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24 <strong>Applied</strong> <strong>Diff</strong>erential <strong>Geom</strong>etry: A Modern Introductionand (q 1 , t 1 ) fixed. 29The total energy function called Hamiltonian, denoted by H, is obtainedby performing a Legendre transformation on the Lagrangian. 30 TheHamiltonian is the basis for an alternative formulation of classical mechanicsknown as Hamiltonian mechanics (see below).In 1948, R.P. Feynman invented the path–integral formulation extendingthe principle of least action to quantum mechanics for electrons and pho-29 More generally, a Lagrangian L[ϕ i ] of a dynamical system is a function of the dynamicalvariables ϕ i (x) and concisely describes the equations of motion of the systemin coordinates x i , (i = 1, ..., n). The equations of motion are obtained by means of anaction principle, written asδSδϕ i= 0,where the action is a functionalZS[ϕ i ] = L[ϕ i (s)] d n x,(d n x = dx 1 ...dx n ).The equations of motion obtained by means of the functional derivative are identical tothe usual Euler–Lagrange equations. Dynamical system whose equations of motion areobtainable by means of an action principle on a suitably chosen Lagrangian are knownas Lagrangian dynamical systems. Examples of Lagrangian dynamical systems rangefrom the (classical version of the) Standard Model, to Newton’s equations, to purelymathematical problems such as geodesic equations and the Plateau’s problem.The Lagrangian mechanics is important not just for its broad applications, but alsofor its role in advancing deep understanding of physics. Although Lagrange sought todescribe classical mechanics, the action principle that is used to derive the Lagrange’sequation is now recognized to be deeply tied to quantum mechanics: physical action andquantum–mechanical phase (waves) are related via Planck’s constant, and the Principleof stationary action can be understood in terms of constructive interference of wavefunctions. The same principle, and the Lagrangian formalism, are tied closely to NoetherTheorem, which relates physical conserved quantities to continuous symmetries of aphysical system; and Lagrangian mechanics and Noether’s Theorem together yield anatural formalism for first quantization by including commutators between certain termsof the Lagrange’s equations of motion for a physical system.More specifically, in field theory, occasionally a distinction is made between the LagrangianL, of which the action is the time integral S = R Ldt and the Lagrangiandensity L, which one integrates over all space–time to get the 4D action:ZS[ϕ i ] = L[ϕ i (x)] d 4 x.The Lagrangian is then the spatial integral of the Lagrangian density.30 The Hamiltonian is the Legendre transform of the Lagrangian:H (q, p, t) = X i˙q i p i − L(q, ˙q, t).
Introduction 23Such a force is independent of third– or higher–order derivatives of ⃗r, soNewton’s Second Law forms a set of 3 second–order ODEs. Therefore,the motion of the particle can be completely described by 6 independentvariables, or degrees of freedom (DOF). An obvious set of variables is theCartesian components of ⃗r and their time derivatives, at a given instant oftime, that is position (x, y, z) and velocity (v x , v y , v z ).More generally, we can work with a set of generalized coordinates,q i , (i = 1, ..., n), and their time derivatives, the generalized velocities, ˙q i .The position vector ⃗r is related to the generalized coordinates by sometransformation equation: ⃗r = ⃗r(q i , t). The term ‘generalized coordinates’is really a leftover from the period when Cartesian coordinates were thedefault coordinate system. In the q i −coordinates the Lagrange’s equationsread:∂L∂q i = d ∂Ldt ∂ ˙q i ,where L = T − V is the system’s Lagrangian.The time integral of the Lagrangian L, denoted S is called the action: 28∫S = L dt.Let q 0 and q 1 be the coordinates at respective initial and final times t 0 andt 1 . Using the calculus of variations, it can be shown the Lagrange’s equationsare equivalent to the Hamilton’s principle: “The system undergoesthe trajectory between t 0 and t 1 whose action has a stationary value.” Thisis formally written:δS = 0,where by ‘stationary’, we mean that the action does not vary to first–orderfor infinitesimal deformations of the trajectory, with the end–points (q 0 , t 0 )28 The action principle is an assertion about the nature of motion, from which thetrajectory of a dynamical system subject to some forces can be determined. The pathof an object is the one that yields a stationary value for a quantity called the action.Thus, instead of thinking about an object accelerating in response to applied forces,one might think of them picking out the path with a stationary action. The action isa scalar (a number) with the unit of measure for Action as Energy × Time. Althoughequivalent in classical mechanics with Newton’s laws, the action principle is better suitedfor generalizations and plays an important role in modern physics. Indeed, this principleis one of the great generalizations in physical science. In particular, it is fully appreciatedand best understood within quantum mechanics. Richard Feynman’s path integralformulation of quantum mechanics is based on a stationary–action principle, using pathintegrals. Maxwell’s equations can be derived as conditions of stationary action.
24 <strong>Applied</strong> <strong>Diff</strong>erential <strong>Geom</strong>etry: A Modern Introductionand (q 1 , t 1 ) fixed. 29The total energy function called Hamiltonian, denoted by H, is obtainedby performing a Legendre transformation on the Lagrangian. 30 TheHamiltonian is the basis for an alternative formulation of classical mechanicsknown as Hamiltonian mechanics (see below).In 1948, R.P. Feynman invented the path–integral formulation extendingthe principle of least action to quantum mechanics for electrons and pho-29 More generally, a Lagrangian L[ϕ i ] of a dynamical system is a function of the dynamicalvariables ϕ i (x) and concisely describes the equations of motion of the systemin coordinates x i , (i = 1, ..., n). The equations of motion are obtained by means of anaction principle, written asδSδϕ i= 0,where the action is a functionalZS[ϕ i ] = L[ϕ i (s)] d n x,(d n x = dx 1 ...dx n ).The equations of motion obtained by means of the functional derivative are identical tothe usual Euler–Lagrange equations. Dynamical system whose equations of motion areobtainable by means of an action principle on a suitably chosen Lagrangian are knownas Lagrangian dynamical systems. Examples of Lagrangian dynamical systems rangefrom the (classical version of the) Standard Model, to Newton’s equations, to purelymathematical problems such as geodesic equations and the Plateau’s problem.The Lagrangian mechanics is important not just for its broad applications, but alsofor its role in advancing deep understanding of physics. Although Lagrange sought todescribe classical mechanics, the action principle that is used to derive the Lagrange’sequation is now recognized to be deeply tied to quantum mechanics: physical action andquantum–mechanical phase (waves) are related via Planck’s constant, and the Principleof stationary action can be understood in terms of constructive interference of wavefunctions. The same principle, and the Lagrangian formalism, are tied closely to NoetherTheorem, which relates physical conserved quantities to continuous symmetries of aphysical system; and Lagrangian mechanics and Noether’s Theorem together yield anatural formalism for first quantization by including commutators between certain termsof the Lagrange’s equations of motion for a physical system.More specifically, in field theory, occasionally a distinction is made between the LagrangianL, of which the action is the time integral S = R Ldt and the Lagrangiandensity L, which one integrates over all space–time to get the 4D action:ZS[ϕ i ] = L[ϕ i (x)] d 4 x.The Lagrangian is then the spatial integral of the Lagrangian density.30 The Hamiltonian is the Legendre transform of the Lagrangian:H (q, p, t) = X i˙q i p i − L(q, ˙q, t).
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APPLIEDDIFFERENTIALGEOMETRYA Modern
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APPLIEDDIFFERENTIALGEOMETRYA Modern
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Dedicated to:Nitya, Atma and Kali
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Chapter 3Applied Manifold Geometry3
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Applied Manifold Geometry 139where
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Applied Manifold Geometry 141In loc
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Applied Manifold Geometry 143on the
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Applied Manifold Geometry 145exist
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Applied Manifold Geometry 147then E
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Applied Manifold Geometry 1490D man
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Applied Manifold Geometry 151Fig. 3
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Applied Manifold Geometry 153the fo
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Applied Manifold Geometry 155contin
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Applied Manifold Geometry 157In oth
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Applied Manifold Geometry 159scalar
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Applied Manifold Geometry 161where
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Applied Manifold Geometry 1633.6 Te
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Applied Manifold Geometry 165Let ϕ
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Applied Manifold Geometry 1673.6.1.
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Applied Manifold Geometry 169or, if
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Applied Manifold Geometry 171If X i
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Applied Manifold Geometry 173along
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Applied Manifold Geometry 177where
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Applied Manifold Geometry 179(1) α
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Applied Manifold Geometry 181diagra
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Applied Manifold Geometry 1853.6.3.
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Applied Manifold Geometry 187duces,
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Applied Manifold Geometry 2273.8.5
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Applied Manifold Geometry 229in whi
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Applied Manifold Geometry 233itly:(
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Applied Manifold Geometry 239Basic
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Applied Manifold Geometry 241orthog
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Applied Manifold Geometry 243is adm
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Applied Manifold Geometry 245Root S
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Applied Manifold Geometry 247groups
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Applied Manifold Geometry 2493.9.1.
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Applied Manifold Geometry 251second
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Applied Manifold Geometry 253where
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Applied Manifold Geometry 255be a v
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Applied Manifold Geometry 257Now, a
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Applied Manifold Geometry 259This i
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Applied Manifold Geometry 2613.9.4
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Applied Manifold Geometry 263other
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Applied Manifold Geometry 265where
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Applied Manifold Geometry 267result
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Applied Manifold Geometry 269where
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Applied Manifold Geometry 2713.10 R
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Applied Manifold Geometry 273More p
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Applied Manifold Geometry 275Y (t 0
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Applied Manifold Geometry 277which
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Applied Manifold Geometry 279for an
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Applied Manifold Geometry 281where
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Applied Manifold Geometry 283Thus w
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Applied Manifold Geometry 2853.10.2
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Applied Manifold Geometry 287evolut
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Applied Manifold Geometry 2893.10.3
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Applied Manifold Geometry 2913.10.3
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Applied Manifold Geometry 293‘ren
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Applied Manifold Geometry 295mute;
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Applied Manifold Geometry 297Rather
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Applied Manifold Geometry 299of the
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Applied Manifold Geometry 301clidea
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Applied Manifold Geometry 305i.e.,
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Applied Manifold Geometry 307invari
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Applied Manifold Geometry 309The Ba
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Applied Manifold Geometry 311Let us
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Applied Manifold Geometry 313all cu
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Applied Manifold Geometry 315ders.
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Applied Manifold Geometry 317this c
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Applied Manifold Geometry 319In par
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Applied Manifold Geometry 323the fo
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Applied Manifold Geometry 325per (1
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Applied Manifold Geometry 327Genera
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Applied Manifold Geometry 331pretat
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Applied Manifold Geometry 333byMDL
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Applied Manifold Geometry 3353.12 S
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Applied Manifold Geometry 337H 2 (M
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Applied Manifold Geometry 339bracke
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Applied Manifold Geometry 341Any so
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Applied Manifold Geometry 343All on
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Applied Manifold Geometry 345The ac
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Applied Manifold Geometry 347linear
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Applied Manifold Geometry 349such t
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Applied Manifold Geometry 351Let γ
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Applied Manifold Geometry 353This i
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Applied Manifold Geometry 355chart
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Applied Manifold Geometry 357phase-
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Applied Manifold Geometry 359( ) 0
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Applied Manifold Geometry 361hetero
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Applied Manifold Geometry 363thus c
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Applied Manifold Geometry 365Action
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Applied Manifold Geometry 367and we
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Applied Manifold Geometry 369action
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Applied Manifold Geometry 371and we
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Applied Manifold Geometry 373by (J(
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Applied Manifold Geometry 375fields
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Applied Manifold Geometry 377Let Tx
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Applied Manifold Geometry 379where
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Applied Manifold Geometry 381The ve
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Applied Manifold Geometry 383and gi
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Applied Manifold Geometry 385The co
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Applied Manifold Geometry 387tor Ca
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Applied Manifold Geometry 389where
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Applied Manifold Geometry 3912 −
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Applied Manifold Geometry 393comple
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Applied Manifold Geometry 395F[C]
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Applied Manifold Geometry 397become
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Applied Manifold Geometry 399of fuz
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Applied Manifold Geometry 401In our
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Applied Manifold Geometry 403Biodyn
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Applied Manifold Geometry 405for T
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Applied Manifold Geometry 407A vari
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Applied Manifold Geometry 409such t
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Applied Manifold Geometry 411of neg
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Applied Manifold Geometry 413where
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Applied Manifold Geometry 415Follow
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Applied Manifold Geometry 417and J
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Applied Manifold Geometry 419This t
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Applied Manifold Geometry 421above,
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Applied Manifold Geometry 423As a L
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Applied Manifold Geometry 425Theore
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Applied Manifold Geometry 427real v
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Applied Manifold Geometry 429indepe
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Applied Manifold Geometry 431In ter
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Applied Manifold Geometry 433M, we
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Applied Manifold Geometry 435The cu
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Applied Manifold Geometry 437(iv) (
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Applied Manifold Geometry 439where
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. ..Applied Manifold Geometry 441in
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Applied Manifold Geometry 443⌋ is
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Applied Manifold Geometry 445Now, a
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Applied Manifold Geometry 447symmet
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Applied Manifold Geometry 449study
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Applied Manifold Geometry 451togeth
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Applied Manifold Geometry 453result
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Applied Manifold Geometry 455will b
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Applied Manifold Geometry 457g 11 =
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Applied Manifold Geometry 459is def
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Applied Manifold Geometry 461ordere
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Applied Manifold Geometry 463Now, r
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Applied Manifold Geometry 465Simila
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Applied Manifold Geometry 467which
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Applied Manifold Geometry 469From t
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Applied Manifold Geometry 471where
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Applied Manifold Geometry 4733.17.2
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Applied Manifold Geometry 475On the
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Applied Manifold Geometry 477ing di
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Applied Manifold Geometry 479Note t
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Applied Manifold Geometry 481Then w
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Applied Manifold Geometry 483consis
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Chapter 4Applied Bundle Geometry4.1
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Applied Bundle Geometry 487is calle
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Applied Bundle Geometry 489V and an
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Applied Bundle Geometry 4914.3 Vect
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Applied Bundle Geometry 493dimensio
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Applied Bundle Geometry 495orthogon
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Applied Bundle Geometry 497t, is a
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Applied Bundle Geometry 499Let T Y
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Applied Bundle Geometry 501In other
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Applied Bundle Geometry 503we shall
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Applied Bundle Geometry 505Pendulum
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Applied Bundle Geometry 507Using L
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Applied Bundle Geometry 509the isom
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Applied Bundle Geometry 511theory f
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Applied Bundle Geometry 513of the c
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Applied Bundle Geometry 515where Ω
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Applied Bundle Geometry 517nals, th
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Applied Bundle Geometry 519(1985)])
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Applied Bundle Geometry 521finite-d
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Applied Bundle Geometry 523and thes
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Applied Bundle Geometry 525Now, let
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Applied Bundle Geometry 527gravity
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Applied Bundle Geometry 529a Dp−b
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Applied Bundle Geometry 531A princi
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Applied Bundle Geometry 533is the r
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Applied Bundle Geometry 535Now, let
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Applied Bundle Geometry 537By cycli
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Applied Bundle Geometry 5394.9.2 Fe
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Applied Bundle Geometry 541This lin
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Applied Bundle Geometry 543(1) L g
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Applied Bundle Geometry 545and ω 0
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Applied Bundle Geometry 547isfiesω
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Applied Bundle Geometry 549of the c
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Applied Bundle Geometry 551Controll
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Applied Bundle Geometry 553Foliatio
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Applied Bundle Geometry 555ψ : (x,
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Applied Bundle Geometry 557from res
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Applied Bundle Geometry 559Motion a
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Applied Bundle Geometry 561where Z
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Applied Bundle Geometry 563Systems
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Applied Bundle Geometry 5653. Recal
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Applied Bundle Geometry 567Therefor
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Applied Bundle Geometry 569smooth m
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Applied Bundle Geometry 571state va
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Applied Bundle Geometry 573The geom
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Applied Bundle Geometry 575Y ∋ y
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Applied Bundle Geometry 577autonomo
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Applied Bundle Geometry 579(see 3.1
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Applied Bundle Geometry 583potentia
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Applied Bundle Geometry 585effector
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Applied Bundle Geometry 587Fig. 4.9
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Applied Bundle Geometry 589physical
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Applied Bundle Geometry 591open Lio
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Applied Bundle Geometry 593et al. (
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Applied Bundle Geometry 595(2) Acti
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Applied Bundle Geometry 597˙Φ ≤
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Applied Bundle Geometry 599number o
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Applied Bundle Geometry 601where Ψ
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Applied Bundle Geometry 603ing to c
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Applied Bundle Geometry 605where [
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Applied Bundle Geometry 607interval
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Applied Bundle Geometry 609The foll
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Applied Bundle Geometry 611In parti
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Applied Bundle Geometry 615(2) δ
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Applied Bundle Geometry 617of unita
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Applied Bundle Geometry 621of H pre
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Applied Bundle Geometry 623It follo
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Applied Bundle Geometry 625a produc
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Applied Bundle Geometry 627B Ham(M)
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Applied Bundle Geometry 629b.Let us
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Applied Bundle Geometry 631If P is
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Applied Bundle Geometry 633question
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Applied Bundle Geometry 635spheres
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Applied Bundle Geometry 639of the b
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Applied Bundle Geometry 641in C 1 (
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Applied Bundle Geometry 643from a t
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Applied Bundle Geometry 645Now let
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Applied Bundle Geometry 647for t <
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Applied Bundle Geometry 649is a Ham
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Applied Bundle Geometry 651The foll
- Page 688 and 689:
Applied Bundle Geometry 653is the u
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Applied Bundle Geometry 655every no
- Page 692 and 693:
Applied Bundle Geometry 657P × P w
- Page 694 and 695:
Applied Bundle Geometry 659that the
- Page 696 and 697:
Applied Bundle Geometry 661extend t
- Page 698 and 699:
Applied Bundle Geometry 663identity
- Page 700 and 701:
Applied Bundle Geometry 665The mapw
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Applied Bundle Geometry 667It is a
- Page 704 and 705:
Applied Bundle Geometry 669I Q is i
- Page 706 and 707:
Applied Bundle Geometry 671whereC i
- Page 708 and 709:
Applied Bundle Geometry 673form Q.
- Page 710 and 711:
Applied Bundle Geometry 675τ( exp(
- Page 712 and 713:
Applied Bundle Geometry 6774.13.2.2
- Page 714 and 715:
Applied Bundle Geometry 679Therefor
- Page 716 and 717:
Applied Bundle Geometry 6814.13.2.3
- Page 718 and 719:
Applied Bundle Geometry 683⎛⎞co
- Page 720 and 721:
Applied Bundle Geometry 685Usually,
- Page 722 and 723:
Applied Bundle Geometry 687on the s
- Page 724 and 725:
Applied Bundle Geometry 6894.13.2.5
- Page 726 and 727:
Applied Bundle Geometry 6914.13.3 P
- Page 728 and 729:
Applied Bundle Geometry 693Since T
- Page 730 and 731:
Applied Bundle Geometry 695On the o
- Page 732 and 733:
Applied Bundle Geometry 697may writ
- Page 734 and 735:
Applied Bundle Geometry 699determin
- Page 736 and 737:
Applied Bundle Geometry 701section,
- Page 738 and 739:
Applied Bundle Geometry 703for deri
- Page 740 and 741:
Applied Bundle Geometry 705dictions
- Page 742 and 743:
Applied Bundle Geometry 707too sing
- Page 744 and 745:
Applied Bundle Geometry 709In (4.18
- Page 746 and 747:
Applied Bundle Geometry 7114.13.4.4
- Page 748 and 749:
Applied Bundle Geometry 713sis repr
- Page 750 and 751:
Applied Bundle Geometry 7154.13.4.6
- Page 752 and 753:
Applied Bundle Geometry 717a (linea
- Page 754 and 755:
Applied Bundle Geometry 719past few
- Page 756 and 757:
Applied Bundle Geometry 721global q
- Page 758 and 759:
Applied Bundle Geometry 7239. It ha
- Page 760 and 761:
Applied Bundle Geometry 725needs to
- Page 762 and 763:
Applied Bundle Geometry 727pear at
- Page 764 and 765:
Applied Bundle Geometry 729X, F A i
- Page 766 and 767:
Applied Bundle Geometry 731ifolds,
- Page 768 and 769:
Applied Bundle Geometry 733and not
- Page 770 and 771:
Applied Bundle Geometry 735under U(
- Page 772 and 773:
Applied Bundle Geometry 737N = 2 br
- Page 774 and 775:
Applied Bundle Geometry 739thusa
- Page 776 and 777:
Applied Bundle Geometry 741antisymm
- Page 778 and 779:
Applied Bundle Geometry 743see this
- Page 780 and 781:
Applied Bundle Geometry 745Coupling
- Page 782 and 783:
Applied Bundle Geometry 747acts as
- Page 784 and 785:
Applied Bundle Geometry 749good coo
- Page 786 and 787:
Applied Bundle Geometry 751Φ. Brea
- Page 788 and 789:
Applied Bundle Geometry 753equating
- Page 790 and 791:
Applied Bundle Geometry 755differen
- Page 792 and 793:
Applied Bundle Geometry 7574.14.10
- Page 794 and 795:
Applied Bundle Geometry 759deformed
- Page 796 and 797:
Applied Bundle Geometry 761phenomen
- Page 798 and 799:
Applied Bundle Geometry 763appears
- Page 800 and 801:
Applied Bundle Geometry 7654.14.10.
- Page 802 and 803:
Applied Bundle Geometry 767For J >
- Page 804 and 805:
Applied Bundle Geometry 769K 1/2
- Page 806 and 807:
Applied Bundle Geometry 771(i) The
- Page 808 and 809:
Applied Bundle Geometry 773by α =
- Page 810 and 811:
Applied Bundle Geometry 775(4.279)
- Page 812 and 813:
Applied Bundle Geometry 777Λ ⊗
- Page 814 and 815:
Applied Bundle Geometry 779coupling
- Page 816 and 817:
Applied Bundle Geometry 781where we
- Page 818 and 819:
Applied Bundle Geometry 783The posi
- Page 820 and 821:
Applied Bundle Geometry 785Although
- Page 822 and 823:
Applied Bundle Geometry 787Let us a
- Page 824 and 825:
Applied Bundle Geometry 789of the t
- Page 826 and 827:
Applied Bundle Geometry 791The Riem
- Page 828 and 829:
Applied Bundle Geometry 793To compl
- Page 830 and 831:
Applied Bundle Geometry 795Ω 1 :Ω
- Page 832 and 833:
Chapter 5Applied Jet GeometryModern
- Page 834 and 835:
Applied Jet Geometry 799x, with coe
- Page 836 and 837:
Applied Jet Geometry 801s i (x), as
- Page 838 and 839:
Applied Jet Geometry 803It is conve
- Page 840 and 841:
Applied Jet Geometry 805has the 1
- Page 842 and 843:
Applied Jet Geometry 807which split
- Page 844 and 845:
Applied Jet Geometry 809gives the h
- Page 846 and 847:
Applied Jet Geometry 811Furthermore
- Page 848 and 849:
Applied Jet Geometry 813This is a c
- Page 850 and 851:
Applied Jet Geometry 815Due to this
- Page 852 and 853:
Applied Jet Geometry 817The affine
- Page 854 and 855:
Applied Jet Geometry 819In particul
- Page 856 and 857:
Applied Jet Geometry 821Building on
- Page 858 and 859:
Applied Jet Geometry 823on J ∞ (X
- Page 860 and 861:
Applied Jet Geometry 825its tangent
- Page 862 and 863:
Applied Jet Geometry 827Fig. 5.3 Hi
- Page 864 and 865:
Applied Jet Geometry 829It follows
- Page 866 and 867:
Applied Jet Geometry 831affine, whi
- Page 868 and 869:
Applied Jet Geometry 833where Γ i
- Page 870 and 871:
Applied Jet Geometry 8355.6.5 Jacob
- Page 872 and 873:
Applied Jet Geometry 837is also the
- Page 874 and 875:
Applied Jet Geometry 839A generic m
- Page 876 and 877:
Applied Jet Geometry 841where ϑ eH
- Page 878 and 879:
Applied Jet Geometry 843Since E L|
- Page 880 and 881:
Applied Jet Geometry 845Thus, every
- Page 882 and 883:
Applied Jet Geometry 847along u. Ev
- Page 884 and 885:
Applied Jet Geometry 849equations c
- Page 886 and 887:
Applied Jet Geometry 851A Hamiltoni
- Page 888 and 889:
Applied Jet Geometry 853Connections
- Page 890 and 891:
Applied Jet Geometry 855Lagrangian
- Page 892 and 893:
Applied Jet Geometry 857on V ∗ M.
- Page 894 and 895:
Applied Jet Geometry 859contain no
- Page 896 and 897:
Applied Jet Geometry 861(ii) For an
- Page 898 and 899:
Applied Jet Geometry 8635.6.16 Lyap
- Page 900 and 901:
Applied Jet Geometry 865for any t >
- Page 902 and 903:
Applied Jet Geometry 867vertical co
- Page 904 and 905:
Applied Jet Geometry 869difficulty,
- Page 906 and 907:
Applied Jet Geometry 871the energy
- Page 908 and 909:
Applied Jet Geometry 873by the tran
- Page 910 and 911:
Applied Jet Geometry 875The conditi
- Page 912 and 913:
Applied Jet Geometry 877Let Γ = (M
- Page 914 and 915:
Applied Jet Geometry 879geometrical
- Page 916 and 917:
Applied Jet Geometry 881Let (M, I)
- Page 918 and 919:
Applied Jet Geometry 883so referrin
- Page 920 and 921:
Applied Jet Geometry 885The Poincar
- Page 922 and 923:
Applied Jet Geometry 887where N n
- Page 924 and 925:
Applied Jet Geometry 889be a projec
- Page 926 and 927:
Applied Jet Geometry 891to differen
- Page 928 and 929:
Applied Jet Geometry 893for every c
- Page 930 and 931:
Applied Jet Geometry 895vector-fiel
- Page 932 and 933:
Applied Jet Geometry 897This is phr
- Page 934 and 935:
Applied Jet Geometry 899where a, b
- Page 936 and 937:
Applied Jet Geometry 901represent t
- Page 938 and 939:
Applied Jet Geometry 903̂L : J 1 (
- Page 940 and 941:
Applied Jet Geometry 905space J 1 (
- Page 942 and 943:
Applied Jet Geometry 907First of al
- Page 944 and 945:
Applied Jet Geometry 909Therefore,
- Page 946 and 947:
Applied Jet Geometry 911Recall that
- Page 948 and 949:
Applied Jet Geometry 9135.11 Applic
- Page 950 and 951:
Applied Jet Geometry 915the vector
- Page 952 and 953:
Applied Jet Geometry 9175.11.4 Gaug
- Page 954 and 955:
Applied Jet Geometry 9195.11.5 Lagr
- Page 956 and 957:
Applied Jet Geometry 921are represe
- Page 958 and 959:
Applied Jet Geometry 923which are t
- Page 960 and 961:
Applied Jet Geometry 925of this fun
- Page 962 and 963:
Applied Jet Geometry 927a principal
- Page 964 and 965:
Applied Jet Geometry 929Recall that
- Page 966 and 967:
Applied Jet Geometry 931Moreover, t
- Page 968 and 969:
Applied Jet Geometry 933other only
- Page 970 and 971:
Applied Jet Geometry 935In particul
- Page 972 and 973:
Applied Jet Geometry 937Note that,
- Page 974 and 975:
Applied Jet Geometry 939This equali
- Page 976 and 977:
Applied Jet Geometry 941Then, the s
- Page 978 and 979:
Applied Jet Geometry 943In the Lagr
- Page 980 and 981:
Applied Jet Geometry 945Let τ be a
- Page 982 and 983:
Applied Jet Geometry 947Using (5.42
- Page 984 and 985:
Applied Jet Geometry 949associated
- Page 986 and 987:
Applied Jet Geometry 951parts,F α
- Page 988 and 989:
Applied Jet Geometry 953of the Hilb
- Page 990 and 991:
Applied Jet Geometry 955For example
- Page 992 and 993:
Applied Jet Geometry 957the convent
- Page 994 and 995:
Applied Jet Geometry 959left ideal
- Page 996 and 997:
Applied Jet Geometry 961LX → Σ i
- Page 998 and 999:
Applied Jet Geometry 963Levi-Civita
- Page 1000 and 1001:
Applied Jet Geometry 965see this fr
- Page 1002 and 1003:
Applied Jet Geometry 967curves. And
- Page 1004 and 1005:
Applied Jet Geometry 969One sees th
- Page 1006 and 1007:
Applied Jet Geometry 971sums over a
- Page 1008 and 1009:
Applied Jet Geometry 973all this to
- Page 1010 and 1011:
Applied Jet Geometry 975universe ev
- Page 1012 and 1013:
Applied Jet Geometry 977on Σ. If M
- Page 1014 and 1015:
Applied Jet Geometry 979(1) Lorentz
- Page 1016 and 1017:
Applied Jet Geometry 981terpret thi
- Page 1018 and 1019:
Chapter 6Geometrical Path Integrals
- Page 1020 and 1021:
Geometrical Path Integrals and Thei
- Page 1022 and 1023:
Geometrical Path Integrals and Thei
- Page 1024 and 1025:
Geometrical Path Integrals and Thei
- Page 1026 and 1027:
Geometrical Path Integrals and Thei
- Page 1028 and 1029:
Geometrical Path Integrals and Thei
- Page 1030 and 1031:
Geometrical Path Integrals and Thei
- Page 1032 and 1033:
Geometrical Path Integrals and Thei
- Page 1034 and 1035:
Geometrical Path Integrals and Thei
- Page 1036 and 1037:
Geometrical Path Integrals and Thei
- Page 1038 and 1039:
Geometrical Path Integrals and Thei
- Page 1040 and 1041:
Geometrical Path Integrals and Thei
- Page 1042 and 1043:
Geometrical Path Integrals and Thei
- Page 1044 and 1045:
Geometrical Path Integrals and Thei
- Page 1046 and 1047:
Geometrical Path Integrals and Thei
- Page 1048 and 1049:
Geometrical Path Integrals and Thei
- Page 1050 and 1051:
Geometrical Path Integrals and Thei
- Page 1052 and 1053:
Geometrical Path Integrals and Thei
- Page 1054 and 1055:
Geometrical Path Integrals and Thei
- Page 1056 and 1057:
Geometrical Path Integrals and Thei
- Page 1058 and 1059:
Geometrical Path Integrals and Thei
- Page 1060 and 1061:
Geometrical Path Integrals and Thei
- Page 1062 and 1063:
Geometrical Path Integrals and Thei
- Page 1064 and 1065:
Geometrical Path Integrals and Thei
- Page 1066 and 1067:
Geometrical Path Integrals and Thei
- Page 1068 and 1069:
Geometrical Path Integrals and Thei
- Page 1070 and 1071:
Geometrical Path Integrals and Thei
- Page 1072 and 1073:
Geometrical Path Integrals and Thei
- Page 1074 and 1075:
Geometrical Path Integrals and Thei
- Page 1076 and 1077:
Geometrical Path Integrals and Thei
- Page 1078 and 1079:
Geometrical Path Integrals and Thei
- Page 1080 and 1081:
Geometrical Path Integrals and Thei
- Page 1082 and 1083:
Geometrical Path Integrals and Thei
- Page 1084 and 1085:
Geometrical Path Integrals and Thei
- Page 1086 and 1087:
Geometrical Path Integrals and Thei
- Page 1088 and 1089:
Geometrical Path Integrals and Thei
- Page 1090 and 1091:
Geometrical Path Integrals and Thei
- Page 1092 and 1093:
Geometrical Path Integrals and Thei
- Page 1094 and 1095:
Geometrical Path Integrals and Thei
- Page 1096 and 1097:
Geometrical Path Integrals and Thei
- Page 1098 and 1099:
Geometrical Path Integrals and Thei
- Page 1100 and 1101:
Geometrical Path Integrals and Thei
- Page 1102 and 1103:
Geometrical Path Integrals and Thei
- Page 1104 and 1105:
Geometrical Path Integrals and Thei
- Page 1106 and 1107:
Geometrical Path Integrals and Thei
- Page 1108 and 1109:
Geometrical Path Integrals and Thei
- Page 1110 and 1111:
Geometrical Path Integrals and Thei
- Page 1112 and 1113:
Geometrical Path Integrals and Thei
- Page 1114 and 1115:
Geometrical Path Integrals and Thei
- Page 1116 and 1117:
Geometrical Path Integrals and Thei
- Page 1118 and 1119:
Geometrical Path Integrals and Thei
- Page 1120 and 1121:
Geometrical Path Integrals and Thei
- Page 1122 and 1123:
Geometrical Path Integrals and Thei
- Page 1124 and 1125:
Geometrical Path Integrals and Thei
- Page 1126 and 1127:
Geometrical Path Integrals and Thei
- Page 1128 and 1129:
Geometrical Path Integrals and Thei
- Page 1130 and 1131:
Geometrical Path Integrals and Thei
- Page 1132 and 1133:
Geometrical Path Integrals and Thei
- Page 1134 and 1135:
Geometrical Path Integrals and Thei
- Page 1136 and 1137:
Geometrical Path Integrals and Thei
- Page 1138 and 1139:
Geometrical Path Integrals and Thei
- Page 1140 and 1141:
Geometrical Path Integrals and Thei
- Page 1142 and 1143:
Geometrical Path Integrals and Thei
- Page 1144 and 1145:
Geometrical Path Integrals and Thei
- Page 1146 and 1147:
Geometrical Path Integrals and Thei
- Page 1148 and 1149:
Geometrical Path Integrals and Thei
- Page 1150 and 1151:
Geometrical Path Integrals and Thei
- Page 1152 and 1153:
Geometrical Path Integrals and Thei
- Page 1154 and 1155:
Geometrical Path Integrals and Thei
- Page 1156 and 1157:
Geometrical Path Integrals and Thei
- Page 1158 and 1159:
Geometrical Path Integrals and Thei
- Page 1160 and 1161:
Geometrical Path Integrals and Thei
- Page 1162 and 1163:
Geometrical Path Integrals and Thei
- Page 1164 and 1165:
Geometrical Path Integrals and Thei
- Page 1166 and 1167:
Geometrical Path Integrals and Thei
- Page 1168 and 1169:
Geometrical Path Integrals and Thei
- Page 1170 and 1171:
Geometrical Path Integrals and Thei
- Page 1172 and 1173:
Geometrical Path Integrals and Thei
- Page 1174 and 1175:
Geometrical Path Integrals and Thei
- Page 1176 and 1177:
Geometrical Path Integrals and Thei
- Page 1178 and 1179:
Geometrical Path Integrals and Thei
- Page 1180 and 1181:
Geometrical Path Integrals and Thei
- Page 1182 and 1183:
Geometrical Path Integrals and Thei
- Page 1184 and 1185:
Geometrical Path Integrals and Thei
- Page 1186 and 1187:
Geometrical Path Integrals and Thei
- Page 1188 and 1189:
Geometrical Path Integrals and Thei
- Page 1190 and 1191:
Geometrical Path Integrals and Thei
- Page 1192 and 1193:
Geometrical Path Integrals and Thei
- Page 1194 and 1195:
Geometrical Path Integrals and Thei
- Page 1196 and 1197:
Geometrical Path Integrals and Thei
- Page 1198 and 1199:
Geometrical Path Integrals and Thei
- Page 1200 and 1201:
Geometrical Path Integrals and Thei
- Page 1202 and 1203:
Geometrical Path Integrals and Thei
- Page 1204 and 1205:
Geometrical Path Integrals and Thei
- Page 1206 and 1207:
Geometrical Path Integrals and Thei
- Page 1208 and 1209:
Geometrical Path Integrals and Thei
- Page 1210 and 1211:
Geometrical Path Integrals and Thei
- Page 1212 and 1213:
Geometrical Path Integrals and Thei
- Page 1214 and 1215:
Geometrical Path Integrals and Thei
- Page 1216 and 1217:
Geometrical Path Integrals and Thei
- Page 1218 and 1219:
Geometrical Path Integrals and Thei
- Page 1220 and 1221:
Geometrical Path Integrals and Thei
- Page 1222 and 1223:
Geometrical Path Integrals and Thei
- Page 1224 and 1225:
Geometrical Path Integrals and Thei
- Page 1226 and 1227:
Geometrical Path Integrals and Thei
- Page 1228 and 1229:
Geometrical Path Integrals and Thei
- Page 1230 and 1231:
Geometrical Path Integrals and Thei
- Page 1232 and 1233:
Geometrical Path Integrals and Thei
- Page 1234 and 1235:
Geometrical Path Integrals and Thei
- Page 1236 and 1237:
Geometrical Path Integrals and Thei
- Page 1238 and 1239:
Geometrical Path Integrals and Thei
- Page 1240 and 1241:
Geometrical Path Integrals and Thei
- Page 1242 and 1243:
Geometrical Path Integrals and Thei
- Page 1244 and 1245:
Geometrical Path Integrals and Thei
- Page 1246 and 1247:
Geometrical Path Integrals and Thei
- Page 1248 and 1249:
Geometrical Path Integrals and Thei
- Page 1250 and 1251:
Geometrical Path Integrals and Thei
- Page 1252 and 1253:
Geometrical Path Integrals and Thei
- Page 1254 and 1255:
Geometrical Path Integrals and Thei
- Page 1256 and 1257:
Geometrical Path Integrals and Thei
- Page 1258 and 1259:
Geometrical Path Integrals and Thei
- Page 1260 and 1261:
Geometrical Path Integrals and Thei
- Page 1262 and 1263:
Geometrical Path Integrals and Thei
- Page 1264 and 1265:
Geometrical Path Integrals and Thei
- Page 1266 and 1267:
Geometrical Path Integrals and Thei
- Page 1268 and 1269:
Geometrical Path Integrals and Thei
- Page 1270 and 1271:
Geometrical Path Integrals and Thei
- Page 1272 and 1273:
Geometrical Path Integrals and Thei
- Page 1274 and 1275:
Geometrical Path Integrals and Thei
- Page 1276 and 1277:
Geometrical Path Integrals and Thei
- Page 1278 and 1279:
Geometrical Path Integrals and Thei
- Page 1280 and 1281:
Geometrical Path Integrals and Thei
- Page 1282 and 1283:
Geometrical Path Integrals and Thei
- Page 1284 and 1285:
Geometrical Path Integrals and Thei
- Page 1286 and 1287:
Geometrical Path Integrals and Thei
- Page 1288 and 1289:
BibliographyAblowitz, M.J., Clarkso
- Page 1290 and 1291:
Bibliography 1255equations. Springe
- Page 1292 and 1293:
Bibliography 1257connections, Phys.
- Page 1294 and 1295:
Bibliography 1259JHEP 0003:007.Bran
- Page 1296 and 1297:
Bibliography 1261Clementi, C. Petti
- Page 1298 and 1299:
Bibliography 1263Domany, E., Van He
- Page 1300 and 1301:
Bibliography 1265geometrical signat
- Page 1302 and 1303:
Bibliography 1267Bergmann theory of
- Page 1304 and 1305:
Bibliography 1269to Complex Systems
- Page 1306 and 1307:
Bibliography 1271Action for String
- Page 1308 and 1309:
Bibliography 1273Ivancevic, V., Bea
- Page 1310 and 1311:
Bibliography 1275Krener, A. (1984).
- Page 1312 and 1313:
Bibliography 1277Li, M., Vitanyi, P
- Page 1314 and 1315:
Bibliography 1279Nature, 261(5560),
- Page 1316 and 1317:
Bibliography 1281Lambert and R. Gol
- Page 1318 and 1319:
Bibliography 1283dynamics of human
- Page 1320 and 1321:
Bibliography 1285Reshetikhin, N.Yu,
- Page 1322 and 1323:
Bibliography 1287Rev. E, 58, 6333-6
- Page 1324 and 1325:
Bibliography 1289Stasheff, J.D. (19
- Page 1326 and 1327:
Bibliography 1291matical Physics. S
- Page 1328 and 1329:
Bibliography 1293Witten, E. (1991).
- Page 1330 and 1331:
Index1−jet bundle, 8271−jet lif
- Page 1332 and 1333:
Index 1297Chan-Paton factor, 1238Ch
- Page 1334 and 1335:
Index 1299Dirac-Born-Infeld theory,
- Page 1336 and 1337:
Index 1301Gauss, 5Gauss map, 284Gau
- Page 1338 and 1339:
Index 1303isometric, 17isomorphic,
- Page 1340 and 1341:
Index 1305Maxwell Lagrangian, 34Max
- Page 1342 and 1343:
Index 1307probability manifold, 333
- Page 1344 and 1345:
Index 1309super-space, 1215supercov
- Page 1346:
Index 1311Yang-Mills theory, 39, 29
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