Wheeler, Mechanics

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II Motion: Lagrangian mechanics 61 5 Covariance of the Euler-Lagrangian equation 61 6 Symmetries and the Euler-Lagrange equation 64 6.1 Noether’s theorem for the generalized Euler-Lagrange equation . . . . . . . . . . . . . . . . . 64 6.2 Conserved quantities in restricted Euler-Lagrange systems . . . . . . . . . . . . . . . . . . . . 66 6.2.1 Cyclic coordinates and conserved momentum . . . . . . . . . . . . . . . . . . . . . . . 66 6.2.2 Rotational symmetry and conservation of angular momentum . . . . . . . . . . . . . . 67 6.2.3 Conservation of energy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71 6.2.4 Scale Invariance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71 6.3 Consequences of Newtonian dynamical and measurement theories . . . . . . . . . . . . . . . . 73 6.4 Conserved quantities in generalized Euler-Lagrange systems . . . . . . . . . . . . . . . . . . . 76 6.4.1 Conserved momenta . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76 6.4.2 Angular momentum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77 6.4.3 Energy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79 6.4.4 Scale invariance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79 6.5 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80 7 The physical Lagrangian 81 7.1 Galilean symmetry and the invariance of Newton’s Law . . . . . . . . . . . . . . . . . . . . . 81 7.2 Galileo, Lagrange and inertia . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83 7.3 Gauging Newton’s law . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87 8 Motion in central forces 91 8.1 Regularization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93 8.1.1 Euler’s regularization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93 8.1.2 Higher dimensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94 8.2 General central potentials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96 8.3 Energy, angular momentum and convexity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98 8.4 Bertrand’s theorem: closed orbits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100 8.5 Symmetries of motion for the Kepler problem . . . . . . . . . . . . . . . . . . . . . . . . . . . 102 8.5.1 Conic sections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105 8.6 Newtonian gravity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106 9 Constraints 108 10 Rotating coordinates 112 10.1 Rotations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112 10.2 The Coriolis theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114 11 Inequivalent Lagrangians 115 11.1 General free particle Lagrangians . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116 11.2 Inequivalent Lagrangians . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118 11.2.1 Are inequivalent Lagrangians equivalent? . . . . . . . . . . . . . . . . . . . . . . . . . 120 11.3 Inequivalent Lagrangians in higher dimensions . . . . . . . . . . . . . . . . . . . . . . . . . . . 120 III Conformal gauge theory 121 2
12 Special Relativity 122 12.1 Spacetime . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122 12.2 Relativistic dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123 12.3 Acceleration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127 12.4 Equations of motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128 12.5 Relativistic action with a potential . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128 13 The symmetry of Newtonian mechanics 131 13.1 The conformal group of Euclidean 3-space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132 13.2 The relativisic conformal group . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136 13.3 A linear representation for conformal transformations . . . . . . . . . . . . . . . . . . . . . . 137 14 A new arena for mechanics 139 14.1 Dilatation covariant derivative . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140 14.2 Consequences of the covariant derivative . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141 14.3 Biconformal geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142 14.4 Motion in biconformal space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144 14.5 Hamiltonian dynamics and phase space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144 14.5.1 Multiparticle mechanics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 146 14.6 Measurement and Hamilton’s principal function . . . . . . . . . . . . . . . . . . . . . . . . . . 147 14.7 A second proof of the existence of Hamilton’s principal function . . . . . . . . . . . . . . . . . 150 14.8 Phase space and the symplectic form . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152 14.9 Poisson brackets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155 14.9.1 Example 1: Coordinate transformations . . . . . . . . . . . . . . . . . . . . . . . . . . 158 14.9.2 Example 2: Interchange of x and p. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 160 14.9.3 Example 3: Momentum transformations . . . . . . . . . . . . . . . . . . . . . . . . . . 160 14.10Generating functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161 15 General solution in Hamiltonian dynamics 162 15.1 The Hamilton-Jacobi Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162 15.2 Quantum <strong>Mechanics</strong> and the Hamilton-Jacobi equation . . . . . . . . . . . . . . . . . . . . . 162 15.3 Trivialization of the motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163 15.3.1 Example 1: Free particle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165 15.3.2 Example 2: Simple harmonic oscillator . . . . . . . . . . . . . . . . . . . . . . . . . . . 166 15.3.3 Example 3: One dimensional particle motion . . . . . . . . . . . . . . . . . . . . . . . 167 IV Bonus sections 169 16 Classical spin, statistics and pseudomechanics 169 16.1 Spin . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 169 16.2 Statistics and pseudomechanics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171 16.3 Spin-statistics theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173 17 Gauge theory 175 17.1 Group theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 175 17.2 Lie algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 178 17.2.1 The Lie algebra so(3) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 180 17.2.2 The Lie algebras so(p, q) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183 17.2.3 Lie algebras: a general approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 184 17.3 Differential forms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 187 17.4 The exterior derivative . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 191 3
Page 1: Mechanics James T. Wheeler Septembe
Page 5 and 6: Part I Preliminaries Geometry is ph
Page 7 and 8: The first case is immediate because
Page 9 and 10: in contradiction with x and y being
Page 11 and 12: 1. For all A, B in τ, their inters
Page 13 and 14: 1.3 The Euclidean group We now retu
Page 15 and 16: Now, we see that the double sum of
Page 17 and 18: To recover 3-dim space, we first id
Page 19 and 20: Sometimes it is useful to think of
Page 21 and 22: functional. Whereas a function retu
Page 23 and 24: = ∣ ∣ ∣∣∣∣∣ ∣∣∣
Page 25 and 26: 2.3.2 Formal definition of function
Page 27 and 28: The development of the theory of di
Page 29 and 30: where h(λ) is arbitrary. In partic
Page 31 and 32: Suppose g (x (β)) = ∫ h (x, x
Page 33 and 34: = 1 ∫ ∫ dλ 1 2 + 1 ∫ ∫ dλ
Page 35 and 36: ( ) If we want to know the position
Page 37 and 38: 4.1.2 Vector transformations To hel
Page 39 and 40: 4.1.4 Some second rank tensors Mome
Page 41 and 42: The squared separation is therefore
Page 43 and 44: 1. V is closed under addition. If u
Page 45 and 46: 2. Functions on M. A function on a
Page 47 and 48: together with the coordinate invari
Page 49 and 50: 4.3 The metric We have already intr
Page 51 and 52: Formally, we are treating the map g
Page 53 and 54:
When we do this, the only distincti
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( 2. Express the orthonormal basis
Page 57 and 58:
By constrast, suppose we know that
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and so on so that e i ⊗ e j has a
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y a Lie group. In the next sections
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= = ∫ ∫ ( ∂L ∂x k ∂x k
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where ε i (x) is a fixed function
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is ∫ ( ) ∂L ∂L δS = δq +
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Now consider the (vanishing) variat
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so x i and v i are always parallel
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Using scalings of the variables, we
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We map this point into a 1-dim cont
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for all δx. This p i is the genera
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is constant in time for all antisym
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7 The physical Lagrangian We have s
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where M i j and ai are constant. Th
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in the last step. We can give this
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7.3 Gauging Newton’s law Newton
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There are distinct advantages to th
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8 Motion in central forces While th
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Alternatively, after solving for
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Notice that now any transform of r
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1. At any given value of the energy
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so the condition requires −Ak 2 s
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Since j must be constant while x =
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Using the force law and the angular
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8.5.1 Conic sections We can charact
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Now, for the integral ∫ I = lim a
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do this. Since the force that maint
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Notice that the magnitude is mg cos
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Finally, substituting the expressio
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so The kinetic energy is therefore
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Newtonian mechanics. this is the on
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Now, dividing by ẋ and integratin
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where v = √ v 2 then the momenta
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Our most important tool is the inva
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and τ is invariant, we can find u
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12.3 Acceleration Next we consider
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Notice that P α β = δ β α + 1
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4. Show, using the constraint freel
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Add the first two and subtract the
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Once again cycling the indices, add
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The argument is the same as above,
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and the metric is ⎛ 1 η AB = ⎜
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and demanding that W m transform to
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or in a basis (dx α , dy α )as W
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acket we find 0 = δ { t, ξ A} = {
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This means that each p n 0 is suffi
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[ V i ] A dξA = − ∂H ∂p i dt
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is just 0 = dθ = dF i ∧ dx i =
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Note that Hamilton’s equations ar
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Consider what happens to Hamilton
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= = N∑ j=1 N∑ j=1 = ∂H ∂p i
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and let π j be the momentum variab
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This establishes that replacing the
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Write the quantum wave function as
Page 165 and 166:
arbitrary functions, but we need to
Page 167 and 168:
Hamilton’s principal function is
Page 169 and 170:
Substituting into the expression fo
Page 171 and 172:
16.2 Statistics and pseudomechanics
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we see that B b ε a bc is an eleme
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17 Gauge theory Recall our progress
Page 177 and 178:
The identity is present because = A
Page 179 and 180:
in the infinitesimal transformation
Page 181 and 182:
y inserting identities to write [ A
Page 183 and 184:
These properties - a vector space w
Page 185 and 186:
matrix that depend on, say, N conti
Page 187 and 188:
−J a (J c J b ) + (J a J c ) J b
Page 189 and 190:
then For any odd-order form, ω, we
Page 191 and 192:
17.4 The exterior derivative We may
Page 193 and 194:
and further require the star to be
Page 195 and 196:
where ( ) ∂x m J = det ∂y n is
Page 197 and 198:
so ∇f = ∂f ∂ρ ˆρ + 1 ∂f
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= d ( e ijk ω i dx j dx k) = d (
Page 201 and 202:
From Maxwell’s equations, ∗ d
Page 203 and 204:
[33] Gorringe, V. M. and Leach, P.
Page 205 and 206:
[79] Morgan, J. A., Spin and statis
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Wheeler, Mechanics

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