 + 1 2 2 0 2 (3.40)<br />

where = 2 0 The average total energy is given by substitution for and ˙ and tak<strong>in</strong>g the average over<br />

one cycle. S<strong>in</strong>ce<br />

() = −( Γ 2 ) cos ( 1 − ) (3.41)<br />

Then the velocity is given by<br />

˙() =− − Γ 2<br />

∙ 1 s<strong>in</strong> ( 1 − )+ Γ ¸<br />

2 cos ( 1 − )<br />

(3.42)<br />

Insert<strong>in</strong>g equations 341 and 342 <strong>in</strong>to 340 gives a small amplitude oscillation aboutDan exponential decay for<br />

the energy . Averag<strong>in</strong>g over one cycle and us<strong>in</strong>g the fact that hs<strong>in</strong> cos i =0,and [s<strong>in</strong> ] 2E D<br />

= [cos ] 2E =<br />

1<br />

2<br />

, gives the time-averaged total energy as<br />

Ã<br />

hi = −Γ 1<br />

4 2 2 1 + 1 µ 2 Γ<br />

4 2 + 1 2 4 2 0!<br />

2 (3.43)<br />

which can be written as<br />

hi = 0 −Γ (3.44)<br />

Note that the energy of the l<strong>in</strong>early damped free oscillator decays away with a time constant = 1 Γ That<br />

is, the <strong>in</strong>tensity has a time constant that is half the time constant for the decay of the amplitude of the<br />

transient response. Note that the average k<strong>in</strong>etic and potential energies are identical, as implied by the<br />

Virial theorem, and both decay away with the same time constant. This relation between the mean life <br />

for decay of the damped harmonic oscillator and the damp<strong>in</strong>g width term Γ occurs frequently <strong>in</strong> physics.<br />

The damp<strong>in</strong>g of an oscillator usually is characterized by a s<strong>in</strong>gle parameter called the Quality Factor<br />

where<br />

Energy stored <strong>in</strong> the oscillator<br />

≡ (3.45)<br />

Energy dissipated per radian<br />

The energy loss per radian is given by<br />

∆ = <br />

<br />

1<br />

1<br />

= Γ<br />

1<br />

=<br />

q<br />

Γ<br />

2 − ¡ ¢<br />

(3.46)<br />

Γ 2<br />

2<br />

q<br />

where the numerator 1 = 2 − ¡ ¢<br />

Γ 2<br />

2 is the frequency of the free damped l<strong>in</strong>ear oscillator.<br />

Thus the Quality factor equals<br />

.<br />

= ∆ = 1<br />

(3.47)<br />

Γ<br />

The larger the factor, the less damped is the system, and the<br />

greater is the number of cycles of the oscillation <strong>in</strong> the damped<br />

wave tra<strong>in</strong>. Chapter 3113 shows that the longer the wave tra<strong>in</strong>,<br />

that is the higher is the factor, the narrower is the frequency<br />

distribution around the central value. The Mössbauer effect <strong>in</strong><br />

nuclear physics provides a remarkably long wave tra<strong>in</strong> that can<br />

be used to make high precision measurements. The high- precision<br />

of the LIGO laser <strong>in</strong>terferometer was used <strong>in</strong> the first successful<br />

observation of gravity waves <strong>in</strong> 2015.<br />

Typical Q factors<br />

Earth, for earthquake wave 250-1400<br />

Piano str<strong>in</strong>g 3000<br />

Crystal <strong>in</strong> digital watch 10 4<br />

Microwave cavity 10 4<br />

Excited atom 10 7<br />

Neutron star 10 12<br />

LIGO laser 10 13<br />

Mössbauer effect <strong>in</strong> nucleus 10 14<br />

Table 3.1: Typical Q factors <strong>in</strong> nature.

60 CHAPTER 3. LINEAR OSCILLATORS<br />

3.6 S<strong>in</strong>usoidally-drive, l<strong>in</strong>early-damped, l<strong>in</strong>ear oscillator<br />

The l<strong>in</strong>early-damped l<strong>in</strong>ear oscillator, driven by a harmonic driv<strong>in</strong>g force, is of considerable importance to<br />

all branches of science and eng<strong>in</strong>eer<strong>in</strong>g. The equation of motion can be written as<br />

¨ + Γ ˙ + 2 0 = ()<br />

<br />

(3.48)<br />

where () is the driv<strong>in</strong>g force. For mathematical simplicity the driv<strong>in</strong>g force is chosen to be a s<strong>in</strong>usoidal<br />

harmonic force. The solution of this second-order differential equation comprises two components, the<br />

complementary solution (transient response), and the particular solution (steady-state response).<br />

3.6.1 Transient response of a driven oscillator<br />

The transient response of a driven oscillator is given by the complementary solution of the above second-order<br />

differential equation<br />

¨ + Γ ˙ + 2 0 =0 (3.49)<br />

which is identical to the solution of the free l<strong>in</strong>early-damped harmonic oscillator. As discussed <strong>in</strong> section 35<br />

the solution of the l<strong>in</strong>early-damped free oscillator is given by the real part of the complex variable where<br />

= − Γ 2 £ 1 1 + 2 −1¤ (3.50)<br />

and<br />

s<br />

µ 2 Γ<br />

1 ≡ 2 −<br />

(3.51)<br />

2<br />

Underdamped motion 2 1 ≡ 2 − Γ 2<br />

2<br />

0: When <br />

2<br />

1 0 then the square root is real so the transient<br />

solution can be written tak<strong>in</strong>g the real part of which gives<br />

() = 0<br />

− Γ 2 cos ( 1 ) (3.52)<br />

The solution has the follow<strong>in</strong>g characteristics:<br />

a) The amplitude of the transient solution decreases exponentially with a time constant = 2 Γ while<br />

the energy decreases with a time constant of 1 Γ <br />

b) There is a small downward frequency shift <strong>in</strong> that 1 =<br />

q<br />

2 − ¡ Γ<br />

2<br />

¢ 2<br />

Overdamped case 2 1 ≡ 2 − ¡ ¢<br />

Γ 2<br />

2 0: Inthiscasethesquarerootisimag<strong>in</strong>ary,whichcanbeexpressed<br />

q ¡<br />

as 0 1 ≡ Γ<br />

¢ 2<br />

2 − <br />

2 which is real and the solution is just an exponentially damped one<br />

() = h 0<br />

− Γ 2 0 1 + −0 i 1<br />

(3.53)<br />

There is no oscillatory motion for the overdamped case, it slowly moves monotonically to zero. The total<br />

energy decays away with two time constants greater than 1 Γ <br />

Critically damped 2 1 ≡ 2 − ¡ Γ<br />

2<br />

¢ 2<br />

=0: For this case, as mentioned for the damped free oscillator, the<br />

solution is of the form<br />

() =( + ) − Γ 2 (3.54)<br />

The critically-damped system has the shortest time constant.

3.6. SINUSOIDALLY-DRIVE, LINEARLY-DAMPED, LINEAR OSCILLATOR 61<br />

3.6.2 Steady state response of a driven oscillator<br />

The particular solution of the differential equation gives the important steady state response, () to the<br />

forc<strong>in</strong>g function. Consider that the forc<strong>in</strong>g term is a s<strong>in</strong>gle frequency s<strong>in</strong>usoidal oscillation.<br />

() = 0 cos() (3.55)<br />

Thus the particular solution is the real part of the complex variable which is a solution of<br />

A trial solution is<br />

¨ + Γ ˙ + 2 0 = 0<br />

(3.56)<br />

= 0 (3.57)<br />

This leads to the relation<br />

− 2 0 + Γ 0 + 2 0 0 = 0<br />

<br />

Multiply<strong>in</strong>g the numerator and denom<strong>in</strong>ator by the factor ¡ 2 0 − 2¢ − Γ gives<br />

0<br />

<br />

(3.58)<br />

0<br />

0 =<br />

( 2 0 − 2 )+Γ = £¡<br />

<br />

2<br />

( 2 0 − 2 ) 2 +(Γ) 2 0 − 2¢ − Γ ¤ (3.59)<br />

The steady state solution () thus is given by the real part of , thatis<br />

() <br />

=<br />

0<br />

<br />

( 2 0 − 2 ) 2 +(Γ) 2 £¡<br />

<br />

2<br />

0 − 2¢ cos + Γ s<strong>in</strong> ¤ (3.60)<br />

This can be expressed <strong>in</strong> terms of a phase def<strong>in</strong>ed as<br />

µ Γ<br />

tan ≡<br />

2 0 − 2<br />

(3.61)<br />

As shown <strong>in</strong> figure 37 the hypotenuse of the triangle equals<br />

q<br />

( 2 0 − 2 ) 2 +(Γ) 2 .Thus<br />

cos =<br />

2 0 − 2<br />

q( 2 0 − 2 ) 2 +(Γ) 2 (3.62)<br />

and<br />

s<strong>in</strong> =<br />

Γ<br />

q( 2 0 − 2 ) 2 +(Γ) 2 (3.63)<br />

The phase represents the phase difference between the<br />

driv<strong>in</strong>g force and the resultant motion. For a fixed 0 the<br />

phase = 0 when = 0 and <strong>in</strong>creases to = 2<br />

when<br />

= 0 .For 0 the phase → as →∞.<br />

The steady state solution can be re-expressed <strong>in</strong> terms of<br />

the phase shift as<br />

Figure 3.7: Phase between driv<strong>in</strong>g force and<br />

resultant motion.<br />

() <br />

=<br />

0<br />

<br />

q( 2 0 − 2 ) 2 +(Γ) 2 [cos cos +s<strong>in</strong> s<strong>in</strong> ]<br />

=<br />

0<br />

<br />

q( 2 0 − 2 ) 2 +(Γ) 2 cos ( − ) (3.64)

62 CHAPTER 3. LINEAR OSCILLATORS<br />

Figure 3.8: Amplitude versus time, and state space plots of the transient solution (dashed) and total solution<br />

(solid) for two cases. The upper row shows the case where the driv<strong>in</strong>g frequency = 1<br />

5<br />

while the lower row<br />

shows the same for the case where the driv<strong>in</strong>g frequency =5 1 <br />

3.6.3 Complete solution of the driven oscillator<br />

To summarize, the total solution of the s<strong>in</strong>usoidally forced l<strong>in</strong>early-damped harmonic oscillator is the sum<br />

of the transient and steady-state solutions of the equations of motion.<br />

() = () + () (3.65)<br />

For the underdamped case, the transient solution is the complementary solution<br />

where 1 =<br />

() = 0<br />

− Γ 2 cos ( 1 − ) (3.66)<br />

q<br />

2 − ¡ ¢<br />

Γ 2.<br />

2 The steady-state solution is given by the particular solution<br />

() =<br />

0<br />

<br />

q( 2 0 − 2 ) 2 +(Γ) 2 cos ( − ) (3.67)<br />

Note that the frequency of the transient solution is 1 which <strong>in</strong> general differs from the driv<strong>in</strong>g frequency<br />

. The phase shift − for the transient component is set by the <strong>in</strong>itial conditions. The transient response<br />

leads to a more complicated motion immediately after the driv<strong>in</strong>g function is switched on. Figure 38<br />

illustrates the amplitude time dependence and state space diagram for the transient component, and the<br />

total response, when the driv<strong>in</strong>g frequency is either = 1<br />

5<br />

or =5 1 Note that the modulation of the<br />

steady-state response by the transient response is unimportant once the transient response has damped out<br />

lead<strong>in</strong>g to a constant elliptical state space trajectory. For cases where the <strong>in</strong>itial conditions are = ˙ =0<br />

then the transient solution has a relative phase difference − = radians at =0and relative amplitudes<br />

such that the transient and steady-state solutions cancel at =0<br />

The characteristic sounds of different types of musical <strong>in</strong>struments depend very much on the admixture<br />

of transient solutions plus the number and mixture of oscillatory active modes. Percussive <strong>in</strong>struments, such<br />

as the piano, have a large transient component. The mixture of transient and steady-state solutions for<br />

forced oscillations occurs frequently <strong>in</strong> studies of networks <strong>in</strong> electrical circuit analysis.

3.6. SINUSOIDALLY-DRIVE, LINEARLY-DAMPED, LINEAR OSCILLATOR 63<br />

3.6.4 Resonance<br />

The discussion so far has discussed the role of the transient and steady-state solutions of the driven damped<br />

harmonic oscillator which occurs frequently is science, and eng<strong>in</strong>eer<strong>in</strong>g. Another important aspect is resonance<br />

that occurs when the driv<strong>in</strong>g frequency approaches the natural frequency 1 of the damped system.<br />

Considerthecasewherethetimeissufficient for the transient solution to have decayed to zero.<br />

Figure 39 shows the amplitude and phase for the steadystate<br />

response as goes through a resonance as the driv<strong>in</strong>g<br />

frequency is changed. The steady-states solution of the<br />

driven oscillator follows the driv<strong>in</strong>g force when 0 <strong>in</strong><br />

that the phase difference is zero and the amplitude is just<br />

0<br />

<br />

The response of the system peaks at resonance, while<br />

for 0 the harmonic system is unable to follow the<br />

more rapidly oscillat<strong>in</strong>g driv<strong>in</strong>g force and thus the phase of<br />

the <strong>in</strong>duced oscillation is out of phase with the driv<strong>in</strong>g force<br />

and the amplitude of the oscillation tends to zero.<br />

Note that the resonance frequency for a driven damped<br />

oscillator, differs from that for the undriven damped oscillator,<br />

and differs from that for the undamped oscillator. The<br />

natural frequency for an undamped harmonic oscillator<br />

is given by<br />

2 0 = (3.68)<br />

<br />

The transient solution is the same as damped free oscillations<br />

of a damped oscillator and has a frequency of<br />

the system 1 given by<br />

2 1 = 2 0 − µ Γ<br />

2<br />

2<br />

(3.69)<br />

That is, damp<strong>in</strong>g slightly reduces the frequency.<br />

For the driven oscillator the maximum value of the<br />

steady-state amplitude response is obta<strong>in</strong>ed by tak<strong>in</strong>g the<br />

maximum of the function () , that is when <br />

<br />

=0 This<br />

occurs at the resonance angular frequency where<br />

µ 2 Γ<br />

2 = 2 0 − 2<br />

(3.70)<br />

2<br />

Figure 3.9: Resonance behavior for the<br />

l<strong>in</strong>early-damped, harmonically driven, l<strong>in</strong>ear<br />

oscillator.<br />

No resonance occurs if 2 0−2 ¡ Γ<br />

2<br />

¢ 2<br />

0 s<strong>in</strong>ce then is imag<strong>in</strong>ary and the amplitude decreases monotonically<br />

with <strong>in</strong>creas<strong>in</strong>g Note that the above three frequencies are identical if Γ =0but they differ when Γ 0<br />

and 1 0 <br />

For the driven oscillator it is customary to def<strong>in</strong>e the quality factor as<br />

≡ <br />

(3.71)<br />

Γ<br />

When 1 the system has a narrow high resonance peak. As the damp<strong>in</strong>g <strong>in</strong>creases the quality factor<br />

decreases lead<strong>in</strong>g to a wider and lower peak. The resonance disappears when 1 .<br />

3.6.5 Energy absorption<br />

Discussion of energy stored <strong>in</strong> resonant systems is best described us<strong>in</strong>g the steady state solution which is<br />

dom<strong>in</strong>ant after the transient solution has decayed to zero. Then<br />

() <br />

=<br />

0<br />

<br />

( 2 0 − 2 ) 2 +(Γ) 2 £¡<br />

<br />

2<br />

0 − 2¢ cos + Γ s<strong>in</strong> ¤ (3.72)

64 CHAPTER 3. LINEAR OSCILLATORS<br />

This can be rewritten as<br />

where the elastic amplitude<br />

while the absorptive amplitude<br />

() = cos + s<strong>in</strong> (3.73)<br />

=<br />

0<br />

<br />

( 2 0 − 2 ) 2 +(Γ) 2 ¡<br />

<br />

2<br />

0 − 2¢ (3.74)<br />

=<br />

Figure 310 shows the behavior of the absorptive and<br />

elastic amplitudes as a function of angular frequency .<br />

The absorptive amplitude is significant only near resonance<br />

whereas the elastic amplitude goes to zero at<br />

resonance. Note that the full width at half maximum of<br />

the absorptive amplitude peak equals Γ<br />

The work done by the force 0 cos on the oscillator<br />

is<br />

Z Z<br />

= = ˙ (3.76)<br />

Thus the absorbed power () is given by<br />

0<br />

<br />

( 2 0 − 2 ) 2 Γ (3.75)<br />

2<br />

+(Γ)<br />

() = <br />

= ˙ (3.77)<br />

The steady state response gives a velocity<br />

˙() = − s<strong>in</strong> + cos (3.78)<br />

Thus the steady-state <strong>in</strong>stantaneous power <strong>in</strong>put is<br />

() = 0 cos [− s<strong>in</strong> + cos ] (3.79)<br />

Figure 3.10: Elastic (solid) and absorptive<br />

(dashed) amplitudes of the steady-state solution<br />

for Γ =010 0 <br />

The absorptive term steadily absorbs energy while the elastic term oscillates as energy is alternately absorbed<br />

or emitted. The time average over one cycle is given by<br />

h i = 0<br />

h<br />

− hcos s<strong>in</strong> i + <br />

D(cos ) 2Ei (3.80)<br />

where hcos s<strong>in</strong> i and cos 2® are the time average over one cycle. The time averages over one complete<br />

cycle for the first term <strong>in</strong> the bracket is<br />

− hcos s<strong>in</strong> i =0 (3.81)<br />

while for the second term<br />

<br />

cos <br />

2 ® = 1 <br />

Z 0+<br />

<br />

cos 2 = 1 2<br />

Thus the time average power <strong>in</strong>put is determ<strong>in</strong>ed by only the absorptive term<br />

(3.82)<br />

h i = 1 2 0 = 0<br />

2 Γ 2<br />

2 ( 2 0 − 2 ) 2 +(Γ) 2 (3.83)<br />

This shape of the power curve is a classic Lorentzian shape. Note that the maximum of the average k<strong>in</strong>etic<br />

energy occurs at = 0 which is different from the peak of the amplitude which occurs at 2 1 = 2 0 −¡ ¢<br />

Γ 2.<br />

2<br />

The potential energy is proportional to the amplitude squared, i.e. 2 which occurs at the same angular<br />

frequency as the amplitude, that is, 2 = 2 = 2 0 − 2 ¡ ¢<br />

Γ 2.<br />

2 The k<strong>in</strong>etic and potential energies resonate<br />

at different angular frequencies as a result of the fact that the driven damped oscillator is not conservative

3.6. SINUSOIDALLY-DRIVE, LINEARLY-DAMPED, LINEAR OSCILLATOR 65<br />

because energy is cont<strong>in</strong>ually exchanged between the oscillator and the driv<strong>in</strong>g force system <strong>in</strong> addition to<br />

the energy dissipation due to the damp<strong>in</strong>g.<br />

When ∼ 0 Γ, then the power equation simplifies s<strong>in</strong>ce<br />

¡<br />

<br />

2<br />

0 − 2¢ =( 0 + )( 0 − ) ≈ 2 0 ( 0 − ) (3.84)<br />

Therefore<br />

h i' 0<br />

2<br />

8<br />

Γ<br />

( 0 − ) 2 + ¡ Γ<br />

2<br />

¢ 2<br />

(3.85)<br />

This is called the Lorentzian or Breit-Wigner shape. The half power po<strong>in</strong>ts are at a frequency difference<br />

from resonance of ±∆ where<br />

∆ = | 0 − | = ± Γ (3.86)<br />

2<br />

Thus the full width at half maximum of the Lorentzian curve equals Γ Note that the Lorentzian has a<br />

narrower peak but much wider tail relative to a Gaussian shape. At the peak of the absorbed power, the<br />

absorptive amplitude can be written as<br />

( = 0 )= 0<br />

<br />

<br />

2 0<br />

(3.87)<br />

That is, the peak amplitude <strong>in</strong>creases with <strong>in</strong>crease <strong>in</strong> . This expla<strong>in</strong>s the classic comedy scene where the<br />

soprano shatters the crystal glass because the highest quality crystal glass has a high which leads to a<br />

large amplitude oscillation when she s<strong>in</strong>gs on resonance.<br />

The mean lifetime of the free l<strong>in</strong>early-damped harmonic oscillator, that is, the time for the energy of<br />

free oscillations to decay to 1 was shown to be related to the damp<strong>in</strong>g coefficient Γ by<br />

= 1 Γ<br />

(3.88)<br />

Therefore we have the classical uncerta<strong>in</strong>ty pr<strong>in</strong>ciple for the l<strong>in</strong>early-damped harmonic oscillator<br />

that the measured full-width at half maximum of the energy resonance curve for forced oscillation and the<br />

mean life for decay of the energy of a free l<strong>in</strong>early-damped oscillator are related by<br />

Γ =1 (3.89)<br />

This relation is correct only for a l<strong>in</strong>early-damped harmonic system. Comparable relations between the<br />

lifetime and damp<strong>in</strong>g width exist for different forms of damp<strong>in</strong>g.<br />

One can demonstrate the above l<strong>in</strong>e width and decay time relationship us<strong>in</strong>g an acoustically driven<br />

electric guitar str<strong>in</strong>g. Similarily, the width of the electromagnetic radiation is related to the lifetime for<br />

decay of atomic or nuclear electromagnetic decay. This classical uncerta<strong>in</strong>ty pr<strong>in</strong>ciple is exactly the same<br />

as the one encountered <strong>in</strong> quantum physics due to wave-particle duality. In nuclear physics it is difficult to<br />

measure the lifetime of states when 10 −13 For shorter lifetimes the value of Γ can be determ<strong>in</strong>ed from<br />

the shape of the resonance curve which can be measured directly when the damp<strong>in</strong>g is large.<br />

3.1 Example: Harmonically-driven series RLC circuit<br />

The harmonically-driven, resonant, series circuit, is encountered frequently<br />

<strong>in</strong> AC circuits. Kirchhoff’s Rules applied to the series circuit<br />

lead to the differential equation<br />

¨ + ˙ + = 0 s<strong>in</strong> <br />

where is charge, L is the <strong>in</strong>ductance, is the capacitance, is the resistance,<br />

and the applied voltage across the circuit is () = 0 s<strong>in</strong> . The l<strong>in</strong>earity of<br />

the network allows use of the phasor approach which assumes that the current<br />

= 0 the voltage = 0 (+) and the impedance is a complex number

66 CHAPTER 3. LINEAR OSCILLATORS<br />

= 0<br />

0<br />

where is the phase difference between the voltage and the current. For this circuit the impedance<br />

is given by<br />

µ<br />

= + − 1 <br />

<br />

Because of the phases <strong>in</strong>volved <strong>in</strong> this circuit, at resonance the maximum voltage across the resistor<br />

occurs at a frequency of = 0 across the capacitor the maximum voltage occurs at a frequency 2 =<br />

2 0 − 2<br />

2<br />

and across the <strong>in</strong>ductor the maximum voltage occurs at a frequency 2 2 = 2 0<br />

where 2 1− 2<br />

0 = 1<br />

<br />

2<br />

is the resonance angular frequency when =0. Thus these resonance frequencies differ when 2 0.<br />

3.7 Wave equation<br />

Wave motion is a ubiquitous feature <strong>in</strong> nature. Mechanical wave motion is manifest by transverse waves<br />

on fluid surfaces, longitud<strong>in</strong>al and transverse seismic waves travell<strong>in</strong>g through the Earth, and vibrations of<br />

mechanical structures such as suspended cables. Acoustical wave motion occurs on the stretched str<strong>in</strong>gs of<br />

the viol<strong>in</strong>, as well as the cavities of w<strong>in</strong>d <strong>in</strong>struments. Wave motion occurs for deformable bodies where<br />

elastic forces act<strong>in</strong>g between the nearest-neighbor atoms of the body exert time-dependent forces on one<br />

another. Electromagnetic wave motion <strong>in</strong>cludes wavelengths rang<strong>in</strong>g from 10 5 radiowaves, to 10 −13 rays.<br />

Matter waves are a prom<strong>in</strong>ent feature of quantum physics. All these manifestations of waves exhibit<br />

the same general features of wave motion. Chapter 14 will <strong>in</strong>troduce the collective modes of motion, called<br />

the normal modes, of coupled, many-body, l<strong>in</strong>ear oscillators which act as <strong>in</strong>dependent modes of motion.<br />

The basic elements of wavemotion are <strong>in</strong>troduced at this juncture because the equations of wave motion are<br />

simple, and wave motion features prom<strong>in</strong>ently <strong>in</strong> several chapters throughout this book.<br />

Consider a travell<strong>in</strong>g wave <strong>in</strong> one dimension for a l<strong>in</strong>ear system. If the wave is mov<strong>in</strong>g, then the wave<br />

function Ψ ( ) describ<strong>in</strong>g the shape of the wave, is a function of both and . The <strong>in</strong>stantaneous amplitude<br />

of the wave Ψ ( ) could correspond to the transverse displacement of a wave on a str<strong>in</strong>g, the longitud<strong>in</strong>al<br />

amplitude of a wave on a spr<strong>in</strong>g, the pressure of a longitud<strong>in</strong>al sound wave, the transverse electric or magnetic<br />

fields <strong>in</strong> an electromagnetic wave, a matter wave, etc. If the wave tra<strong>in</strong> ma<strong>in</strong>ta<strong>in</strong>s its shape as it moves, then<br />

one can describe the wave tra<strong>in</strong> by the function () where the coord<strong>in</strong>ate is measured relative to the<br />

shape of the wave, that is, it could correspond to the phase of a crest of the wave. Consider that ( =0)<br />

corresponds to a constant phase, e.g. the peak of the travell<strong>in</strong>g pulse, then assum<strong>in</strong>g that the wave travels<br />

at a phase velocity <strong>in</strong> the direction and the peak is at =0for =0 then it is at = at time .<br />

That is, a po<strong>in</strong>t with phase fixedwithrespecttothewaveformshapeofthewaveprofile () moves <strong>in</strong><br />

the + direction for = − and <strong>in</strong> − direction for = + .<br />

General wave motion can be described by solutions of a wave equation. The wave equation can be<br />

written <strong>in</strong> terms of the spatial and temporal derivatives of the wave function Ψ() Consider the first partial<br />

derivatives of Ψ() =( ∓ ) =()<br />

and<br />

Ψ<br />

= Ψ <br />

= Ψ<br />

<br />

Ψ<br />

= Ψ <br />

<br />

Factor<strong>in</strong>g out Ψ<br />

for the first derivatives gives Ψ Ψ<br />

= ∓<br />

<br />

= ∓<br />

Ψ<br />

<br />

(3.90)<br />

(3.91)<br />

(3.92)<br />

The sign <strong>in</strong> this equation depends on the sign of the wave velocity mak<strong>in</strong>g it not a generally useful formula.<br />

Consider the second derivatives<br />

and<br />

2 Ψ<br />

2<br />

2 Ψ<br />

2<br />

= 2 Ψ<br />

2 <br />

= 2 Ψ<br />

2 (3.93)<br />

=<br />

2 Ψ<br />

2 <br />

=+2 2 Ψ<br />

2 (3.94)

3.8. TRAVELLING AND STANDING WAVE SOLUTIONS OF THE WAVE EQUATION 67<br />

Factor<strong>in</strong>g out 2 Ψ<br />

gives<br />

2<br />

2 Ψ<br />

2 = 1 2 Ψ<br />

2 2 (3.95)<br />

This wave equation <strong>in</strong> one dimension for a l<strong>in</strong>ear system is <strong>in</strong>dependent of the sign of the velocity. There<br />

are an <strong>in</strong>f<strong>in</strong>ite number of possible shapes of waves both travell<strong>in</strong>g and stand<strong>in</strong>g <strong>in</strong> one dimension, all of these<br />

must satisfy this one-dimensional wave equation. The converse is that any function that satisfies this one<br />

dimensional wave equation must be a wave <strong>in</strong> this one dimension.<br />

The Wave Equation <strong>in</strong> three dimensions is<br />

∇ 2 Ψ ≡ 2 Ψ<br />

2 + 2 Ψ<br />

2 + 2 Ψ<br />

2 = 1 2 Ψ<br />

2 2 (3.96)<br />

There are an unlimited number of possible solutions Ψ to this wave equation, any one of which corresponds<br />

toawavemotionwithvelocity.<br />

The Wave Equation is applicable to all manifestations of wave motion, both transverse and longitud<strong>in</strong>al,<br />

for l<strong>in</strong>ear systems. That is, it applies to waves on a str<strong>in</strong>g, water waves, seismic waves, sound waves,<br />

electromagnetic waves, matter waves, etc. If it can be shown that a wave equation can be derived for any<br />

system, discrete or cont<strong>in</strong>uous, then this is equivalent to prov<strong>in</strong>g the existence of waves of any waveform,<br />

frequency, or wavelength travell<strong>in</strong>g with the phase velocity given by the wave equation.[Cra65]<br />

3.8 Travell<strong>in</strong>g and stand<strong>in</strong>g wave solutions of the wave equation<br />

The wave equation can exhibit both travell<strong>in</strong>g and stand<strong>in</strong>g-wave solutions. Consider a one-dimensional<br />

travell<strong>in</strong>g wave with velocity hav<strong>in</strong>g a specific wavenumber ≡ 2 . Then the travell<strong>in</strong>g wave is best<br />

written <strong>in</strong> terms of the phase of the wave as<br />

Ψ( ) =() 2 (∓) = () (∓) (3.97)<br />

where the wave number ≡ 2 <br />

with be<strong>in</strong>g the wave length, and angular frequency ≡ . Thisparticular<br />

solution satisfies the wave equation and corresponds to a travell<strong>in</strong>g wave with phase velocity = <br />

<br />

<strong>in</strong> the<br />

positive or negative direction depend<strong>in</strong>g on whether the sign is negative or positive. Assum<strong>in</strong>g that the<br />

superposition pr<strong>in</strong>ciple applies, then the superposition of these two particular solutions of the wave equation<br />

can be written as<br />

Ψ( ) =()( (−) + (+) )=() ( − + )=2() cos (3.98)<br />

Thus the superposition of two identical s<strong>in</strong>gle wavelength travell<strong>in</strong>g waves propagat<strong>in</strong>g <strong>in</strong> opposite directions<br />

cancorrespondtoastand<strong>in</strong>g wave solution. Note that a stand<strong>in</strong>g wave is identical to a stationary normal<br />

modeofthesystemdiscussed<strong>in</strong>chapter14. This transformation between stand<strong>in</strong>g and travell<strong>in</strong>g waves can<br />

be reversed, that is, the superposition of two stand<strong>in</strong>g waves, i.e. normal modes, can lead to a travell<strong>in</strong>g<br />

wave solution of the wave equation.<br />

Discussion of waveforms is simplified when us<strong>in</strong>g either of the follow<strong>in</strong>g two limits.<br />

1)Thetimedependenceofthewaveformatagivenlocation = 0 which can be expressed us<strong>in</strong>g a<br />

Fourier decomposition, appendix 2, of the time dependence as a function of angular frequency = 0 .<br />

∞X<br />

∞X<br />

Ψ( 0 )= ( 0 0 − 0 ) = ( 0 ) − 0<br />

(3.99)<br />

=−∞<br />

=−∞<br />

2) The spatial dependence of the waveform at a given <strong>in</strong>stant = 0 which can be expressed us<strong>in</strong>g a<br />

Fourier decomposition of the spatial dependence as a function of wavenumber = 0<br />

∞X<br />

∞X<br />

Ψ( 0 )= (0−10) = ( 0 ) 0 (3.100)<br />

=−∞<br />

=−∞<br />

The above is applicable both to discrete, or cont<strong>in</strong>uous l<strong>in</strong>ear oscillator systems, e.g. waves on a str<strong>in</strong>g.<br />

In summary, stationary normal modes of a system are obta<strong>in</strong>ed by a superposition of travell<strong>in</strong>g waves<br />

travell<strong>in</strong>g <strong>in</strong> opposite directions, or equivalently, travell<strong>in</strong>g waves can result from a superposition of stationary<br />

normal modes.

68 CHAPTER 3. LINEAR OSCILLATORS<br />

3.9 Waveform analysis<br />

3.9.1 Harmonic decomposition<br />

As described <strong>in</strong> appendix , when superposition applies, then a<br />

Fourier series decomposition of the form 3101 canbemadeof<br />

any periodic function where<br />

() =<br />

X<br />

cos( 0 + ) (3.101)<br />

=1<br />

A more general Fourier Transform can be made for an aperiodic<br />

function where<br />

Z<br />

() = ()cos( + ()) (3.102)<br />

Any l<strong>in</strong>ear system that is subject to the forc<strong>in</strong>g function ()<br />

has an output that can be expressed as a l<strong>in</strong>ear superposition<br />

of the solutions of the <strong>in</strong>dividual harmonic components of the<br />

forc<strong>in</strong>g function. Fourier analysis of periodic waveforms <strong>in</strong> terms<br />

of harmonic trigonometric functions plays a key role <strong>in</strong> describ<strong>in</strong>g<br />

oscillatory motion <strong>in</strong> classical mechanics and signal process<strong>in</strong>g<br />

for l<strong>in</strong>ear systems. Fourier’s theorem states that any arbitrary<br />

forc<strong>in</strong>g function () canbedecomposed<strong>in</strong>toasumofharmonic<br />

Figure 3.11: The time and frequency representations<br />

of a system exhibit<strong>in</strong>g beats.<br />

terms. As a consequence two equivalent representations can be used to describe signals and waves; the first<br />

is <strong>in</strong> the time doma<strong>in</strong> which describes the time dependence of the signal. The second is <strong>in</strong> the frequency<br />

doma<strong>in</strong> which describes the frequency decomposition of the signal. Fourier analysis relates these equivalent<br />

representations.<br />

For example, the superposition of two equal <strong>in</strong>tensity harmonic<br />

oscillators <strong>in</strong> the time doma<strong>in</strong> is given by<br />

() = cos ( 1 )+ cos ( 2 )<br />

∙µ ¸ ∙µ ¸<br />

1 + 2 1 − 2<br />

= 2 cos<br />

cos<br />

(3.103)<br />

2<br />

2<br />

which leads to the phenomenon of beats as illustrated for both<br />

the time doma<strong>in</strong> and frequency doma<strong>in</strong> <strong>in</strong> figure 311<br />

3.9.2 The free l<strong>in</strong>early-damped l<strong>in</strong>ear oscillator<br />

The response of the free, l<strong>in</strong>early-damped, l<strong>in</strong>ear oscillator is one<br />

of the most frequently encountered waveforms <strong>in</strong> science and thus<br />

it is useful to <strong>in</strong>vestigate the Fourier transform of this waveform.<br />

The waveform amplitude for the underdamped case, shown <strong>in</strong><br />

figure 35 is given by equation (335), thatis<br />

() = − Γ 2 cos ( 1 − ) ≥ 0 (3.104)<br />

() = 0 0 (3.105)<br />

where 2 1 = 2 0 − ¡ Γ<br />

2<br />

¢ 2<br />

and where 0 is the angular frequency of<br />

the undamped system. The Fourier transform is given by<br />

0<br />

£¡<br />

() =<br />

2<br />

( 2 − 2 1 )2 +(Γ) 2 − 2 ¢ ¤<br />

1 − Γ<br />

which is complex and has the famous Lorentz form.<br />

(3.106)<br />

Figure 3.12: The <strong>in</strong>tensity () 2 and<br />

Fourier transform |()| 2 of the free<br />

l<strong>in</strong>early-underdamped harmonic oscillator<br />

with 0 =10and damp<strong>in</strong>g Γ =1.

3.9. WAVEFORM ANALYSIS 69<br />

The<strong>in</strong>tensityofthewavegives<br />

| ()| 2 = 2 −Γ cos 2 ( 1 − ) (3.107)<br />

| ()| 2 =<br />

2 0<br />

( 2 − 2 1 )2 +(Γ) 2 (3.108)<br />

Note that s<strong>in</strong>ce the average over 2 of cos 2 = 1 2 then the average over the cos2 ( 1 − ) term gives the<br />

<strong>in</strong>tensity () = 2<br />

2 −Γ which has a mean lifetime for the decay of = 1 Γ The | ()|2 distribution has the<br />

classic Lorentzian shape, shown <strong>in</strong> figure 312, which has a full width at half-maximum, FWHM, equal to Γ.<br />

Note that () is complex and thus one also can determ<strong>in</strong>e the phase shift which is given by the ratio of<br />

the imag<strong>in</strong>ary to real parts of equation 3105 i.e. tan =<br />

Γ<br />

( 2 − 2 1) .<br />

The mean lifetime of the exponential decay of the <strong>in</strong>tensity can be determ<strong>in</strong>ed either by measur<strong>in</strong>g <br />

from the time dependence, or measur<strong>in</strong>g the FWHM Γ = 1 <br />

of the Fourier transform | ()|2 . In nuclear<br />

and atomic physics excited levels decay by photon emission with the wave form of the free l<strong>in</strong>early-damped,<br />

l<strong>in</strong>ear oscillator. Typically the mean lifetime usually can be measured when & 10 −12 whereas for<br />

shorter lifetimes the radiation width Γ becomes sufficiently large to be measured. Thus the two experimental<br />

approaches are complementary.<br />

3.9.3 Damped l<strong>in</strong>ear oscillator subject to an arbitrary periodic force<br />

Fourier’s theorem states that any arbitrary forc<strong>in</strong>g function () can be decomposed <strong>in</strong>to a sum of harmonic<br />

terms. Consider the response of a damped l<strong>in</strong>ear oscillator to an arbitrary periodic force.<br />

() =<br />

X<br />

0 ( )cos( + ) (3.109)<br />

=0<br />

For each harmonic term the response of a l<strong>in</strong>early-damped l<strong>in</strong>ear oscillator to the forc<strong>in</strong>g function<br />

() = 0 ()cos( ) is given by equation (365 − 67) to be<br />

() = () + () <br />

⎡<br />

⎤<br />

= 0 ( )<br />

⎣ − Γ 2 1<br />

cos ( 1 − )+<br />

cos ( − ) ⎦ (3.110)<br />

<br />

q( 2 0 − 2 ) 2 +(Γ ) 2<br />

The amplitude is obta<strong>in</strong>ed by substitut<strong>in</strong>g <strong>in</strong>to (3110) the derived values 0()<br />

<br />

3.2 Example: Vibration isolation<br />

from the Fourier analysis.<br />

Frequently it is desired to isolate <strong>in</strong>strumentation from the<br />

<strong>in</strong>fluence of horizontal and vertical external vibrations that exist<br />

<strong>in</strong> the environment. One arrangement to achieve this isolation<br />

is to mount a heavy base of mass on weak spr<strong>in</strong>gs of spr<strong>in</strong>g<br />

constant plus weak damp<strong>in</strong>g. The response of this system is<br />

given by equation 3109 which exhibits a resonance at the angular<br />

frequency 2 = 2 0 − 2 ¡ ¢<br />

Γ 2<br />

2 associated with each resonant<br />

frequency 0 of the system. For each resonant frequency the system<br />

amplifies the vibrational amplitude for angular frequencies<br />

close to resonance that is, below √ 2 0 while it attenuates the Seismic isolation of an optical bench.<br />

vibration roughly by a factor of ¡ 0<br />

¢ 2<br />

<br />

at higher frequencies. To<br />

avoid the amplification near the resonance it is necessary to make 0 very much smaller than the frequency<br />

range of the vibrational spectrum and have a moderately high value. This is achieved by use a very heavy<br />

base and weak spr<strong>in</strong>g constant so that 0 is very small. A typical table may have the resonance frequency<br />

at 05 which is well below typical perturb<strong>in</strong>g vibrational frequencies, and thus the table attenuates the<br />

vibration by 99% at 5 and even more attenuation for higher frequency perturbations. This pr<strong>in</strong>ciple is<br />

used extensively <strong>in</strong> design of vibration-isolation tables for optics or microbalance equipment.

70 CHAPTER 3. LINEAR OSCILLATORS<br />

3.10 Signal process<strong>in</strong>g<br />

It has been shown that the response of the l<strong>in</strong>early-damped l<strong>in</strong>ear oscillator, subject to any arbitrary periodic<br />

force, can be calculated us<strong>in</strong>g a frequency decomposition, (Fourier analysis), of the force, appendix . The<br />

response also can be calculated us<strong>in</strong>g a time-ordered discrete-time sampl<strong>in</strong>g of the pulse shape; that is, the<br />

Green’s function approach, appendix . The l<strong>in</strong>early-damped, l<strong>in</strong>ear oscillator is the simplest example of<br />

a l<strong>in</strong>ear system that exhibits both resonance and frequency-dependent response. Typically physical l<strong>in</strong>ear<br />

systems exhibit far more complicated response functions hav<strong>in</strong>g multiple resonances. For example, an automobile<br />

suspension system <strong>in</strong>volves four wheels and associated spr<strong>in</strong>gs plus dampers allow<strong>in</strong>g the car to<br />

rock sideways, or forward and backward, <strong>in</strong> addition to the up-down motion, when subject to the forces<br />

produced by a rough road. Similarly a suspension bridge or aircraft w<strong>in</strong>g can twist as well as bend due to<br />

air turbulence, or a build<strong>in</strong>g can undergo complicated oscillations due to seismic waves. An acoustic system<br />

exhibits similar complexity. Signal analysis and signal process<strong>in</strong>g is of pivotal importance to elucidat<strong>in</strong>g the<br />

response of complicated l<strong>in</strong>ear systems to complicated periodic forc<strong>in</strong>g functions. Signal process<strong>in</strong>g is used<br />

extensively <strong>in</strong> eng<strong>in</strong>eer<strong>in</strong>g, acoustics, and science.<br />

Theresponseofalow-passfilter, such as an R-C circuit or a coaxial cable, to a <strong>in</strong>put square wave,<br />

shown <strong>in</strong> figure 313, provides a simple example of the relative advantages of us<strong>in</strong>g the complementary<br />

Fourier analysis <strong>in</strong> the frequency doma<strong>in</strong>, or the Green’s discrete-function analysis <strong>in</strong> the time doma<strong>in</strong>. The<br />

response of a repetitive square-wave <strong>in</strong>put signal is shown <strong>in</strong> the time doma<strong>in</strong> plus the Fourier transform to<br />

the frequency doma<strong>in</strong>. The middle curves show the time dependence for the response of the low-pass filter<br />

to an impulse () and the correspond<strong>in</strong>g Fourier transform (). The output of the low-pass filter can<br />

be calculated by fold<strong>in</strong>g the <strong>in</strong>put square wave and impulse time dependence <strong>in</strong> the time doma<strong>in</strong> as shown<br />

on the left or by fold<strong>in</strong>g of their Fourier transforms shown on the right. Work<strong>in</strong>g <strong>in</strong> the frequency doma<strong>in</strong><br />

the response of l<strong>in</strong>ear mechanical systems, such as an automobile suspension or a musical <strong>in</strong>strument, as<br />

well as l<strong>in</strong>ear electronic signal process<strong>in</strong>g systems such as amplifiers, loudspeakers and microphones, can<br />

be treated as black boxes hav<strong>in</strong>g a certa<strong>in</strong> transfer function () describ<strong>in</strong>g the ga<strong>in</strong> and phase shift<br />

versus frequency. That is, the output wave frequency decomposition is<br />

() = ( ) · () (3.111)<br />

Work<strong>in</strong>g <strong>in</strong> the time doma<strong>in</strong>, the the low-pass system has an impulse response () = − ,whichisthe<br />

Fourier transform of the transfer function ( ). In the time doma<strong>in</strong><br />

() =<br />

Z ∞<br />

−∞<br />

() · ( − ) (3.112)<br />

This is shown schematically <strong>in</strong> figure 313. The Fourier transformation connects the three quantities <strong>in</strong> the<br />

time doma<strong>in</strong> with the correspond<strong>in</strong>g three <strong>in</strong> the frequency doma<strong>in</strong>. For example, the impulse response of<br />

the low-pass filter has a fall time of which is related by a Fourier transform to the width of the transfer<br />

function. Thus the time and frequency doma<strong>in</strong> approaches are closely related and give the same result for<br />

the output signal for the low-pass filter to the applied square-wave <strong>in</strong>put signal. The result is that the<br />

higher-frequency components are attenuated lead<strong>in</strong>g to slow rise and fall times <strong>in</strong> the time doma<strong>in</strong>.<br />

Analog signal process<strong>in</strong>g and Fourier analysis were the primary tools to analyze and process all forms of<br />

periodic motion dur<strong>in</strong>g the 20 century. For example, musical <strong>in</strong>struments, mechanical systems, electronic<br />

circuits, all employed resonant systems to enhance the desired frequencies and suppress the undesirable<br />

frequencies and the signals could be observed us<strong>in</strong>g analog oscilloscopes. The remarkable development of<br />

comput<strong>in</strong>g has enabled use of digital signal process<strong>in</strong>g lead<strong>in</strong>g to a revolution <strong>in</strong> signal process<strong>in</strong>g that has<br />

had a profound impact on both science and eng<strong>in</strong>eer<strong>in</strong>g. The digital oscilloscope, which can sample at frequencies<br />

above 10 9 has replaced the analog oscilloscope because it allows sophisticated analysis of each<br />

<strong>in</strong>dividual signal that was not possible us<strong>in</strong>g analog signal process<strong>in</strong>g. For example, the analog approach <strong>in</strong><br />

nuclear physics used t<strong>in</strong>y analog electric signals, produced by many <strong>in</strong>dividual radiation detectors, that were<br />

transmitted hundreds of meters via carefully shielded and expensive coaxial cables to the data room where<br />

the signals were amplified and signal processed us<strong>in</strong>g analog filters to maximize the signal to noise <strong>in</strong> order to<br />

separate the signal from the background noise. Stray electromagnetic radiation picked up via the cables significantly<br />

degraded the signals. The performance and limitations of the analog electronics severely restricted<br />

the pulse process<strong>in</strong>g capabilities. Digital signal process<strong>in</strong>g has rapidly replaced analog signal process<strong>in</strong>g.

3.11. WAVE PROPAGATION 71<br />

Figure 3.13: Response of an electrical circuit to an <strong>in</strong>put square wave. The upper row shows the time<br />

and the exponential-form frequency representations of the square-wave <strong>in</strong>put signal. The middle row gives<br />

the impulse response, and correspond<strong>in</strong>g transfer function for the circuit. The bottom row shows the<br />

correspond<strong>in</strong>g output properties <strong>in</strong> both the time and frequency doma<strong>in</strong>s<br />

Analog to digital detector circuits are built directly <strong>in</strong>to the electronics for each <strong>in</strong>dividual detector so that<br />

only digital <strong>in</strong>formation needs to be transmitted from each detector to the analysis computers. Computer<br />

process<strong>in</strong>g provides unlimited and flexible process<strong>in</strong>g capabilities for the digital signals greatly enhanc<strong>in</strong>g<br />

the response and sensitivity of our detector systems. Digital CD and DVD disks are common application of<br />

digital signal process<strong>in</strong>g.<br />

3.11 Wave propagation<br />

Wave motion typically <strong>in</strong>volves a packet of waves encompass<strong>in</strong>g a f<strong>in</strong>ite number of wave cycles. Information<br />

<strong>in</strong> a wave only can be transmitted by start<strong>in</strong>g, stopp<strong>in</strong>g, or modulat<strong>in</strong>g the amplitude of a wave tra<strong>in</strong>, which<br />

is equivalent to form<strong>in</strong>g a wave packet. For example, a musician will play a note for a f<strong>in</strong>ite time, and this<br />

wave tra<strong>in</strong> propagates out as a wave packet of f<strong>in</strong>ite length. You have no <strong>in</strong>formation as to the frequency<br />

and amplitude of the sound prior to the wave packet reach<strong>in</strong>g you, or after the wave packet has passed you.<br />

The velocity of the wavelets conta<strong>in</strong>ed with<strong>in</strong> the wave packet is called the phase velocity. Foradispersive<br />

system the phase velocity of the wavelets conta<strong>in</strong>ed with<strong>in</strong> the wave packet is frequency dependent and the<br />

shape of the wave packet travels at the group velocity which usually differs from the phase velocity. If<br />

the shape of the wave packet is time dependent, then neither the phase velocity, which is the velocity of the<br />

wavelets, nor the group velocity, which is the velocity of an <strong>in</strong>stantaneous po<strong>in</strong>t fixed to the shape of the<br />

wave packet envelope, represent the actual velocity of the overall wavepacket.<br />

A third wavepacket velocity, the signal velocity, isdef<strong>in</strong>ed to be the velocity of the lead<strong>in</strong>g edge of the<br />

energy distribution, and correspond<strong>in</strong>g <strong>in</strong>formation content, of the wave packet. For most l<strong>in</strong>ear systems<br />

the shape of the wave packet is not time dependent and then the group and signal velocities are identical.<br />

However, the group and signal velocities can be very different for non-l<strong>in</strong>ear systems as discussed <strong>in</strong> chapter<br />

47. Note that even when the phase velocity of the waves with<strong>in</strong> the wave packet travels faster than the group<br />

velocity of the shape, or the signal velocity of the energy content of the envelope of the wave packet, the<br />

<strong>in</strong>formation conta<strong>in</strong>ed <strong>in</strong> a wave packet is only manifest when the wave packet envelope reaches the detector<br />

and this energy and <strong>in</strong>formation travel at the signal velocity. The modern ideas of wave propagation,<br />

<strong>in</strong>clud<strong>in</strong>g Hamilton’s concept of group velocity, were developed by Lord Rayleigh when applied to the theory<br />

of sound[Ray1887]. The concept of phase, group, and signal velocities played a major role <strong>in</strong> discussion of<br />

electromagnetic waves as well as de Broglie’s development of wave-particle duality <strong>in</strong> quantum mechanics.

72 CHAPTER 3. LINEAR OSCILLATORS<br />

3.11.1 Phase, group, and signal velocities of wave packets<br />

The concepts of wave packets, as well as their phase, group, and signal velocities, are of considerable importance<br />

for propagation of <strong>in</strong>formation and other manifestations of wave motion <strong>in</strong> science and eng<strong>in</strong>eer<strong>in</strong>g.<br />

This importance warrants further discussion at this juncture.<br />

Consider a particular component of a one-dimensional wave,<br />

The argument of the exponential is called the phase of the wave where<br />

( ) = (±) (3.113)<br />

≡ − (3.114)<br />

If we move along the axis at a velocity such that the phase is constant then we perceive a stationary<br />

pattern <strong>in</strong> this mov<strong>in</strong>g frame. The velocity of this wave is called the phase velocity. To ensure constant<br />

phase requires that is constant, or assum<strong>in</strong>g real and <br />

Therefore the phase velocity is def<strong>in</strong>ed to be<br />

= (3.115)<br />

= <br />

(3.116)<br />

The velocity discussed so far is just the phase velocity of the <strong>in</strong>dividual wavelets at the carrier frequency. If<br />

or are complex then one must take the real parts to ensure that the velocity is real.<br />

If the phase velocity of a wave is dependent on the wavelength, that is, () then the system is<br />

said to be dispersive <strong>in</strong> that the wave is dispersed accord<strong>in</strong>g the wavelength. The simplest illustration of<br />

dispersion is the refraction of light <strong>in</strong> glass prism which leads to dispersion of the light <strong>in</strong>to the spectrum of<br />

wavelengths. Dispersion leads to development of wave packets that travel at group and signal velocities that<br />

usually differ from the phase velocity. To illustrate this behavior, consider two equal amplitude travell<strong>in</strong>g<br />

waves hav<strong>in</strong>g slightly different wave number and angular frequency . Superposition of these waves gives<br />

( ) = ( [−] + [(+∆)−(+∆)] ) (3.117)<br />

=<br />

∆<br />

∆<br />

[(+<br />

2 )−(+ 2 )] ∆ ∆<br />

−[ ·{ 2 − 2 ] ∆ ∆<br />

[<br />

+ 2 − 2 ] }<br />

=<br />

∆<br />

∆<br />

[(+<br />

2 2 )−(+ 2 )] cos[ ∆<br />

2 − ∆<br />

2 ]<br />

This corresponds to a wave with the average carrier frequency modulated by the cos<strong>in</strong>e term which has a<br />

wavenumber of ∆<br />

2<br />

and angular frequency ∆<br />

2<br />

, that is, this is the usual example of beats The cos<strong>in</strong>e term<br />

modulates the average wave produc<strong>in</strong>g wave packets as shown <strong>in</strong> figure 311. The velocity of these wave<br />

packetsiscalledthegroup velocity given by requir<strong>in</strong>g that the phase of the modulat<strong>in</strong>g term is constant,<br />

that is<br />

∆<br />

2<br />

∆<br />

= (3.118)<br />

2<br />

Thus the group velocity is given by<br />

= <br />

= ∆<br />

(3.119)<br />

∆<br />

If dispersion is present then the group velocity = ∆<br />

∆ does not equal the phase velocity = <br />

Expand<strong>in</strong>g the above example to superposition of waves gives<br />

( ) =<br />

X<br />

(±) (3.120)<br />

=1<br />

In the event that →∞and the frequencies are cont<strong>in</strong>uously distributed, then the summation is replaced<br />

by an <strong>in</strong>tegral<br />

( ) =<br />

Z ∞<br />

−∞<br />

() (±) (3.121)

3.11. WAVE PROPAGATION 73<br />

where the factor () represents the distribution amplitudes of the component waves, that is the spectral<br />

decomposition of the wave. This is the usual Fourier decomposition of the spatial distribution of the wave.<br />

Consider an extension of the l<strong>in</strong>ear superposition of two waves to a well def<strong>in</strong>ed wave packet where the<br />

amplitude is nonzero only for a small range of wavenumbers 0 ± ∆<br />

( ) =<br />

Z 0 +∆<br />

0−∆<br />

() (−) (3.122)<br />

This functional shape is called a wave packet which only has mean<strong>in</strong>g if ∆ 0 . The angular frequency<br />

can be expressed by mak<strong>in</strong>g a Taylor expansion around 0<br />

µ <br />

() =( 0 )+ ( − 0 )+ (3.123)<br />

<br />

0<br />

For a l<strong>in</strong>ear system the phase then reduces to<br />

µ <br />

− =( 0 − 0 )+( − 0 ) − ( − 0 ) (3.124)<br />

<br />

0<br />

The summation of terms <strong>in</strong> the exponent given by 3124 leads to the amplitude 3122 hav<strong>in</strong>gtheformofa<br />

product where the <strong>in</strong>tegral becomes<br />

( ) = (0−0) Z 0 +∆<br />

0−∆<br />

() (− 0)[−( <br />

) 0<br />

]<br />

(3.125)<br />

The <strong>in</strong>tegral term modulates the (0−0) first term.<br />

The group velocity is def<strong>in</strong>ed to be that for which the phase of the exponential term <strong>in</strong> the <strong>in</strong>tegral is<br />

constant. Thus<br />

µ <br />

=<br />

(3.126)<br />

<br />

0<br />

S<strong>in</strong>ce = then<br />

= + <br />

(3.127)<br />

<br />

For non-dispersive systems the phase velocity is <strong>in</strong>dependent of the wave number or angular frequency <br />

and thus = The case discussed earlier, equation (3103) for beat<strong>in</strong>g of two waves gives the<br />

same relation <strong>in</strong> the limit that ∆ and ∆ are <strong>in</strong>f<strong>in</strong>itessimal.<br />

¡ The group velocity of a wave packet is of physical significance for dispersive media where =<br />

<br />

¢<br />

0<br />

6= = . Every wave tra<strong>in</strong> has a f<strong>in</strong>ite extent and thus we usually observe the motion of a<br />

group of waves rather than the wavelets mov<strong>in</strong>g with<strong>in</strong> the wave packet. In general, for non-l<strong>in</strong>ear dispersive<br />

systems the derivative <br />

<br />

can be either positive or negative and thus <strong>in</strong> pr<strong>in</strong>ciple the group velocity<br />

can either be greater than, or less than, the phase velocity. Moreover, if the group velocity is frequency<br />

dependent, that is, when group velocity dispersion occurs, then the overall shape of the wave packet is time<br />

dependent and thus the speed of a specific relativelocationdef<strong>in</strong>ed by the shape of the envelope of the wave<br />

packet does not represent the signal velocity of the wave packet. Brillou<strong>in</strong> showed that the distribution<br />

of the energy, and correspond<strong>in</strong>g <strong>in</strong>formation content, for any wave packet, travels at the signal velocity<br />

which can be different from the group velocity if the shape of the envelope of the wave packet is time<br />

dependent. For electromagnetic waves one has the possibility that the group velocity = In<br />

1914 Brillou<strong>in</strong>[Bri14][Bri60] showed that the signal velocity of electromagnetic waves, def<strong>in</strong>ed by the lead<strong>in</strong>g<br />

edge of the time-dependent envelope of the wave packet, never exceeds even though the group velocity<br />

correspond<strong>in</strong>g to the velocity of the <strong>in</strong>stantaneous shape of the wave packet may exceed . Thus, there is<br />

no violation of E<strong>in</strong>ste<strong>in</strong>’s fundamental pr<strong>in</strong>ciple of relativity that the velocity of an electromagnetic wave<br />

cannot exceed .

74 CHAPTER 3. LINEAR OSCILLATORS<br />

3.3 Example: Water waves break<strong>in</strong>g on a beach<br />

The concepts of phase and group velocity are illustrated by the example of water waves mov<strong>in</strong>g at velocity<br />

<strong>in</strong>cident upon a straight beach at an angle to the shorel<strong>in</strong>e. Consider that the wavepacket comprises<br />

many wavelengths of wavelength . Dur<strong>in</strong>g the time it takes the wave to travel a distance the po<strong>in</strong>t where<br />

the crest of one wave breaks on the beach travels a distance<br />

crest of the one wavelet <strong>in</strong> the wave packet is<br />

=<br />

The velocity of the wave packet along the beach equals<br />

<br />

cos <br />

= cos <br />

<br />

cos <br />

along beach. Thus the phase velocity of the<br />

Note that for the wave mov<strong>in</strong>g parallel to the beach =0and = = . However, for = 2<br />

→∞and → 0. In general for waves break<strong>in</strong>g on the beach<br />

= 2<br />

The same behavior is exhibited by surface waves bounc<strong>in</strong>g off the sides of the Erie canal, sound waves <strong>in</strong><br />

a trombone, and electromagnetic waves transmitted down a rectangular wave guide. In the latter case the<br />

phase velocity exceeds the velocity of light <strong>in</strong> apparent violation of E<strong>in</strong>ste<strong>in</strong>’s theory of relativity. However,<br />

the <strong>in</strong>formation travels at the signal velocity which is less than .<br />

3.4 Example: Surface waves for deep water<br />

In the “Theory of Sound”[Ray1887] Rayleigh discusses the example of surface waves for water. He derives<br />

a dispersion relation for the phase velocity and wavenumber whicharerelatedtothedensity, depth<br />

, gravity , and surface tension , by<br />

2 = + 3<br />

<br />

tanh()<br />

For deep water where the wavelength is short compared with the depth, that is kl 1 then tanh() → 1<br />

and the dispersion relation is given approximately by<br />

2 = + 3<br />

<br />

For long surface waves for deep water, that is, small , then the gravitational first term <strong>in</strong> the dispersion<br />

relation dom<strong>in</strong>ates and the group velocity is given by<br />

µ <br />

= = 1 2<br />

r <br />

= 1 <br />

2 = <br />

2<br />

That is, the group velocity is half of the phase velocity. Here the wavelets are build<strong>in</strong>g at the back of the wave<br />

packet, progress through the wave packet and dissipate at the front. This can be demonstrated by dropp<strong>in</strong>g a<br />

pebble <strong>in</strong>to a calm lake. It will be seen that the surface disturbance comprises a wave packet mov<strong>in</strong>g outwards<br />

at the group velocity with the <strong>in</strong>dividual waves with<strong>in</strong> the wave packet expand<strong>in</strong>g at twice the group velocity<br />

of the wavepacket, that is, they are created at the <strong>in</strong>ner radius of the wave packet and disappear at the outer<br />

radius of the wave packet.<br />

For small wavelength ripples, where is large, then the surface tension term dom<strong>in</strong>ates and the dispersion<br />

relation is approximately given by<br />

2 ' 3<br />

<br />

lead<strong>in</strong>g to a group velocity of<br />

µ <br />

= = 3 2 <br />

Here the group velocity exceeds the phase velocity and wavelets are build<strong>in</strong>g at the front of the wave packet and<br />

dissipate at the back. Note that for this l<strong>in</strong>ear system, the Brillion signal velocity equals the group velocity<br />

for both gravity and surface tension waves for deep water.

3.11. WAVE PROPAGATION 75<br />

3.5 Example: Electromagnetic waves <strong>in</strong> ionosphere<br />

The response to radio waves, <strong>in</strong>cident upon a free electron plasma <strong>in</strong> the ionosphere, provides an excellent<br />

example that <strong>in</strong>volves cut-off frequency, complex wavenumber as well as the phase, group, and signal<br />

velocities. Maxwell’s equations give the most general wave equation for electromagnetic waves to be<br />

∇ 2 E − 2 E<br />

2 = j ³<br />

<br />

´<br />

+ ∇·<br />

<br />

<br />

∇ 2 H − 2 H<br />

2 = −∇ × j <br />

where and j are the unbound charge and current densities. The effect of the bound charges and<br />

currents are absorbed <strong>in</strong>to and . Ohm’s Law can be written <strong>in</strong> terms of the electrical conductivity which<br />

is a constant<br />

j =E<br />

Assum<strong>in</strong>g Ohm’s Law plus assum<strong>in</strong>g =0, <strong>in</strong> the plasma gives the relations<br />

∇ 2 E − 2 E<br />

2<br />

∇ 2 H − 2 H<br />

2<br />

− E <br />

− H <br />

The third term <strong>in</strong> both of these wave equations is a damp<strong>in</strong>g term that leads to a damped solution of an<br />

electromagnetic wave <strong>in</strong> a good conductor.<br />

The solution of these damped wave equations can be solved by consider<strong>in</strong>g an <strong>in</strong>cident wave<br />

E = ˆx (−)<br />

Substitut<strong>in</strong>g for E <strong>in</strong> the first damped wave equation gives<br />

That is<br />

− 2 + 2 − =0<br />

∙<br />

2 = 2 1 − ¸<br />

<br />

In general is complex, that is, it has real and imag<strong>in</strong>ary parts that lead to a solution of the form<br />

E = − (−)<br />

The first exponential term is an exponential damp<strong>in</strong>g term while the second exponential term is the oscillat<strong>in</strong>g<br />

term.<br />

Consider that the plasma <strong>in</strong>volves the motion of a bound damped electron, of charge of mass bound<br />

<strong>in</strong> a one dimensional atom or lattice subject to an oscillatory electric field of frequency . Assume that the<br />

electromagnetic wave is travell<strong>in</strong>g <strong>in</strong> the ˆ direction with the transverse electric field <strong>in</strong> the ˆ direction. The<br />

equation of motion of an electron can be written as<br />

= 0<br />

= 0<br />

ẍ + Γẋ + 2 0 = ˆx 0 (−)<br />

where Γ is the damp<strong>in</strong>g factor. The <strong>in</strong>stantaneous displacement of the oscillat<strong>in</strong>g charge equals<br />

x = 1<br />

( 2 0 − 2 )+Γ ˆx 0 (−)<br />

and the velocity is<br />

Thus the <strong>in</strong>stantaneous current density is given by<br />

ẋ = <br />

( 2 0 − 2 )+Γ ˆx 0 (−)<br />

j = ẋ = 2<br />

<br />

<br />

( 2 0 − 2 )+Γ ˆx 0 (−)

76 CHAPTER 3. LINEAR OSCILLATORS<br />

Therefore the electrical conductivity is given by<br />

= 2<br />

<br />

<br />

( 2 0 − 2 )+Γ<br />

Let us consider only unbound charges <strong>in</strong> the plasma, that is let 0 =0. Then the conductivity is given by<br />

= 2<br />

<br />

<br />

Γ − 2<br />

For a low density ionized plasma Γ thus the conductivity is given approximately by<br />

≈− 2<br />

<br />

<br />

S<strong>in</strong>ce is pure imag<strong>in</strong>ary, then j and E have a phase difference of<br />

2<br />

which implies that the average of<br />

the Joule heat<strong>in</strong>g over a complete period is hj · Ei =0 Thus there is no energy loss due to Joule heat<strong>in</strong>g<br />

imply<strong>in</strong>g that the electromagnetic energy is conserved.<br />

Substitution of <strong>in</strong>to the relation for 2<br />

∙<br />

2 = 2 1 − ¸<br />

¸<br />

= 2 <br />

∙1 − 2<br />

<br />

2<br />

Def<strong>in</strong>e the Plasma oscillation frequency to be<br />

r<br />

<br />

2<br />

≡<br />

<br />

then 2 can be written as<br />

∙ ³<br />

2 = 2 <br />

´2¸<br />

1 −<br />

()<br />

<br />

For a low density plasma the dielectric constant ' 1 and the relative permeability ' 1 and thus<br />

= 0 ' 0 and = 0 ' 0 . The velocity of light <strong>in</strong> vacuum = √ 1<br />

0 <br />

. Thus for low density<br />

0<br />

equation can be written as<br />

2 = 2 + 2 2<br />

()<br />

Differentiation of equation with respect to gives 2 <br />

=22 That is, = 2 and the phase<br />

velocity is<br />

r<br />

= 2 + 2 <br />

2<br />

Therearethreecasestoconsider.<br />

1) : For this case<br />

h1 − ¡ <br />

¢ i 2<br />

1 and thus is a pure real number. Therefore the electromagnetic<br />

wave is transmitted with a phase velocity that exceeds while the group velocity is less than<br />

.<br />

2) : For this case<br />

h1 − ¡ <br />

¢ i 2<br />

1 and thus is a pure imag<strong>in</strong>ary number. Therefore the<br />

electromagnetic wave is not transmitted <strong>in</strong> the ionosphere and is attenuated rapidly as −( <br />

) . However,<br />

s<strong>in</strong>ce there are no Joule heat<strong>in</strong>g losses, then the electromagnetic wave must be complete reflected. Thus the<br />

Plasma oscillation frequency serves as a cut-off frequency. For this example the signal and group velocities<br />

are identical.<br />

For the ionosphere = 10 −11 electrons/m 3 ,whichcorrespondstoaPlasmaoscillationfrequencyof<br />

= 2 =3. Thus electromagnetic waves <strong>in</strong> the AM waveband ( 16) are totally reflected by<br />

the ionosphere and bounce repeatedly around the Earth, whereas for VHF frequencies above 3,thewaves<br />

are transmitted and refracted pass<strong>in</strong>g through the atmosphere. Thus light is transmitted by the ionosphere.<br />

By contrast, for a good conductor like silver, the Plasma oscillation frequency is around 10 16 which is<br />

<strong>in</strong> the far ultraviolet part of the spectrum. Thus, all lower frequencies, such as light, are totally reflected<br />

by such a good conductor, whereas X-rays have frequencies above the Plasma oscillation frequency and are<br />

transmitted.

3.11. WAVE PROPAGATION 77<br />

3.11.2 Fourier transform of wave packets<br />

The relation between the time distribution and the correspond<strong>in</strong>g<br />

frequency distribution, or equivalently, the<br />

spatial distribution and the correspond<strong>in</strong>g wave-number<br />

distribution, are of considerable importance <strong>in</strong> discussion<br />

of wave packets and signal process<strong>in</strong>g. It directly<br />

relates to the uncerta<strong>in</strong>ty pr<strong>in</strong>ciple that is a characteristic<br />

of all forms of wave motion. The relation between the<br />

time and correspond<strong>in</strong>g frequency distribution is given<br />

via the Fourier transform discussed <strong>in</strong> appendix . The<br />

follow<strong>in</strong>g are two examples of the Fourier transforms of<br />

typical but rather differentwavepacketshapesthatare<br />

encountered often <strong>in</strong> science and eng<strong>in</strong>eer<strong>in</strong>g.<br />

3.6 Example: Fourier transform of a<br />

Gaussian wave packet:<br />

Assum<strong>in</strong>g that the amplitude of the wave is a<br />

Gaussian wave packet shown <strong>in</strong> the adjacent figure where<br />

() = − (− 0 )2<br />

2 2 <br />

This leads to the Fourier transform<br />

() = √ 2 − 2 2<br />

2 cos ( 0 )<br />

Fourier transform of a Gaussian frequency<br />

distribution.<br />

Note that the wavepacket has a standard deviation for the amplitude of the wavepacket of = 1<br />

<br />

,that<br />

is · =1. The Gaussian wavepacket results <strong>in</strong> the m<strong>in</strong>imum product of the standard deviations of the<br />

frequency and time representations for a wavepacket. This has profound importance for all wave phenomena,<br />

and especially to quantum mechanics. Because matter exhibits wave-like behavior, the above property of wave<br />

packet leads to Heisenberg’s Uncerta<strong>in</strong>ty Pr<strong>in</strong>ciple. For signal process<strong>in</strong>g, it shows that if you truncate a<br />

wavepacket you will broaden the frequency distribution.<br />

3.7 Example: Fourier transform of a rectangular wave packet:<br />

Assume unity amplitude of the frequency distribution between 0 − ∆ ≤ ≤ 0 + ∆ ,thatis,as<strong>in</strong>gle<br />

isolated square pulse of width that is described by the rectangular function Π def<strong>in</strong>ed as<br />

½ 1<br />

Π() =<br />

0<br />

| − 0 | ∆<br />

| − 0 | ∆<br />

Then the Fourier transform us given by<br />

() =<br />

∙ ¸ s<strong>in</strong> ∆<br />

cos 0 <br />

∆<br />

That is, the transform of a rectangular wavepacket gives a cos<strong>in</strong>e wave modulated by an unnormalized<br />

function which is a nice example of a simple wave packet. That is, on the right hand side we have<br />

a wavepacket ∆ = ± 2<br />

∆<br />

wide. Note that the product of the two measures of the widths ∆ · ∆ = ±<br />

Example 2 considers a rectangular<br />

³ ´<br />

pulse of unity amplitude between − 2 ≤ ≤ 2 whichresulted<strong>in</strong>a<br />

s<strong>in</strong> <br />

Fourier transform () = 2<br />

. That is, for a pulse of width ∆ = ± <br />

2<br />

2<br />

the frequency envelope has<br />

the first zero at ∆ = ± <br />

. Note that this is the complementary system to the one considered here which has<br />

∆ · ∆ = ± illustrat<strong>in</strong>g the symmetry of the Fourier transform and its <strong>in</strong>verse.

78 CHAPTER 3. LINEAR OSCILLATORS<br />

3.11.3 Wave-packet Uncerta<strong>in</strong>ty Pr<strong>in</strong>ciple<br />

The Uncerta<strong>in</strong>ty Pr<strong>in</strong>ciple states that wavemotion exhibits a m<strong>in</strong>imum product of the uncerta<strong>in</strong>ty <strong>in</strong> the<br />

simultaneously measured width <strong>in</strong> time of a wave packet, and the distribution width of the frequency decomposition<br />

of this wave packet. This was illustrated by the Fourier transforms of wave packets discussed<br />

above where it was shown the product of the widths is m<strong>in</strong>imized for a Gaussian-shaped wave packet. The<br />

Uncerta<strong>in</strong>ty Pr<strong>in</strong>ciple implies that to make a precise measurement of the frequency of a s<strong>in</strong>usoidal wave<br />

requiresthatthewavepacketbe<strong>in</strong>f<strong>in</strong>itely long. If the duration of the wave packet is reduced then the<br />

frequency distribution broadens. The crucial aspect qneeded for this discussion, is that, for the amplitudes<br />

of any wavepacket, the standard deviations () = h 2 i − hi 2 characteriz<strong>in</strong>g the width of the spectral<br />

distribution <strong>in</strong> the angular frequency doma<strong>in</strong>, (), and the width for the conjugate variable <strong>in</strong> time ()<br />

are related :<br />

() · () > 1<br />

(Relation between amplitude uncerta<strong>in</strong>ties.)<br />

This product of the standard deviations equals unity only for the special case of Gaussian-shaped spectral<br />

distributions, and it is greater than unity for all other shaped spectral distributions.<br />

The <strong>in</strong>tensity of the wave is the square of the amplitude lead<strong>in</strong>g to standard deviation widths for a<br />

Gaussian distribution where () 2 = 1 2 () 2 ,thatis, () = () √<br />

2<br />

. Thus the standard deviations for the<br />

spectral distribution and width of the <strong>in</strong>tensity of the wavepacket are related by:<br />

() · () > 1 2<br />

(Uncerta<strong>in</strong>ty pr<strong>in</strong>ciple for frequency-time <strong>in</strong>tensities)<br />

This states that the uncerta<strong>in</strong>ties with which you can simultaneously measure the time and frequency<br />

for the <strong>in</strong>tensity of a given wavepacket are related. If you try to measure the frequency with<strong>in</strong> a short time<br />

<strong>in</strong>terval () then the uncerta<strong>in</strong>ty <strong>in</strong> the frequency measurement () > 1<br />

2 ()<br />

Accurate measurement<br />

of the frequency requires measurement times that encompass many cycles of oscillation, that is, a long<br />

wavepacket.<br />

Exactly the same relations exist between the spectral distribution as a function of wavenumber and<br />

the correspond<strong>in</strong>g spatial dependence of a wave which are conjugate representations. Thus the spectral<br />

distribution plotted versus is directly related to the amplitude as a function of position ; the spectral<br />

distribution versus is related to the amplitude as a function of ; and the spectral distribution is related<br />

to the spatial dependence on Follow<strong>in</strong>g the same arguments discussed above, the standard deviation,<br />

( ) characteriz<strong>in</strong>g the width of the spectral <strong>in</strong>tensity distribution of , and the standard deviation<br />

() characteriz<strong>in</strong>g the spatial width of the wave packet <strong>in</strong>tensity as a function of are related by the<br />

Uncerta<strong>in</strong>ty Pr<strong>in</strong>ciple for position-wavenumber. Thus <strong>in</strong> summary the temporal and spatial uncerta<strong>in</strong>ty<br />

pr<strong>in</strong>ciples of the <strong>in</strong>tensity of wave motion is,<br />

() · () > 1 2<br />

(3.128)<br />

() · ( ) > 1 2<br />

() · ( ) > 1 2<br />

() · ( ) > 1 2<br />

This applies to all forms of wave motion, be they, sound waves, water waves, electromagnetic waves, or<br />

matter waves.<br />

As discussed <strong>in</strong> chapter 18, the transition to quantum mechanics <strong>in</strong>volves relat<strong>in</strong>g the matter-wave properties<br />

to the energy and momentum of the correspond<strong>in</strong>g particle. That is, <strong>in</strong> the case of matter waves,<br />

multiply<strong>in</strong>g both sides of equation 3129 by ~ and us<strong>in</strong>g the de Broglie relations gives that the particle energy<br />

is related to the angular frequency by = ~ and the particle momentum is related to the wavenumber,<br />

that is −→ p = ~ −→ k . These lead to the Heisenberg Uncerta<strong>in</strong>ty Pr<strong>in</strong>ciple:<br />

() · () > ~ 2<br />

(3.129)<br />

() · ( ) > ~ 2<br />

() · ( ) > ~ 2<br />

() · ( ) > ~ 2<br />

This uncerta<strong>in</strong>ty pr<strong>in</strong>ciple applies equally to the wavefunction of the electron <strong>in</strong> the<br />

hydrogen atom, proton <strong>in</strong> a nucleus, as well as to a wavepacket describ<strong>in</strong>g a particle wave mov<strong>in</strong>g along some

3.11. WAVE PROPAGATION 79<br />

trajectory. This implies that, for a particle of given momentum, the wavefunction is spread out spatially.<br />

Planck’s constant ~ =105410 −34 · =658210 −16 · is extremely small compared with energies and<br />

times encountered <strong>in</strong> normal life, and thus the effects due to the Uncerta<strong>in</strong>ty Pr<strong>in</strong>ciple are not important for<br />

macroscopic dimensions.<br />

Conf<strong>in</strong>ement of a particle, of mass , with<strong>in</strong>±() of a fixed location implies that there is a correspond<strong>in</strong>g<br />

uncerta<strong>in</strong>ty <strong>in</strong> the momentum<br />

( ) ≥ ~<br />

2()<br />

Now the variance <strong>in</strong> momentum p is given by the difference <strong>in</strong> the average of the square<br />

square of the average of hpi 2 .Thatis<br />

(p) 2 =<br />

(3.130)<br />

D<br />

(p · p) 2E ,andthe<br />

D<br />

(p · p) 2E − hpi 2 (3.131)<br />

Assum<strong>in</strong>g a fixed average location implies that hpi =0,then<br />

D<br />

(p · p) 2E µ 2 ~<br />

= () 2 ≥<br />

(3.132)<br />

2()<br />

S<strong>in</strong>ce the k<strong>in</strong>etic energy is given by:<br />

K<strong>in</strong>etic energy = 2<br />

2 ≥ ~ 2<br />

8() 2<br />

(Zero-po<strong>in</strong>t energy)<br />

This zero-po<strong>in</strong>t energy is the m<strong>in</strong>imum k<strong>in</strong>etic energy that a particle of mass can have if conf<strong>in</strong>ed with<strong>in</strong> a<br />

distance ±() This zero-po<strong>in</strong>t energy is a consequence of wave-particle duality and the uncerta<strong>in</strong>ty between<br />

the size and wavenumber for any wave packet. It is a quantal effect <strong>in</strong> that the classical limit has ~ → 0 for<br />

which the zero-po<strong>in</strong>t energy → 0<br />

Insert<strong>in</strong>g numbers for the zero-po<strong>in</strong>t energy gives that an electron conf<strong>in</strong>ed to the radius of the atom,<br />

that is () =10 −10 has a zero-po<strong>in</strong>t k<strong>in</strong>etic energy of ∼ 1 .Conf<strong>in</strong><strong>in</strong>g this electron to 3 × 10 −15 the<br />

size of a nucleus, gives a zero-po<strong>in</strong>t energy of 10 9 (1 ) Conf<strong>in</strong><strong>in</strong>g a proton to the size of the nucleus<br />

gives a zero-po<strong>in</strong>t energy of 05. These values are typical of the level spac<strong>in</strong>g observed <strong>in</strong> atomic and<br />

nuclear physics. If ~ was a large number, then a billiard ball conf<strong>in</strong>ed to a billiard table would be a blur<br />

as it oscillated with the m<strong>in</strong>imum zero-po<strong>in</strong>t k<strong>in</strong>etic energy. The smaller the spatial region that the ball<br />

was conf<strong>in</strong>ed, the larger would be its zero-po<strong>in</strong>t energy and momentum caus<strong>in</strong>g it to rattle back and forth<br />

between the boundaries of the conf<strong>in</strong>ed region. Life would be dramatically different if ~ was a large number.<br />

In summary, Heisenberg’s Uncerta<strong>in</strong>ty Pr<strong>in</strong>ciple is a well-known and crucially important aspect of quantum<br />

physics. What is less well known, is that the Uncerta<strong>in</strong>ty Pr<strong>in</strong>ciple applies for all forms of wave motion,<br />

that is, it is not restricted to matter waves. The follow<strong>in</strong>g three examples illustrate application of the<br />

Uncerta<strong>in</strong>ty Pr<strong>in</strong>ciple to acoustics, the nuclear Mössbauer effect, and quantum mechanics.<br />

3.8 Example: Acoustic wave packet<br />

A viol<strong>in</strong>ist plays the note middle C (261625) with constant <strong>in</strong>tensity for precisely 2 seconds. Us<strong>in</strong>g<br />

the fact that the velocity of sound <strong>in</strong> air is 3432 calculate the follow<strong>in</strong>g:<br />

1) The wavelength of the sound wave <strong>in</strong> air: = 3432261625 = 1312.<br />

2) The length of the wavepacket <strong>in</strong> air: Wavepacket length = 3432 × 2 = 6864<br />

3) The fractional frequency width of the note: S<strong>in</strong>ce the wave packet has a square pulse shape of length<br />

=2, then the Fourier transform is a s<strong>in</strong>c function hav<strong>in</strong>g the first zeros when s<strong>in</strong> <br />

2 =0,thatis,∆ = 1 .<br />

Therefore the fractional width is ∆<br />

= 1<br />

=00019. Note that to achieve a purity of ∆<br />

<br />

=10−6 the viol<strong>in</strong>ist<br />

would have to play the note for 106.<br />

3.9 Example: Gravitational red shift<br />

The Mössbauer effect <strong>in</strong> nuclear physics provides a wave packet that has an exceptionally small fractional<br />

width <strong>in</strong> frequency. For example, the 57 Fe nucleus emits a 144 deexcitation-energy photon which corresponds<br />

to ≈ 2 × 10 25 withadecaytimeof ≈ 10 −7 . Thus the fractional width is ∆<br />

≈ 3 × 10−18 .

80 CHAPTER 3. LINEAR OSCILLATORS<br />

In 1959 Pound and Rebka used this to test E<strong>in</strong>ste<strong>in</strong>’s general theory of relativity by measurement of the<br />

gravitational red shift between the attic and basement of the 225 high physics build<strong>in</strong>g at Harvard. The<br />

magnitude of the predicted relativistic red shift is ∆<br />

<br />

=25 × 15 10− which is what was observed with a<br />

fractional precision of about 1%.<br />

3.10 Example: Quantum baseball<br />

George Gamow, <strong>in</strong> his book ”Mr. Tompk<strong>in</strong>s <strong>in</strong> Wonderland”, describes the strange world that would exist<br />

if ~ was a large number. As an example, consider you play baseball <strong>in</strong> a universe where ~ is a large number.<br />

The pitcher throws a 150 ball 20 to the batter at a speed of 40. For a strike to be thrown, the ball’s<br />

position must be pitched with<strong>in</strong> the 30 radius of the strike zone, that is, it is required that ∆ ≤ 03.<br />

The uncerta<strong>in</strong>ty relation tells us that the transverse velocity of the ball cannot be less than ∆ =<br />

<br />

2∆ The<br />

time of flight of the ball from the mound to batter is =05. Because of the transverse velocity uncerta<strong>in</strong>ty,<br />

∆ theballwilldeviate∆ transversely from the strike zone. This also must not exceed the size of the<br />

strike zone, that is;<br />

∆ = ~<br />

2∆ ≤ 03<br />

(Due to transverse velocity uncerta<strong>in</strong>ty)<br />

Comb<strong>in</strong><strong>in</strong>g both of these requirements gives<br />

~ ≤ 2∆2<br />

<br />

=54 10 −2 · <br />

This is 32 orders of magnitude larger than ~ so quantal effects are negligible. However, if ~ exceeded the<br />

above value, then the pitcher would have difficulty throw<strong>in</strong>g a reliable strike.<br />

3.12 Summary<br />

L<strong>in</strong>ear systems have the feature that the solutions obey the Pr<strong>in</strong>ciple of Superposition, thatis,theamplitudes<br />

add l<strong>in</strong>early for the superposition of different oscillatory modes. Applicability of the Pr<strong>in</strong>ciple of<br />

Superposition to a system provides a tremendous advantage for handl<strong>in</strong>g and solv<strong>in</strong>g the equations of motion<br />

of oscillatory systems.<br />

Geometric representations of the motion of dynamical systems provide sensitive probes of periodic motion.<br />

Configuration space (q q), statespace(q ˙q) and phase space (q p), are powerful geometric<br />

representations that are used extensively for recogniz<strong>in</strong>g periodic motion where q ˙q and p are vectors <strong>in</strong><br />

-dimensional space.<br />

L<strong>in</strong>early-damped free l<strong>in</strong>ear oscillator<br />

the equation<br />

The free l<strong>in</strong>early-damped l<strong>in</strong>ear oscillator is characterized by<br />

¨ + Γ ˙ + 2 0 =0<br />

(326)<br />

The solutions of the l<strong>in</strong>early-damped free l<strong>in</strong>ear oscillator are of the form<br />

s<br />

= −( Γ 2 ) £ 1 1 + 2 − 1 ¤<br />

µ 2 Γ<br />

1 ≡ 2 − (333)<br />

2<br />

The solutions of the l<strong>in</strong>early-damped free l<strong>in</strong>ear oscillator have the follow<strong>in</strong>g characteristic frequencies correspond<strong>in</strong>g<br />

to the three levels of l<strong>in</strong>ear damp<strong>in</strong>g<br />

() = −( q<br />

Γ<br />

2 ) cos ( 1 − ) underdamped 1 = 2 − ¡ ¢<br />

Γ 2<br />

2 0<br />

∙ q ¡<br />

() =[ 1 − + + 2 −− ] overdamped ± = − − Γ 2 ± Γ<br />

¢ 2<br />

2 − <br />

2 ¸<br />

() =( + ) −( q<br />

Γ<br />

2 )<br />

critically damped 1 = 2 − ¡ ¢<br />

Γ 2<br />

2 =0

3.12. SUMMARY 81<br />

The energy dissipation for the l<strong>in</strong>early-damped free l<strong>in</strong>ear oscillator time averaged over one period is<br />

given by<br />

hi = 0 −Γ<br />

(344)<br />

The quality factor characteriz<strong>in</strong>g the damp<strong>in</strong>g of the free oscillator is def<strong>in</strong>ed to be<br />

= ∆ = 1<br />

Γ<br />

where ∆ is the energy dissipated per radian.<br />

(347)<br />

S<strong>in</strong>usoidally-driven, l<strong>in</strong>early-damped, l<strong>in</strong>ear oscillator The l<strong>in</strong>early-damped l<strong>in</strong>ear oscillator, driven<br />

by a harmonic driv<strong>in</strong>g force, is of considerable importance to all branches of physics, and eng<strong>in</strong>eer<strong>in</strong>g. The<br />

equation of motion can be written as<br />

¨ + Γ ˙ + 2 0 = ()<br />

(349)<br />

<br />

where () is the driv<strong>in</strong>g force. The complete solution of this second-order differential equation comprises<br />

two components, the complementary solution (transient response), and the particular solution (steady-state<br />

response). Thatis,<br />

() = () + () <br />

(365)<br />

For the underdamped case, the transient solution is the complementary solution<br />

() = 0<br />

− Γ 2 cos ( 1 − )<br />

and the steady-state solution is given by the particular solution<br />

(366)<br />

() =<br />

0<br />

<br />

q( 2 0 − 2 ) 2 +(Γ) 2 cos ( − ) (367)<br />

Resonance A detailed discussion of resonance and energy absorption for the driven l<strong>in</strong>early-damped l<strong>in</strong>ear<br />

oscillator was given. For resonance of the l<strong>in</strong>early-damped l<strong>in</strong>ear oscillator the maximum amplitudes occur<br />

at the follow<strong>in</strong>g resonant frequencies<br />

Resonant system<br />

Resonant<br />

q<br />

frequency<br />

<br />

undamped free l<strong>in</strong>ear oscillator 0 =<br />

<br />

q<br />

l<strong>in</strong>early-damped free l<strong>in</strong>ear oscillator 1 = 2 0 − ¡ ¢<br />

Γ 2<br />

q 2<br />

driven l<strong>in</strong>early-damped l<strong>in</strong>ear oscillator = 2 0 − 2 ¡ ¢<br />

Γ 2<br />

2<br />

The energy absorption for the steady-state solution for resonance is given by<br />

() = cos + s<strong>in</strong> <br />

(373)<br />

where the elastic amplitude<br />

while the absorptive amplitude<br />

=<br />

0<br />

<br />

( 2 0 − 2 ) 2 +(Γ) 2 ¡<br />

<br />

2<br />

0 − 2¢ (374)<br />

0<br />

<br />

=<br />

( 2 0 − 2 ) 2 Γ (375)<br />

2<br />

+(Γ)<br />

The time average power <strong>in</strong>put is given by only the absorptive term<br />

h i = 1 2 0 = 0<br />

2 Γ 2<br />

2 ( 2 0 − 2 ) 2 +(Γ) 2 (3.133)<br />

This power curve has the classic Lorentzian shape.

82 CHAPTER 3. LINEAR OSCILLATORS<br />

Wave propagation The wave equation was <strong>in</strong>troduced and both travell<strong>in</strong>g and stand<strong>in</strong>g wave solutions<br />

of the wave equation were discussed. Harmonic wave-form analysis, and the complementary time-sampled<br />

wave form analysis techniques, were <strong>in</strong>troduced <strong>in</strong> this chapter and <strong>in</strong> appendix . The relative merits of<br />

Fourier analysis and the digital Green’s function waveform analysis were illustrated for signal process<strong>in</strong>g.<br />

The concepts of phase velocity, group velocity, and signal velocity were <strong>in</strong>troduced. The phase velocity<br />

is given by<br />

= (3117)<br />

<br />

and group velocity<br />

µ <br />

= = + <br />

(3128)<br />

<br />

0<br />

<br />

If the group velocity is frequency dependent then the <strong>in</strong>formation content of a wave packet travels at the<br />

signal velocity which can differ from the group velocity.<br />

The Wave-packet Uncerta<strong>in</strong>ty Pr<strong>in</strong>ciple implies that mak<strong>in</strong>g a precise measurement of the frequency q of a<br />

s<strong>in</strong>usoidal wave requires that the wave packet be <strong>in</strong>f<strong>in</strong>itely long. The standard deviation () = h 2 i − hi 2<br />

characteriz<strong>in</strong>g the width of the amplitude of the wavepacket spectral distribution <strong>in</strong> the angular frequency<br />

doma<strong>in</strong>, (), and the correspond<strong>in</strong>g width <strong>in</strong> time () are related by :<br />

() · () > 1<br />

(Relation between amplitude uncerta<strong>in</strong>ties.)<br />

The standard deviations for the spectral distribution and width of the <strong>in</strong>tensity ofthewavepacketare<br />

related by:<br />

() · () > 1 (3.134)<br />

2<br />

() · ( ) > 1 2<br />

() · ( ) > 1 2<br />

() · ( ) > 1 2<br />

This applies to all forms of wave motion, <strong>in</strong>clud<strong>in</strong>g sound waves, water waves, electromagnetic waves, or<br />

matter waves.

3.12. SUMMARY 83<br />

Problems<br />

1. Consider a simple harmonic oscillator consist<strong>in</strong>g of a mass attached to a spr<strong>in</strong>g of spr<strong>in</strong>g constant . For<br />

this oscillator () = s<strong>in</strong>( 0 − ).<br />

(a) F<strong>in</strong>d an expression for ˙().<br />

(b) Elim<strong>in</strong>ate between () and ˙() to arrive at one equation similar to that for an ellipse.<br />

(c) Rewrite the equation <strong>in</strong> part (b) <strong>in</strong> terms of , ˙, , , and the total energy .<br />

(d) Give a rough sketch of the phase space diagram ( ˙ versus ) for this oscillator. Also, on the same set of<br />

axes, sketch the phase space diagram for a similar oscillator with a total energy that is larger than the<br />

first oscillator.<br />

(e) What direction are the paths that you have sketched? Expla<strong>in</strong> your answer.<br />

(f) Would different trajectories for the same oscillator ever cross paths? Why or why not?<br />

2. Consider a damped, driven oscillator consist<strong>in</strong>g of a mass attachedtoaspr<strong>in</strong>gofspr<strong>in</strong>gconstant.<br />

(a) What is the equation of motion for this system?<br />

(b) Solve the equation <strong>in</strong> part (a). The solution consists of two parts, the complementary solution and the<br />

particular solution. When might it be possible to safely neglect one part of the solution?<br />

(c) What is the difference between amplitude resonance and k<strong>in</strong>etic energy resonance?<br />

(d) How might phase space diagrams look for this type of oscillator? What variables would affect the diagram?<br />

3. A particle of mass is subject to the follow<strong>in</strong>g force<br />

where is a constant.<br />

F = ( 3 − 4 2 +3)ˆx<br />

(a) Determ<strong>in</strong>e the po<strong>in</strong>ts when the particle is <strong>in</strong> equilibrium.<br />

(b) Which of these po<strong>in</strong>ts is stable and which are unstable?<br />

(c) Is the motion bounded or unbounded?<br />

4. A very long cyl<strong>in</strong>drical shell has a mass density that depends upon the radial distance such that () = ,<br />

where is a constant. The <strong>in</strong>ner radius of the shell is and the outer radius is .<br />

(a) Determ<strong>in</strong>e the direction and the magnitude of the gravitational field for all regions of space.<br />

(b) If the gravitational potential is zero at the orig<strong>in</strong>, what is the difference between the gravitational potential<br />

at = and = ?<br />

5. A mass is constra<strong>in</strong>ed to move along one dimension. Two identical spr<strong>in</strong>gs are attached to the mass, one on<br />

each side, and each spr<strong>in</strong>g is <strong>in</strong> turn attached to a wall. Both spr<strong>in</strong>gs have the same spr<strong>in</strong>g constant .<br />

(a) Determ<strong>in</strong>e the frequency of the oscillation, assum<strong>in</strong>g no damp<strong>in</strong>g.<br />

(b) Now consider damp<strong>in</strong>g. It is observed that after oscillations, the amplitude of the oscillation has<br />

dropped to one-half of its <strong>in</strong>itial value. F<strong>in</strong>d an expression for the damp<strong>in</strong>g constant.<br />

(c) How long does it take for the amplitude to decrease to one-quarter of its <strong>in</strong>itial value?<br />

6. Discussthemotionofacont<strong>in</strong>uousstr<strong>in</strong>gwhenpluckedat ½<br />

one third of the length of the str<strong>in</strong>g. That is, the<br />

3<br />

<strong>in</strong>itial condition is ˙( 0) = 0, and ( 0) = 0 ≤ ≤ ¾<br />

<br />

3<br />

3<br />

<br />

2<br />

( − )<br />

3 ≤ ≤ <br />

7. When a particular driv<strong>in</strong>g force is applied to a stretched str<strong>in</strong>g it is observed that the str<strong>in</strong>g vibration <strong>in</strong> purely<br />

of the harmonic. F<strong>in</strong>d the driv<strong>in</strong>g force.

84 CHAPTER 3. LINEAR OSCILLATORS<br />

8. Consider the two-mass system pivoted at its vertex where 6= . It undergoes oscillations of the angle <br />

with respect to the vertical <strong>in</strong> the plane of the triangle.<br />

l<br />

l<br />

M<br />

l<br />

m<br />

(a) Determ<strong>in</strong>e the angular frequency of small oscillations.<br />

(b) Use your result from part (a) to show 2 ≈ <br />

for À .<br />

(c) Show that your result from part (a) agrees with 2 = 00 ( )<br />

<br />

where is the equilibrium angle and is<br />

the moment of <strong>in</strong>ertia.<br />

(d) Assume the system has energy . Setup an <strong>in</strong>tegral that determ<strong>in</strong>es the period of oscillation.<br />

9. An unusual pendulum is made by fix<strong>in</strong>gastr<strong>in</strong>gtoahorizontalcyl<strong>in</strong>derofradius wrapp<strong>in</strong>g the str<strong>in</strong>g<br />

several times around the cyl<strong>in</strong>der, and then ty<strong>in</strong>g a mass to the loose end. In equilibrium the mass hangs a<br />

distance 0 vertically below the edge of the cyl<strong>in</strong>der. F<strong>in</strong>d the potential energy if the pendulum has swung to<br />

an angle from the vertical. Show that for small angles, it can be written <strong>in</strong> the Hooke’s Law form = 1 2 2 .<br />

Comment of the value of <br />

10. Consider the two-dimensional anisotropic oscillator with motion with = and = .<br />

a) Prove that if the ratio of the frequencies is rational (that is, <br />

<br />

= <br />

where and are <strong>in</strong>tegers) then the<br />

motion is periodic. What is the period?<br />

b) Prove that if the same ratio is irrational, the motion never repeats itself.<br />

11. A simple pendulum consists of a mass suspended from a fixed po<strong>in</strong>t by a weight-less, extensionless rod of<br />

length .<br />

a) Obta<strong>in</strong> the equation of motion, and <strong>in</strong> the approximation s<strong>in</strong> ≈ show that the natural frequency is<br />

0 = p <br />

<br />

,where is the gravitational field strength.<br />

b) Discuss the motion <strong>in</strong> the event that the motion takes place <strong>in</strong> a viscous medium with retard<strong>in</strong>g force<br />

2 √ ˙.<br />

12. Derive the expression for the State Space paths of the plane pendulum if the total energy is 2. Note<br />

that this is just the case of a particle mov<strong>in</strong>g <strong>in</strong> a periodic potential () =(1−cos) Sketch the State<br />

Space diagram for both 2 and 2<br />

13. Consider the motion of a driven l<strong>in</strong>early-damped harmonic oscillator after the transient solution has died out,<br />

and suppose that it is be<strong>in</strong>g driven close to resonance, = .<br />

a) Show that the oscillator’s total energy is = 1 2 2 2 .<br />

b) Show that the energy ∆ dissipated dur<strong>in</strong>g one cycle by the damp<strong>in</strong>g force Γ ˙ is Γ 2<br />

14. Two masses m 1 and m 2 slide freely on a horizontal frictionless rail and are connected by a spr<strong>in</strong>g whose force<br />

constant is k. F<strong>in</strong>d the frequency of oscillatory motion for this system.<br />

15. A particle of mass moves under the <strong>in</strong>fluence of a resistive force proportional to velocity and a potential ,<br />

that is .<br />

( ˙) =− ˙ − <br />

<br />

where 0 and () =( 2 − 2 ) 2<br />

a) F<strong>in</strong>d the po<strong>in</strong>ts of stable and unstable equilibrium.<br />

b) F<strong>in</strong>d the solution of the equations of motion for small oscillations around the stable equilibrium po<strong>in</strong>ts<br />

c) Show that as →∞the particle approaches one of the stable equilibrium po<strong>in</strong>ts for most choices of <strong>in</strong>itial<br />

conditions. What are the exceptions? (H<strong>in</strong>t: You can prove this without f<strong>in</strong>d<strong>in</strong>g the solutions explicitly.)

Chapter 4<br />

Nonl<strong>in</strong>ear systems and chaos<br />

4.1 Introduction<br />

In nature only a subset of systems have equations of motion that are l<strong>in</strong>ear. Contrary to the impression<br />

given by the analytic solutions presented <strong>in</strong> undergraduate physics courses, most dynamical systems <strong>in</strong><br />

nature exhibit non-l<strong>in</strong>ear behavior that leads to complicated motion. The solutions of non-l<strong>in</strong>ear equations<br />

usually do not have analytic solutions, superposition does not apply, and they predict phenomena such as<br />

attractors, discont<strong>in</strong>uous period bifurcation, extreme sensitivity to <strong>in</strong>itial conditions, roll<strong>in</strong>g motion, and<br />

chaos. Dur<strong>in</strong>g the past four decades, excit<strong>in</strong>g discoveries have been made <strong>in</strong> classical mechanics that are<br />

associated with the recognition that nonl<strong>in</strong>ear systems can exhibit chaos. Chaotic phenomena have been<br />

observed <strong>in</strong> most fields of science and eng<strong>in</strong>eer<strong>in</strong>g such as, weather patterns, fluid flow, motion of planets <strong>in</strong><br />

the solar system, epidemics, chang<strong>in</strong>g populations of animals, birds and <strong>in</strong>sects, and the motion of electrons<br />

<strong>in</strong> atoms. The complicated dynamical behavior predicted by non-l<strong>in</strong>ear differential equations is not limited<br />

to classical mechanics, rather it is a manifestation of the mathematical properties of the solutions of the<br />

differential equations <strong>in</strong>volved, and thus is generally applicable to solutions of first or second-order nonl<strong>in</strong>ear<br />

differential equations. It is important to understand that the systems discussed <strong>in</strong> this chapter follow<br />

a fully determ<strong>in</strong>istic evolution predicted by the laws of classical mechanics, the evolution for which is based<br />

on the prior history. This behavior is completely different from a random walk where each step is based on a<br />

random process. The complicated motion of determ<strong>in</strong>istic non-l<strong>in</strong>ear systems stems <strong>in</strong> part from sensitivity<br />

to the <strong>in</strong>itial conditions.<br />

The French mathematician Po<strong>in</strong>caré is credited with be<strong>in</strong>g the first to recognize the existence of chaos<br />

dur<strong>in</strong>g his <strong>in</strong>vestigation of the gravitational three-body problem <strong>in</strong> celestial mechanics. At the end of the<br />

n<strong>in</strong>eteenth century Po<strong>in</strong>caré noticed that such systems exhibit high sensitivity to <strong>in</strong>itial conditions characteristic<br />

of chaotic motion, and the existence of nonl<strong>in</strong>earity which is required to produce chaos. Po<strong>in</strong>caré’s work<br />

received little notice, <strong>in</strong> part it was overshadowed by the parallel development of the Theory of Relativity<br />

and quantum mechanics at the start of the 20 century. In addition, solv<strong>in</strong>g nonl<strong>in</strong>ear equations of motion<br />

is difficult, which discouraged work on nonl<strong>in</strong>ear mechanics and chaotic motion. The field blossomed dur<strong>in</strong>g<br />

the 1960 0 when computers became sufficiently powerful to solve the nonl<strong>in</strong>ear equations required to calculate<br />

the long-time histories necessary to document the evolution of chaotic behavior. Laplace, and many other<br />

scientists, believed <strong>in</strong> the determ<strong>in</strong>istic view of nature which assumes that if the position and velocities of<br />

all particles are known, then one can unambiguously predict the future motion us<strong>in</strong>g Newtonian mechanics.<br />

Researchers <strong>in</strong> many fields of science now realize that this “clockwork universe” is <strong>in</strong>valid. That is, know<strong>in</strong>g<br />

thelawsofnaturecanbe<strong>in</strong>sufficient to predict the evolution of nonl<strong>in</strong>ear systems <strong>in</strong> that the time evolution<br />

can be extremely sensitive to the <strong>in</strong>itial conditions even though they follow a completely determ<strong>in</strong>istic<br />

development. There are two major classifications of nonl<strong>in</strong>ear systems that lead to chaos <strong>in</strong> nature. The<br />

first classification encompasses nondissipative Hamiltonian systems such as Po<strong>in</strong>caré’s three-body celestial<br />

mechanics system. The other ma<strong>in</strong> classification <strong>in</strong>volves driven, damped, non-l<strong>in</strong>ear oscillatory systems.<br />

Nonl<strong>in</strong>earity and chaos is a broad and active field and thus this chapter will focus only on a few examples<br />

that illustrate the general features of non-l<strong>in</strong>ear systems. Weak non-l<strong>in</strong>earity is used to illustrate bifurcation<br />

and asymptotic attractor solutions for which the system evolves <strong>in</strong>dependent of the <strong>in</strong>itial conditions. The<br />

common s<strong>in</strong>usoidally-driven l<strong>in</strong>early-damped plane pendulum illustrates several features characteristic of the<br />

85

86 CHAPTER 4. NONLINEAR SYSTEMS AND CHAOS<br />

evolution of a non-l<strong>in</strong>ear system from order to chaos. The impact of non-l<strong>in</strong>earity on wavepacket propagation<br />

velocities and the existence of soliton solutions is discussed. The example of the three-body problem is<br />

discussed <strong>in</strong> chapter 11. The transition from lam<strong>in</strong>ar flow to turbulent flow is illustrated by fluid mechanics<br />

discussed <strong>in</strong> chapter 168. Analytic solutions of nonl<strong>in</strong>ear systems usually are not available and thus one<br />

must resort to computer simulations. As a consequence the present discussion focusses on the ma<strong>in</strong> features<br />

of the solutions for these systems and ignores how the equations of motion are solved.<br />

4.2 Weak nonl<strong>in</strong>earity<br />

Most physical oscillators become non-l<strong>in</strong>ear with <strong>in</strong>crease <strong>in</strong> amplitude of the oscillations. Consequences<br />

of non-l<strong>in</strong>earity <strong>in</strong>clude breakdown of superposition, <strong>in</strong>troduction of additional harmonics, and complicated<br />

chaotic motion that has great sensitivity to the <strong>in</strong>itial conditions as illustrated <strong>in</strong> this chapter Weak nonl<strong>in</strong>earity<br />

is <strong>in</strong>terest<strong>in</strong>g s<strong>in</strong>ce perturbation theory can be used to solve the non-l<strong>in</strong>ear equations of motion.<br />

The potential energy function for a l<strong>in</strong>ear oscillator has a pure parabolic shape about the m<strong>in</strong>imum<br />

location, that is, = 1 2 ( − 0) 2 where 0 is the location of the m<strong>in</strong>imum. Weak non-l<strong>in</strong>ear systems have<br />

small amplitude oscillations ∆ about the m<strong>in</strong>imum allow<strong>in</strong>g use of the Taylor expansion<br />

(∆) =( 0 )+∆ ( 0)<br />

<br />

By def<strong>in</strong>ition, at the m<strong>in</strong>imum (0)<br />

<br />

∆ = (∆) − ( 0 )= ∆2<br />

2!<br />

+ ∆2<br />

2!<br />

2 ( 0 )<br />

2<br />

+ ∆3<br />

3!<br />

3 ( 0 )<br />

3<br />

+ ∆4<br />

4!<br />

=0 and thus equation 41 can be written as<br />

2 ( 0 )<br />

2<br />

+ ∆3<br />

3!<br />

3 ( 0 )<br />

3<br />

+ ∆4<br />

4!<br />

4 ( 0 )<br />

4 + (4.1)<br />

4 ( 0 )<br />

4 + (4.2)<br />

2 ( 0 )<br />

For small amplitude oscillations the system is l<strong>in</strong>ear when only the second-order ∆2<br />

2! <br />

term <strong>in</strong> equation<br />

2<br />

42 is significant. The l<strong>in</strong>earity for small amplitude oscillations greatly simplifies description of the oscillatory<br />

motion <strong>in</strong> that superposition applies, and complicated chaotic motion is avoided. For slightly larger amplitude<br />

motion, where the higher-order terms <strong>in</strong> the expansion are still much smaller than the second-order term,<br />

then perturbation theory can be used as illustrated by the simple plane pendulum which is non l<strong>in</strong>ear s<strong>in</strong>ce<br />

the restor<strong>in</strong>g force equals<br />

s<strong>in</strong> ' ( − 3<br />

3! + 5<br />

5! − 7<br />

+ ) (4.3)<br />

7!<br />

This is l<strong>in</strong>ear only at very small angles where the higher-order terms <strong>in</strong> the expansion can be neglected.<br />

Consider the equation of motion at small amplitudes for the harmonically-driven, l<strong>in</strong>early-damped plane<br />

pendulum<br />

¨ + Γ˙ + <br />

2<br />

0 s<strong>in</strong> = ¨ + Γ˙ + 2 0( − 3<br />

6 )= 0 cos () (4.4)<br />

where only the first two terms <strong>in</strong> the expansion 43 have been <strong>in</strong>cluded. It was shown <strong>in</strong> chapter 3 that when<br />

s<strong>in</strong> ≈ then the steady-state solution of equation 44 is of the form<br />

() = cos ( − ) (4.5)<br />

Insert this first-order solution <strong>in</strong>to equation 44, then the cubic term <strong>in</strong> the expansion gives a term 3 =<br />

(cos 3 +3cos). Thus the perturbation expansion to third order <strong>in</strong>volves a solution of the form<br />

1<br />

4<br />

() = cos ( − )+ cos 3( − ) (4.6)<br />

This perturbation solution shows that the non-l<strong>in</strong>ear term has distorted the signal by addition of the third<br />

harmonic of the driv<strong>in</strong>g frequency with an amplitude that depends sensitively on . This illustrates that the<br />

superposition pr<strong>in</strong>ciple is not obeyed for this non-l<strong>in</strong>ear system, but, if the non-l<strong>in</strong>earity is weak, perturbation<br />

theory can be used to derive the solution of a non-l<strong>in</strong>ear equation of motion.<br />

Figure 41 illustrates that for a potential () =2 2 + 4 the 4 non-l<strong>in</strong>ear term are greatest at the<br />

maximum amplitude which makes the total energy contours <strong>in</strong> state-space more rectangular than the<br />

elliptical shape for the harmonic oscillator as shown <strong>in</strong> figure 33. The solution is of the form given <strong>in</strong><br />

equation 46.

4.2. WEAK NONLINEARITY 87<br />

Figure 4.1: The left side shows the potential energy for a symmetric potential () =2 2 + 4 . The right<br />

side shows the contours of constant total energy on a state-space diagram.<br />

4.1 Example: Non-l<strong>in</strong>ear oscillator<br />

Assume that a non-l<strong>in</strong>ear oscillator has a potential given by<br />

() = 2<br />

2 − 3<br />

3<br />

where is small. F<strong>in</strong>d the solution of the equation of motion to first order <strong>in</strong> , assum<strong>in</strong>g =0at =0.<br />

The equation of motion for the nonl<strong>in</strong>ear oscillator is<br />

¨ = − = − + 2<br />

<br />

If the 2 term is neglected, then the second-order equation of motion reduces to a normal l<strong>in</strong>ear oscillator<br />

with<br />

0 = s<strong>in</strong> ( 0 + )<br />

where<br />

r<br />

<br />

0 =<br />

<br />

Assume that the first-order solution has the form<br />

1 = 0 + 1<br />

Substitut<strong>in</strong>g this <strong>in</strong>to the equation of motion, and neglect<strong>in</strong>g terms of higher order than gives<br />

¨ 1 + 2 0 1 = 2 0 = 2<br />

2 [1 − cos (2 0)]<br />

To solve this try a particular <strong>in</strong>tegral<br />

1 = + cos (2 0 )<br />

and substitute <strong>in</strong>to the equation of motion gives<br />

−3 2 0 cos (2 0 )+ 2 0 = 2<br />

2 − 2<br />

2 cos (2 0)<br />

Comparison of the coefficients gives<br />

= 2<br />

2 2 0<br />

= 2<br />

6 2 0

88 CHAPTER 4. NONLINEAR SYSTEMS AND CHAOS<br />

The homogeneous equation is<br />

¨ 1 + 2 0 1 =0<br />

which has a solution of the form<br />

1 = 1 s<strong>in</strong> ( 0 )+ 2 cos ( 0 )<br />

Thus comb<strong>in</strong><strong>in</strong>g the particular and homogeneous solutions gives<br />

∙ ¸<br />

<br />

2<br />

1 =( + 1 )s<strong>in</strong>( 0 )+<br />

2 2 + 2 cos ( 0 )+ 2<br />

0<br />

6 2 cos (2 0 )<br />

0<br />

The <strong>in</strong>itial condition =0at =0then gives<br />

and<br />

1 =( + 1 )s<strong>in</strong>( 0 )+ 2<br />

2 0<br />

2 = − 22<br />

3 2<br />

∙ 1<br />

2 − 2 3 cos ( 0)+ 1 6 cos (2 0)¸<br />

The constant ( + 1 ) is given by the <strong>in</strong>itial amplitude and velocity.<br />

This system is nonl<strong>in</strong>ear <strong>in</strong> that the output amplitude is not proportional to the <strong>in</strong>put amplitude. <strong>Second</strong>ly,<br />

a large amplitude second harmonic component is <strong>in</strong>troduced <strong>in</strong> the output waveform; that is, for a non-l<strong>in</strong>ear<br />

system the ga<strong>in</strong> and frequency decomposition of the output differs from the <strong>in</strong>put. Note that the frequency<br />

composition is amplitude dependent. This particular example of a nonl<strong>in</strong>ear system does not exhibit chaos.<br />

The Laboratory for Laser Energetics uses nonl<strong>in</strong>ear crystals to double the frequency of laser light.<br />

4.3 Bifurcation, and po<strong>in</strong>t attractors<br />

Interest<strong>in</strong>g new phenomena, such as bifurcation, and attractors, occur when the non-l<strong>in</strong>earity is large. In<br />

chapter 3 it was shown that the state-space diagram ( ˙ ) for an undamped harmonic oscillator is an<br />

ellipse with dimensions def<strong>in</strong>ed by the total energy of the system. As shown <strong>in</strong> figure 35 for the damped<br />

harmonic oscillator, the state-space diagram spirals <strong>in</strong>wards to the orig<strong>in</strong> due to dissipation of energy. Nonl<strong>in</strong>earity<br />

distorts the shape of the ellipse or spiral on the state-space diagram, and thus the state-space, or<br />

correspond<strong>in</strong>g phase-space, diagrams, provide useful representations of the motion of l<strong>in</strong>ear and non-l<strong>in</strong>ear<br />

periodic systems.<br />

The complicated motion of non-l<strong>in</strong>ear systems makes it necessary to dist<strong>in</strong>guish between transient and<br />

asymptotic behavior. The damped harmonic oscillator executes a transient spiral motion that asymptotically<br />

approaches the orig<strong>in</strong>. The transient behavior depends on the <strong>in</strong>itial conditions, whereas the asymptotic limit<br />

of the steady-state solution is a specific location, that is called a po<strong>in</strong>t attractor. The po<strong>in</strong>t attractor for<br />

damped motion <strong>in</strong> the anharmonic potential well<br />

() =2 2 + 4 (4.7)<br />

is at the m<strong>in</strong>imum, which is the orig<strong>in</strong> of the state-space diagram as shown <strong>in</strong> figure 41.<br />

The more complicated one-dimensional potential well<br />

() =8− 4 2 +05 4 (4.8)<br />

shown <strong>in</strong> figure 42 has two m<strong>in</strong>ima that are symmetric about =0with a saddle of height 8.<br />

The k<strong>in</strong>etic plus potential energies of a particle with mass =2 released <strong>in</strong> this potential, will be<br />

assumed to be given by<br />

( ˙) = ˙ 2 + () (4.9)<br />

The state-space plot <strong>in</strong> figure 42 shows contours of constant energy with the m<strong>in</strong>ima at ( ˙) =(±2 0).<br />

At slightly higher total energy the contours are closed loops around either of the two m<strong>in</strong>ima at = ±2.<br />

At total energies above the saddle energy of 8 the contours are peanut-shaped and are symmetric about<br />

the orig<strong>in</strong>. Assum<strong>in</strong>g that the motion is weakly damped, then a particle released with total energy <br />

which is higher than will follow a peanut-shaped spiral trajectory centered at ( ˙) =(0 0) <strong>in</strong> the

4.4. LIMIT CYCLES 89<br />

Figure 4.2: The left side shows the potential energy for a bimodal symmetric potential () =8− 4 2 +<br />

05 4 . The right-hand figure shows contours of the sum of k<strong>in</strong>etic and potential energies on a state-space<br />

diagram. For total energies above the saddle po<strong>in</strong>t the particle follows peanut-shaped trajectories <strong>in</strong> statespace<br />

centered around ( ˙) =(0 0). For total energies below the saddle po<strong>in</strong>t the particle will have closed<br />

trajectories about either of the two symmetric m<strong>in</strong>ima located at ( ˙) =(±2 0). Thus the system solution<br />

bifurcates when the total energy is below the saddle po<strong>in</strong>t.<br />

state-space diagram for . For there are two separate solutions for the two<br />

m<strong>in</strong>imum centered at = ±2 and ˙ =0. Thisisanexampleofbifurcation where the one solution for<br />

bifurcates <strong>in</strong>to either of the two solutions for .<br />

For an <strong>in</strong>itial total energy damp<strong>in</strong>g will result <strong>in</strong> spiral trajectories of the particle that<br />

will be trapped <strong>in</strong> one of the two m<strong>in</strong>ima. For the particle trajectories are centered giv<strong>in</strong>g<br />

the impression that they will term<strong>in</strong>ate at ( ˙) =(0 0) when the k<strong>in</strong>etic energy is dissipated. However, for<br />

the particle will be trapped <strong>in</strong> one of the two m<strong>in</strong>imum and the trajectory will term<strong>in</strong>ate<br />

at the bottom of that potential energy m<strong>in</strong>imum occurr<strong>in</strong>g at ( ˙) =(±2 0). These two possible term<strong>in</strong>al<br />

po<strong>in</strong>ts of the trajectory are called po<strong>in</strong>t attractors. This example appears to have a s<strong>in</strong>gle attractor for<br />

which bifurcates lead<strong>in</strong>g to two attractors at ( ˙) =(±2 0) for . The<br />

determ<strong>in</strong>ation as to which m<strong>in</strong>imum traps a given particle depends on exactly where the particle starts <strong>in</strong><br />

state space and the damp<strong>in</strong>g etc. That is, for this case, where there is symmetry about the -axis, the<br />

particle has an <strong>in</strong>itial total energy then the <strong>in</strong>itial conditions with radians of state space<br />

will lead to trajectories that are trapped <strong>in</strong> the left m<strong>in</strong>imum, and the other radians of state space will be<br />

trapped <strong>in</strong> the right m<strong>in</strong>imum. Trajectories start<strong>in</strong>g near the split between these two halves of the start<strong>in</strong>g<br />

state space will be sensitive to the exact start<strong>in</strong>g phase. This is an example of sensitivity to <strong>in</strong>itial conditions.<br />

4.4 Limit cycles<br />

4.4.1 Po<strong>in</strong>caré-Bendixson theorem<br />

Coupled first-order differential equations <strong>in</strong> two dimensions of the form<br />

˙ = ( ) ˙ = ( ) (4.10)<br />

occur frequently <strong>in</strong> physics. The state-space paths do not cross for such two-dimensional autonomous systems,<br />

where an autonomous system is not explicitly dependent on time.<br />

The Po<strong>in</strong>caré-Bendixson theorem states that, state-space, and phase-space, can have three possible paths:<br />

(1) closed paths, like the elliptical paths for the undamped harmonic oscillator,<br />

(2) term<strong>in</strong>ate at an equilibrium po<strong>in</strong>t as →∞, like the po<strong>in</strong>t attractor for a damped harmonic oscillator,<br />

(3) tend to a limit cycle as →∞.<br />

The limit cycle is unusual <strong>in</strong> that the periodic motion tends asymptotically to the limit-cycle attractor<br />

<strong>in</strong>dependent of whether the <strong>in</strong>itial values are <strong>in</strong>side or outside the limit cycle. The balance of dissipative forces<br />

and driv<strong>in</strong>g forces often leads to limit-cycle attractors, especially <strong>in</strong> biological applications. Identification of<br />

limit-cycle attractors, as well as the trajectories of the motion towards these limit-cycle attractors, is more<br />

complicated than for po<strong>in</strong>t attractors.

90 CHAPTER 4. NONLINEAR SYSTEMS AND CHAOS<br />

Figure 4.3: The Po<strong>in</strong>caré-Bendixson theorem allows the follow<strong>in</strong>g three scenarios for two-dimensional autonomous<br />

systems. (1) Closed paths as illustrated by the undamped harmonic oscillator. (2) Term<strong>in</strong>ate at<br />

an equilibrium po<strong>in</strong>t as →∞, as illustrated by the damped harmonic oscillator, and (3) Tend to a limit<br />

cycle as →∞as illustrated by the van der Pol oscillator.<br />

4.4.2 van der Pol damped harmonic oscillator:<br />

The van der Pol damped harmonic oscillator illustrates a non-l<strong>in</strong>ear equation that leads to a well-studied,<br />

limit-cycle attractor that has important applications <strong>in</strong> diverse fields. The van der Pol oscillator has an<br />

equation of motion given by<br />

2 <br />

2 + ¡ 2 − 1 ¢ <br />

+ 2 0 =0 (4.11)<br />

The non-l<strong>in</strong>ear ¡ 2 − 1 ¢ <br />

<br />

damp<strong>in</strong>g term is unusual <strong>in</strong> that the sign changes when =1lead<strong>in</strong>g to<br />

positive damp<strong>in</strong>g for 1 and negative damp<strong>in</strong>g for 1 To simplify equation 411 assume that the term<br />

2 0 = that is, 2 0 =1.<br />

This equation was studied extensively dur<strong>in</strong>g the 1920’s and 1930’s by the Dutch eng<strong>in</strong>eer, Balthazar<br />

van der Pol, for describ<strong>in</strong>g electronic circuits that <strong>in</strong>corporate feedback. The form of the solution can be<br />

simplified by def<strong>in</strong><strong>in</strong>g a variable ≡ <br />

<br />

Then the second-order equation 411 can be expressed as two<br />

coupled first-order equations.<br />

≡ <br />

(4.12)<br />

<br />

<br />

= − − ¡ 2 − 1 ¢ (4.13)<br />

<br />

It is advantageous to transform the ( ˙ ) state space to polar coord<strong>in</strong>ates by sett<strong>in</strong>g<br />

and us<strong>in</strong>g the fact that 2 = 2 + 2 Therefore<br />

= cos (4.14)<br />

= s<strong>in</strong> <br />

<br />

= + <br />

<br />

Similarly for the angle coord<strong>in</strong>ate<br />

<br />

<br />

<br />

<br />

Multiply equation 416 by and 417 by and subtract gives<br />

(4.15)<br />

= <br />

cos − s<strong>in</strong> (4.16)<br />

<br />

= <br />

s<strong>in</strong> + cos (4.17)<br />

<br />

2 <br />

= − <br />

<br />

(4.18)

4.4. LIMIT CYCLES 91<br />

Figure 4.4: Solutions of the van der Pol system for =02 top row and =5bottom row, assum<strong>in</strong>g that<br />

2 0 =1. The left column shows the time dependence (). The right column shows the correspond<strong>in</strong>g ( ˙)<br />

state space plots. Upper: Weak nonl<strong>in</strong>earity, = 02; At large times the solution tends to one limit<br />

cycle for <strong>in</strong>itial values <strong>in</strong>side or outside the limit cycle attractor. The amplitude () for two <strong>in</strong>itial conditions<br />

approaches an approximately harmonic oscillation. Lower: Strong nonl<strong>in</strong>earity, μ = 5; Solutions<br />

approach a common limit cycle attractor for <strong>in</strong>itial values <strong>in</strong>side or outside the limit cycle attractor while<br />

the amplitude () approaches a common approximate square-wave oscillation.<br />

Equations 415 and 418 allow the van der Pol equations of motion to be written <strong>in</strong> polar coord<strong>in</strong>ates<br />

<br />

<br />

= − ¡ 2 cos 2 − 1 ¢ s<strong>in</strong> 2 (4.19)<br />

<br />

<br />

= −1 − ¡ 2 cos 2 − 1 ¢ s<strong>in</strong> cos (4.20)<br />

The non-l<strong>in</strong>ear terms on the right-hand side of equations 419 − 20 have a complicated form.<br />

Weak non-l<strong>in</strong>earity: 1<br />

In the limit that → 0, equations419 420 correspond to a circular state-space trajectory similar to the<br />

harmonic oscillator. That is, the solution is of the form<br />

() = s<strong>in</strong> ( − 0 ) (4.21)<br />

where and 0 are arbitrary parameters. For weak non-l<strong>in</strong>earity, 1 the angular equation 420 has a<br />

rotational frequency that is unity s<strong>in</strong>ce the s<strong>in</strong> cos term changes sign twice per period, <strong>in</strong> addition to the

92 CHAPTER 4. NONLINEAR SYSTEMS AND CHAOS<br />

small value of . For1 and 1 the radial equation 419 has a sign of the ¡ 2 cos 2 − 1 ¢ term that<br />

is positive and thus the radius <strong>in</strong>creases monotonically to unity. For 1 the bracket is predom<strong>in</strong>antly<br />

negative result<strong>in</strong>g <strong>in</strong> a spiral decrease <strong>in</strong> the radius. Thus, for very weak non-l<strong>in</strong>earity, this radial behavior<br />

results <strong>in</strong> the amplitude spirall<strong>in</strong>g to a well def<strong>in</strong>ed limit-cycle attractor value of =2as illustrated by<br />

the state-space plots <strong>in</strong> figure 44 for cases where the <strong>in</strong>itial condition is <strong>in</strong>side or external to the circular<br />

attractor. The f<strong>in</strong>al amplitude for different <strong>in</strong>itial conditions also approach the same asymptotic behavior.<br />

Dom<strong>in</strong>ant non-l<strong>in</strong>earity: 1<br />

For the case where the non-l<strong>in</strong>earity is dom<strong>in</strong>ant, that is 1, then as shown <strong>in</strong> figure 44, the system<br />

approaches a well def<strong>in</strong>ed attractor, but <strong>in</strong> this case it has a significantly skewed shape <strong>in</strong> state-space, while<br />

the amplitude approximates a square wave. The solution rema<strong>in</strong>s close to =+2until = ˙ ≈ +7 and<br />

then it relaxes quickly to = −2 with = ˙ ≈ 0 This is followed by the mirror image. This behavior is<br />

called a relaxed vibration <strong>in</strong> that a tension builds up slowly then dissipates by a sudden relaxation process.<br />

The seesaw is an extreme example of a relaxation oscillator where the seesaw angle switches spontaneously<br />

from one solution to the other when the difference <strong>in</strong> their moment arms changes sign.<br />

The study of feedback <strong>in</strong> electronic circuits was the stimulus for study of this equation by van der<br />

Pol. However, Lord Rayleigh first identified such relaxation oscillator behavior <strong>in</strong> 1880 dur<strong>in</strong>g studies of<br />

vibrations of a str<strong>in</strong>ged <strong>in</strong>strument excited by a bow, or the squeak<strong>in</strong>g of a brake drum. In his discussion of<br />

non-l<strong>in</strong>ear effects <strong>in</strong> acoustics, he derived the equation<br />

¨ − ( − ˙ 2 ) ˙ + 2 0 (4.22)<br />

Differentiation of Rayleigh’s equation 422 gives<br />

...<br />

− ( − 3 ˙ 2 )¨ + 2 0 ˙ =0 (4.23)<br />

Us<strong>in</strong>g the substitution of<br />

leads to the relations<br />

= 0<br />

r<br />

3<br />

<br />

˙ (4.24)<br />

r r r <br />

˙<br />

... ¨<br />

˙ =<br />

¨ = =<br />

(4.25)<br />

3 0 3 0 3 0<br />

Substitut<strong>in</strong>g these relations <strong>in</strong>to equation 423 gives<br />

r r ∙<br />

¨ <br />

− − 3 ˙ 2 ¸ r ˙ <br />

+ 2 <br />

0 =0 (4.26)<br />

3 0 3 0 3 0<br />

Multiply<strong>in</strong>g by 0<br />

q<br />

3<br />

<br />

and rearrang<strong>in</strong>g leads to the van der Pol equation<br />

2 0<br />

¨ − 0<br />

2 (0 2 − 2 ) ˙ − 2 0 =0 (4.27)<br />

The rhythm of a heartbeat driven by a pacemaker is an important application where the self-stabilization of<br />

the attractor is a desirable characteristic to stabilize an irregular heartbeat; the medical term is arrhythmia.<br />

The mechanism that leads to synchronization of the many pacemaker cells <strong>in</strong> the heart and human body due<br />

to the <strong>in</strong>fluence of an implanted pacemaker is discussed <strong>in</strong> chapter 1412. Another biological application of<br />

limit cycles is the time variation of animal populations.<br />

In summary the non-l<strong>in</strong>ear damp<strong>in</strong>g of the van der Pol oscillator leads to a self-stabilized, s<strong>in</strong>gle limitcycle<br />

attractor that is <strong>in</strong>sensitive to the <strong>in</strong>itial conditions. The van der Pol oscillator has many important<br />

applications such as bowed musical <strong>in</strong>struments, electrical circuits, and human anatomy as mentioned above.<br />

The van der Pol oscillator illustrates the complicated manifestations of the motion that can be exhibited by<br />

non-l<strong>in</strong>ear systems

4.5. HARMONICALLY-DRIVEN, LINEARLY-DAMPED, PLANE PENDULUM 93<br />

4.5 Harmonically-driven, l<strong>in</strong>early-damped, plane pendulum<br />

The harmonically-driven, l<strong>in</strong>early-damped, plane pendulum illustrates many of the phenomena exhibited by<br />

non-l<strong>in</strong>ear systems as they evolve from ordered to chaotic motion. It illustrates the remarkable fact that<br />

determ<strong>in</strong>ism does not imply either regular behavior or predictability. The well-known, harmonically-driven<br />

l<strong>in</strong>early-damped pendulum provides an ideal basis for an <strong>in</strong>troduction to non-l<strong>in</strong>ear dynamics 1 .<br />

Consider a harmonically-driven l<strong>in</strong>early-damped plane pendulum of moment of <strong>in</strong>ertia and mass <strong>in</strong><br />

a gravitational field that is driven by a torque due to a force () = cos act<strong>in</strong>g at a moment arm .<br />

The damp<strong>in</strong>g term is and the angular displacement of the pendulum, relative to the vertical, is . The<br />

equation of motion of the harmonically-driven l<strong>in</strong>early-damped simple pendulum can be written as<br />

¨ + ˙ + s<strong>in</strong> = cos (4.28)<br />

Note that the s<strong>in</strong>usoidal restor<strong>in</strong>g force for the plane pendulum is non-l<strong>in</strong>ear for large angles . Thenatural<br />

period of the free pendulum is<br />

r<br />

<br />

0 =<br />

(4.29)<br />

<br />

A dimensionless parameter , which is called the drive strength, is def<strong>in</strong>ed by<br />

≡ <br />

(4.30)<br />

<br />

The equation of motion 428 can be generalized by <strong>in</strong>troduc<strong>in</strong>g dimensionless units for both time ˜ and<br />

relative drive frequency ˜ def<strong>in</strong>ed by<br />

˜ ≡ 0 ˜ ≡ 0<br />

(4.31)<br />

In addition, def<strong>in</strong>e the <strong>in</strong>verse damp<strong>in</strong>g factor as<br />

≡ 0<br />

(4.32)<br />

<br />

These def<strong>in</strong>itions allow equation 428 to be written <strong>in</strong> the dimensionless form<br />

2 <br />

˜ + 1 <br />

2 ˜ +s<strong>in</strong> = cos ˜˜ (4.33)<br />

The behavior of the angle for the driven damped plane pendulum depends on the drive strength <br />

and the damp<strong>in</strong>g factor . Consider the case where equation 433 is evaluated assum<strong>in</strong>g that the damp<strong>in</strong>g<br />

coefficient =2, and that the relative angular frequency ˜ = 2 3 which is close to resonance where chaotic<br />

phenomena are manifest. The Runge-Kutta method is used to solve this non-l<strong>in</strong>ear equation of motion.<br />

4.5.1 Close to l<strong>in</strong>earity<br />

For drive strength =02 the amplitude is sufficiently small that s<strong>in</strong> ' superposition applies, and the<br />

solution is identical to that for the driven l<strong>in</strong>early-damped l<strong>in</strong>ear oscillator. As shown <strong>in</strong> figure 45, once<br />

the transient solution dies away, the steady-state solution asymptotically approaches one attractor that has<br />

an amplitude of ±03 radians and a phase shift with respect to the driv<strong>in</strong>g force. The abscissa is given<br />

<strong>in</strong> units of the dimensionless time ˜ = 0 . The transient solution depends on the <strong>in</strong>itial conditions and<br />

dies away after about 5 periods, whereas the steady-state solution is <strong>in</strong>dependent of the <strong>in</strong>itial conditions<br />

and has a state-space diagram that has an elliptical shape, characteristic of the harmonic oscillator. For all<br />

<strong>in</strong>itial conditions, the time dependence and state space diagram for steady-state motion approaches a unique<br />

solution,calledan“attractor”, that is, the pendulum oscillates s<strong>in</strong>usoidally with a given amplitude at the<br />

frequency of the driv<strong>in</strong>g force and with a constant phase shift , i.e.<br />

() = cos( − ) (4.34)<br />

This solution is identical to that for the harmonically-driven, l<strong>in</strong>early-damped, l<strong>in</strong>ear oscillator discussed <strong>in</strong><br />

chapter 36<br />

1 A similar approach is used by the book "Chaotic Dynamics" by Baker and Gollub[Bak96].

94 CHAPTER 4. NONLINEAR SYSTEMS AND CHAOS<br />

Figure 4.5: Motion of the driven damped pendulum for drive strengths of =02, =09 =105 and<br />

=1078. The left side shows the time dependence of the deflection angle with the time axis expressed<br />

<strong>in</strong> dimensionless units ˜. The right side shows the correspond<strong>in</strong>g state-space plots. These plots assume<br />

˜ = 0<br />

= 2 3<br />

, =2, and the motion starts with = =0.

4.5. HARMONICALLY-DRIVEN, LINEARLY-DAMPED, PLANE PENDULUM 95<br />

20<br />

3<br />

2<br />

10<br />

1<br />

2 4 6 8 10 12 14<br />

t<br />

2 2<br />

1<br />

2<br />

10<br />

20<br />

3<br />

2<br />

10<br />

1<br />

2 4 6 8 10 12 14<br />

t<br />

2 2<br />

1<br />

2<br />

10<br />

20<br />

Figure 4.6: The driven damped pendulum assum<strong>in</strong>g that ˜ = 2 3 , =2, with <strong>in</strong>itial conditions (0) = − 2 ,<br />

(0) = 0. The system exhibits period-two motion for drive strengths of =1078 as shown by the state<br />

space diagram for cycles 10 − 20. For =1081 the system exhibits period-four motion shown for cycles<br />

10 − 30.<br />

4.5.2 Weak nonl<strong>in</strong>earity<br />

Figure 45 shows that for drive strength =09, after the transient solution dies away, the steady-state<br />

solution settles down to one attractor that oscillates at the drive frequency with an amplitude of slightly<br />

more than 2<br />

radians for which the small angle approximation fails. The distortion due to the non-l<strong>in</strong>earity<br />

is exhibited by the non-elliptical shape of the state-space diagram.<br />

The observed behavior can be calculated us<strong>in</strong>g the successive approximation method discussed <strong>in</strong> chapter<br />

42. That is, close to small angles the s<strong>in</strong>e function can be approximated by replac<strong>in</strong>g<br />

s<strong>in</strong> ≈ − 1 6 3<br />

<strong>in</strong> equation 433 to give<br />

1<br />

¨ +<br />

˙ + 2 0<br />

µ − 1 <br />

6 3 = cos ˜˜ (4.35)<br />

As a first approximation assume that<br />

(˜) ≈ cos(˜˜ − )<br />

then the small 3 term <strong>in</strong> equation 435 contributes a term proportional to cos 3 (˜˜ − ). But<br />

cos 3 (˜˜ − ) = 1 ¡<br />

cos 3(˜˜ − )+3cos(˜˜ − ) ¢<br />

4<br />

That is, the nonl<strong>in</strong>earity <strong>in</strong>troduces a small term proportional to cos 3( − ). S<strong>in</strong>ce the right-hand side of<br />

equation 435 is a function of only cos then the terms <strong>in</strong> ˙ and ¨ on the left hand side must conta<strong>in</strong><br />

the third harmonic cos 3( − ) term. Thus a better approximation to the solution is of the form<br />

(˜) = £ cos(˜˜ − )+ cos 3(˜˜ − ) ¤ (4.36)

96 CHAPTER 4. NONLINEAR SYSTEMS AND CHAOS<br />

where the admixture coefficient 1. This successive approximation method can be repeated to add<br />

additional terms proportional to cos ( − ) where is an <strong>in</strong>teger with ≥ 3. Thus the nonl<strong>in</strong>earity<br />

<strong>in</strong>troduces progressively weaker -fold harmonics to the solution. This successive approximation approach<br />

is viable only when the admixture coefficient 1 Note that these harmonics are <strong>in</strong>teger multiples of ,<br />

thus the steady-state response is identical for each full period even though the state space contours deviate<br />

from an elliptical shape.<br />

4.5.3 Onset of complication<br />

Figure 45 shows that for =105 the drive strength is sufficiently strong to cause the transient solution for<br />

the pendulum to rotate through two complete cycles before settl<strong>in</strong>g down to a s<strong>in</strong>gle steady-state attractor<br />

solution at the drive frequency. However, this attractor solution is shifted two complete rotations relative<br />

to the <strong>in</strong>itial condition. The state space diagram clearly shows the roll<strong>in</strong>g motion of the transient solution<br />

for the first two periods prior to the system settl<strong>in</strong>g down to a s<strong>in</strong>gle steady-state attractor. The successive<br />

approximation approach completely fails at this coupl<strong>in</strong>g strength s<strong>in</strong>ce oscillates through large values that<br />

are multiples of <br />

Figure 45 shows that for drive strength =1078 the motion evolves to a much more complicated<br />

periodic motion with a period that is three times the period of the driv<strong>in</strong>g force. Moreover the amplitude<br />

exceeds 2 correspond<strong>in</strong>g to the pendulum oscillat<strong>in</strong>g over top dead center with the centroid of the motion<br />

offset by 3 from the <strong>in</strong>itial condition. Both the state-space diagram, and the time dependence of the motion,<br />

illustrate the complexity of this motion which depends sensitively on the magnitude of the drive strength <br />

<strong>in</strong> addition to the <strong>in</strong>itial conditions, ((0)(0)) and damp<strong>in</strong>g factor as is shown <strong>in</strong> figure 46<br />

4.5.4 Period doubl<strong>in</strong>g and bifurcation<br />

For drive strength =1078 with the <strong>in</strong>itial condition ((0)(0)) = (0 0) the system exhibits a regular<br />

motion with a period that is three times the drive period. In contrast, if the <strong>in</strong>itial condition is [(0) =<br />

− 2<br />

(0) = 0] then, as shown <strong>in</strong> figure 46 the steady-state solution has the drive frequency with no offset<br />

<strong>in</strong> , that is, it exhibits period-one oscillation. This appearance of two separate and very different attractors<br />

for =1078 us<strong>in</strong>g different <strong>in</strong>itial conditions, is called bifurcation.<br />

An additional feature of the system response for =1078 is that chang<strong>in</strong>g the <strong>in</strong>itial conditions to<br />

[(0) = − 2<br />

(0) = 0] shows that the amplitude of the even and odd periods of oscillation differ slightly<br />

<strong>in</strong> shape and amplitude, that is, the system really has period-two oscillation. This period-two motion, i.e.<br />

period doubl<strong>in</strong>g, is clearly illustrated by the state space diagram <strong>in</strong> that, although the motion still is<br />

dom<strong>in</strong>ated by period-one oscillations, the even and odd cycles are slightly displaced. Thus, for different<br />

<strong>in</strong>itial conditions, the system for =1078 bifurcates <strong>in</strong>to either of two attractors that have very different<br />

waveforms, one of which exhibits period doubl<strong>in</strong>g.<br />

The period doubl<strong>in</strong>g exhibited for =1078 is followed by a second period doubl<strong>in</strong>g when =1081 as<br />

shown <strong>in</strong> figure 46 . With <strong>in</strong>crease <strong>in</strong> drive strength this period doubl<strong>in</strong>g keeps <strong>in</strong>creas<strong>in</strong>g <strong>in</strong> b<strong>in</strong>ary multiples<br />

to period 8, 16, 32, 64 etc. Numerically it is found that the threshold for period doubl<strong>in</strong>g is 1 =10663<br />

fromtwotofouroccursat 2 =10793 etc. Feigenbaum showed that this cascade <strong>in</strong>creases with <strong>in</strong>crease <strong>in</strong><br />

drive strength accord<strong>in</strong>g to the relation that obeys<br />

( +1 − ) ' 1 ( − −1 ) (4.37)<br />

where =46692016, is called a Feigenbaum number. As →∞this cascad<strong>in</strong>g sequence goes to a limit<br />

where<br />

=10829 (4.38)<br />

4.5.5 Roll<strong>in</strong>g motion<br />

It was shown that for 105 the transient solution causes the pendulum to have angle excursions exceed<strong>in</strong>g<br />

2, that is, the system rolls over top dead center. For drive strengths <strong>in</strong> the range 13 14 the steadystate<br />

solution for the system undergoes cont<strong>in</strong>uous roll<strong>in</strong>g motion as illustrated <strong>in</strong> figure 47. The time<br />

dependence for the angle exhibits a periodic oscillatory motion superimposed upon a monotonic roll<strong>in</strong>g<br />

motion, whereas the time dependence of the angular frequency = <br />

<br />

is periodic. The state space plots

4.5. HARMONICALLY-DRIVEN, LINEARLY-DAMPED, PLANE PENDULUM 97<br />

Figure 4.7: Roll<strong>in</strong>g motion for the driven damped plane pendulum for =14. (a) The time dependence<br />

of angle () <strong>in</strong>creases by 2 per drive period whereas (b) the angular velocity () exhibits periodicity. (c)<br />

The state space plot for roll<strong>in</strong>g motion is shown with the orig<strong>in</strong> shifted by 2 per revolution to keep the plot<br />

with<strong>in</strong> the bounds − +<br />

for roll<strong>in</strong>g motion corresponds to a cha<strong>in</strong> of loops with a spac<strong>in</strong>g of 2 between each loop. The state space<br />

diagram for roll<strong>in</strong>g motion is more compactly presented if the orig<strong>in</strong> is shifted by 2 per revolution to keep<br />

the plot with<strong>in</strong> bounds as illustrated <strong>in</strong> figure 47.<br />

4.5.6 Onset of chaos<br />

When the drive strength is <strong>in</strong>creased to =1105 then the system does not approach a unique attractor<br />

as illustrated by figure 48 which shows state space orbits for cycles 25 − 200. Note that these orbits do<br />

not repeat imply<strong>in</strong>g the onset of chaos. For drive strengths greater than =10829 the driven damped<br />

plane pendulum starts to exhibit chaotic behavior. The onset of chaotic motion is illustrated by mak<strong>in</strong>g a 3-<br />

dimensional plot which comb<strong>in</strong>es the time coord<strong>in</strong>ate with the state-space coord<strong>in</strong>ates as illustrated <strong>in</strong> figure<br />

48. This plot shows 16 trajectories start<strong>in</strong>g at different <strong>in</strong>itial values <strong>in</strong> the range −015 015<br />

for =1168. Some solutions are erratic <strong>in</strong> that, while try<strong>in</strong>g to oscillate at the drive frequency, they never<br />

settle down to a steady periodic motion which is characteristic of chaotic motion. Figure 48 illustrates<br />

the considerable sensitivity of the motion to the <strong>in</strong>itial conditions. That is, this determ<strong>in</strong>istic system can<br />

exhibit either order, or chaos, dependent on m<strong>in</strong>iscule differences <strong>in</strong> <strong>in</strong>itial conditions.<br />

Figure 4.8: Left: Space-space orbits for the driven damped pendulum with =1105. Note that the orbits<br />

do not repeat for cycles 25 to 200. Right: Time-state-space diagram for =1168. The plot shows 16<br />

trajectories start<strong>in</strong>g with different <strong>in</strong>itial values <strong>in</strong> the range −015 015.

98 CHAPTER 4. NONLINEAR SYSTEMS AND CHAOS<br />

Figure 4.9: State-space plots for the harmonically-driven, l<strong>in</strong>early-damped, pendulum for driv<strong>in</strong>g amplitudes<br />

of =05 and =12. These calculations were performed us<strong>in</strong>g the Runge-Kutta method by E. Shah,<br />

(Private communication)<br />

4.6 Differentiation between ordered and chaotic motion<br />

Chapter 45 showed that motion <strong>in</strong> non-l<strong>in</strong>ear systems can exhibit both order and chaos. The transition<br />

between ordered motion and chaotic motion depends sensitively on both the <strong>in</strong>itial conditions and the model<br />

parameters. It is surpris<strong>in</strong>gly difficult to unambiguously dist<strong>in</strong>guish between complicated ordered motion<br />

and chaotic motion. Moreover, the motion can fluctuate between order and chaos <strong>in</strong> an erratic manner<br />

depend<strong>in</strong>g on the <strong>in</strong>itial conditions. The extremely sensitivity to <strong>in</strong>itial conditions of the motion for nonl<strong>in</strong>ear<br />

systems, makes it essential to have quantitative measures that can characterize the degree of order, and<br />

<strong>in</strong>terpret the complicated dynamical motion of systems. As an illustration, consider the harmonically-driven,<br />

l<strong>in</strong>early-damped, pendulum with =2 and driv<strong>in</strong>g force () = s<strong>in</strong> ˜˜ where ˜ = 2 3<br />

.Figure49 shows<br />

the state-space plots for two driv<strong>in</strong>g amplitudes, =05 which leads to ordered motion, and =12<br />

which leads to possible chaotic motion. It can be seen that for =05 the state-space diagram converges<br />

to a s<strong>in</strong>gle attractor once the transient solution has died away. This is <strong>in</strong> contrast to the case for =12<br />

where the state-space diagram does not converge to a s<strong>in</strong>gle attractor, but exhibits possible chaotic motion.<br />

Three quantitative measures can be used to differentiate ordered motion from chaotic motion for this system;<br />

namely, the Lyapunov exponent, the bifurcation diagram, and the Po<strong>in</strong>caré section, as illustrated below.<br />

4.6.1 Lyapunov exponent<br />

The Lyapunov exponent provides a quantitative and useful measure of the <strong>in</strong>stability of trajectories, and how<br />

quickly nearby <strong>in</strong>itial conditions diverge. It compares two identical systems that start with an <strong>in</strong>f<strong>in</strong>itesimally<br />

small difference <strong>in</strong> the <strong>in</strong>itial conditions <strong>in</strong> order to ascerta<strong>in</strong> whether they converge to the same attractor<br />

at long times, correspond<strong>in</strong>g to a stable system, or whether they diverge to very different attractors, characteristic<br />

of chaotic motion. If the <strong>in</strong>itial separation between the trajectories <strong>in</strong> phase space at =0is | 0 |,<br />

then to first order the time dependence of the difference can be assumed to depend exponentially on time.<br />

That is,<br />

|()| ∼ | 0 | (4.39)<br />

where is the Lyapunov exponent. That is, the Lyapunov exponent is def<strong>in</strong>ed to be<br />

= lim<br />

lim<br />

→∞ 0 →0<br />

1 |()|<br />

ln<br />

| 0 |<br />

(4.40)<br />

Systems for which the Lyapunov exponent 0 (negative), converge exponentially to the same attractor<br />

solution at long times s<strong>in</strong>ce |()| → 0 for →∞. By contrast, systems for which 0 (positive) diverge<br />

to completely different long-time solutions, that is, |()| → ∞ for → ∞. Even for <strong>in</strong>f<strong>in</strong>itesimally

4.6. DIFFERENTIATION BETWEEN ORDERED AND CHAOTIC MOTION 99<br />

Figure 4.10: Lyapunov plots of ∆ versus time for two <strong>in</strong>itial start<strong>in</strong>g po<strong>in</strong>ts differ<strong>in</strong>g by ∆ 0 =0001.<br />

The parameters are =2 and () = s<strong>in</strong>( 2 3 ) and ∆ =004. The Lyapunov exponent for =05<br />

which is drawn as a dashed l<strong>in</strong>e, is convergent with = −0251 For =12 the exponent is divergent as<br />

<strong>in</strong>dicated by the dashed l<strong>in</strong>e which as a slope of =01538 These calculations were performed us<strong>in</strong>g the<br />

Runge-Kutta method by E. Shah, (Private communication)<br />

small differences <strong>in</strong> the <strong>in</strong>itial conditions, systems hav<strong>in</strong>g a positive Lyapunov exponent diverge to different<br />

attractors, whereas when the Lyapunov exponent 0 they correspond to stable solutions.<br />

Figure 410 illustrates Lyapunov plots for the harmonically-driven, l<strong>in</strong>early-damped, plane pendulum,<br />

with the same conditions discussed <strong>in</strong> chapter 45. Note that for the small driv<strong>in</strong>g amplitude =05<br />

the Lyapunov plot converges to ordered motion with an exponent = −0251 whereas for =12 the<br />

plot diverges characteristic of chaotic motion with an exponent =01538 The Lyapunov exponent usually<br />

fluctuates widely at the local oscillator frequency, and thus the time average of the Lyapunov exponent must<br />

be taken over many periods of the oscillation to identify the general trend with time. Some systems near an<br />

order-to-chaos transition can exhibit positive Lyapunov exponents for short times, characteristic of chaos,<br />

and then converge to negative at longer time imply<strong>in</strong>g ordered motion. The Lyapunov exponents are<br />

used extensively to monitor the stability of the solutions for non-l<strong>in</strong>ear systems. For example the Lyapunov<br />

exponent is used to identify whether fluid flow is lam<strong>in</strong>ar or turbulent as discussed <strong>in</strong> chapter 168.<br />

A dynamical system <strong>in</strong> -dimensional phase space will have a set of Lyapunov exponents { 1 2 }<br />

associated with a set of attractors, the importance of which depend on the <strong>in</strong>itial conditions. Typically one<br />

Lyapunov exponent dom<strong>in</strong>ates at one specific location <strong>in</strong> phase space, and thus it is usual to use the maximal<br />

Lyapunov exponent to identify chaos.The Lyapunov exponent is a very sensitive measure of the onset of chaos<br />

and provides an important test of the chaotic nature for the complicated motion exhibited by non-l<strong>in</strong>ear<br />

systems.<br />

4.6.2