Introduction To Mechanics And Symmetry A Basic Exposition of Classical Mechanical Systems

Front
......Page 1
Preface......Page 4
About the Authors......Page 8
Contents
......Page 10
1.1 Lagrangian and Hamiltonian Formalisms......Page 15
1.2 The Rigid Body......Page 20
1.3 Lie–Poisson Brackets,Poisson Manifolds, Momentum Maps......Page 23
Exercises......Page 29
1.4 The Heavy Top......Page 30
1.5 Incompressible Fluids......Page 32
1.6 The Maxwell–Vlasov System......Page 36
1.7 Nonlinear Stability......Page 43
Exercises......Page 56
1.8 Bifurcation......Page 57
Exercises......Page 60
1.9 The Poincaré–Melnikov Method
......Page 61
Exercises......Page 63
1.10 Resonances, Geometric Phases, and Control
......Page 64
2.1 Introduction......Page 75
Exercises......Page 79
2.2 Symplectic Forms on Vector Spaces......Page 80
2.3 Canonical Transformations, or Symplectic Maps
......Page 83
Pull-Back Notation......Page 85
Exercises
......Page 87
2.4 The General Hamilton Equations......Page 88
2.5 When Are Equations Hamiltonian?......Page 91
2.6 Hamiltonian Flows......Page 94
2.7 Poisson Brackets......Page 96
2.8 A Particle in a Rotating Hoop......Page 101
Exercises......Page 106
2.9 The Poincaré–Melnikov Method
......Page 108
3.1 Lagrange’s and Hamilton’s Equations for Field Theory
......Page 119
3.2 Examples: Hamilton’s Equations......Page 121
Exercises......Page 128
3.3 Examples: Poisson Brackets and Conserved Quantities
......Page 129
Exercises......Page 134
4.1 Manifolds......Page 135
Exercises......Page 142
4.2 Differential Forms......Page 143
Vector Calculus and Differential Forms......Page 148
Exercises......Page 149
4.3 The Lie Derivative......Page 151
Exercises......Page 154
4.4 Stokes’ Theorem......Page 155
Identities for Vector Fields and Forms......Page 157
Exercises......Page 159
5.1 Symplectic Manifolds......Page 161
5.2 Symplectic Transformations......Page 164
Exercises......Page 165
5.3 Complex Structures and Kähler Manifolds
......Page 166
Exercises......Page 170
5.4 Hamiltonian Systems......Page 171
Exercises......Page 173
5.5 Poisson Brackets on Symplectic Manifolds
......Page 174
Exercises......Page 177
6.1 The Linear Case......Page 179
6.2 The Nonlinear Case......Page 181
Exercises......Page 183
6.3 Cotangent Lifts......Page 184
Exercises......Page 186
6.4 Lifts of Actions......Page 187
6.5 Generating Functions......Page 188
6.6 Fiber Translations and Magnetic Terms......Page 190
6.7 A Particle in a Magnetic Field......Page 192
Exercises......Page 194
7.1 Hamilton’s Principle of Critical Action......Page 195
7.2 The Legendre Transform......Page 197
7.3 Euler–Lagrange Equations......Page 199
7.4 Hyperregular Lagrangians and Hamiltonians
......Page 202
7.5 Geodesics......Page 209
Exercises......Page 213
7.6 The Kaluza–Klein Approach to Charged Particles
......Page 214
7.7 Motion in a Potential Field......Page 216
7.8 The Lagrange–d’Alembert Principle......Page 219
7.9 The Hamilton–Jacobi Equation......Page 224
Exercises......Page 232
8.1 A Return to Variational Principles......Page 233
8.2 The Geometry of Variational Principles......Page 240
8.3 Constrained Systems......Page 248
Exercises......Page 251
8.4 Constrained Motion in a Potential Field......Page 252
8.5 Dirac Constraints......Page 256
8.6 Centrifugal and Coriolis Forces......Page 262
8.7 The Geometric Phase for a Particle in a Hoop
......Page 267
Exercises......Page 270
8.8 Moving Systems......Page 271
Exercises......Page 273
8.9 Routh Reduction......Page 274
Exercises......Page 277
9 An Introduction to Lie Groups
......Page 279
9.1 Basic Definitions and Properties......Page 281
Exercises......Page 296
9.2 Some Classical Lie Groups......Page 297
9.3 Actions of Lie Groups......Page 323
Exercises......Page 340
10.1 The Definition of Poisson Manifolds......Page 341
Exercises......Page 345
10.2 Hamiltonian Vector Fields and Casimir Functions
......Page 347
10.3 Properties of Hamiltonian Flows......Page 352
10.4 The Poisson Tensor......Page 354
Exercises......Page 362
10.5 Quotients of Poisson Manifolds......Page 363
Exercises......Page 366
10.6 The Schouten Bracket......Page 367
Exercises......Page 373
10.7 Generalities on Lie–Poisson Structures......Page 374
Exercises......Page 378
11.1 Canonical Actions and Their Infinitesimal Generators
......Page 379
11.2 Momentum Maps......Page 381
11.3 An Algebraic Definition of the Momentum Map
......Page 384
11.4 Conservation of Momentum Maps......Page 386
Exercises......Page 391
11.5 Equivariance of Momentum Maps......Page 392
Exercises......Page 395
12.1 Momentum Maps on Cotangent Bundles
......Page 397
Exercises......Page 402
12.2 Examples of Momentum Maps......Page 403
Exercises......Page 409
12.3 Equivariance and Infinitesimal Equivariance
......Page 410
Exercises......Page 416
12.4 Equivariant Momentum Maps Are Poisson
......Page 417
Exercises......Page 425
12.5 Poisson Automorphisms......Page 426
12.6 Momentum Maps and Casimir Functions
......Page 427
Exercises......Page 429
13.1 The Lie–Poisson Reduction Theorem......Page 431
13.2 Proof of the Lie–Poisson Reduction Theorem for GL(n)
......Page 434
13.3 Lie–Poisson Reduction Using Momentum Functions
......Page 435
13.4 Reduction and Reconstruction of Dynamics
......Page 437
13.5 The Euler–Poincaré Equations
......Page 446
13.6 The Lagrange–Poincaré Equations
......Page 456
14 Coadjoint Orbits
......Page 459
14.1 Examples of Coadjoint Orbits......Page 460
Exercises......Page 466
14.2 Tangent Vectors to Coadjoint Orbits......Page 467
14.3 The Symplectic Structure on Coadjoint Orbits
......Page 469
Exercises......Page 474
14.4 The Orbit Bracket via Restriction of the Lie–Poisson Bracket
......Page 475
Exercises......Page 480
14.5 The Special Linear Group of the Plane......Page 481
14.6 The Euclidean Group of the Plane......Page 483
14.7 The Euclidean Group of Three-Space......Page 488
Exercises......Page 495
15.1 Material, Spatial, and Body Coordinates
......Page 497
15.2 The Lagrangian of the Free Rigid Body......Page 499
15.3 The Lagrangian and Hamiltonian for the Rigid Body in Body Representation
......Page 501
15.4 Kinematics on Lie Groups......Page 505
15.5 Poinsot’s Theorem......Page 506
Exercises......Page 508
15.6 Euler Angles......Page 509
15.7 The Hamiltonian of the Free Rigid Body in the Material Description via Euler Angles
......Page 511
Exercises......Page 513
15.8 The Analytical Solution of the Free Rigid-Body Problem
......Page 514
15.9 Rigid-Body Stability......Page 519
Exercises......Page 522
15.10 Heavy Top Stability......Page 523
15.11 The Rigid Body and the Pendulum......Page 528
Exercises......Page 533
References......Page 535
Index......Page 569
Supplement......Page 599
Contents......Page 601
Preface......Page 603
N6.A Linearization of Hamiltonian Systems......Page 605
N7.A The Classical Limit and the MaslovIndex......Page 611
N9.A Automatic Smoothness......Page 625
N9.B Abelian Lie Groups......Page 627
N9.C Lie Subgroups......Page 628
N9.D Lie’s Third Fundamental Theorem......Page 631
N9.E Relations between the Symplectic, Orthogonal, and Unitary Groups
......Page 633
N9.F Generic Coadjoint Isotropy Subalgebras are Abelian
......Page 642
N9.G Some Infinite Dimensional Lie Groups......Page 646
N10.A Proof of the Symplectic Stratification Theorem
......Page 669
N11.A Another Example of a Momentum Map
......Page 673
N13.A Proof of the Lie–Poisson Reduction Theorem for Diff_vol(M)
......Page 675
N13.B Proof of the Lie–Poisson Reduction Theorem for Diff_can(P)
......Page 677
N13.C The Linearized Lie–PoissonEquations......Page 680
Exercises......Page 683
N14.A Casimir Functions do not Determine Orbits
......Page 685
References......Page 689