Geometric Mechanics: Toward a Unification of Classical Physics

Cover......Page 1
Contents......Page 6
Preface......Page 16
Introduction......Page 18
Bibliography......Page 26
1.1 Preview and Rationale......Page 28
1.2 Review of Lagrangians and Hamiltonians......Page 30
1.2.1 Hamilton’s Equations in Multiple Dimensions......Page 31
1.3 Derivation of the Lagrange Equation from Hamilton’s Principle......Page 33
1.4 Linear, Multiparticle Systems......Page 35
1.4.1 The Laplace Transform Method......Page 40
1.4.2 Damped and Driven Simple Harmonic Motion......Page 41
1.5 Effective Potential and the Kepler Problem......Page 43
1.6 Multiparticle Systems......Page 46
1.7 Longitudinal Oscillation of a Beaded String......Page 49
1.7.1 Monofrequency Excitation......Page 50
1.7.2 The Continuum Limit......Page 51
1.7.2.1 Sound Waves in a Long Solid Rod......Page 52
1.8 Field Theoretical Treatment and Lagrangian Density......Page 53
1.9 Hamiltonian Density for Transverse String Motion......Page 56
1.10 String Motion Expressed as Propagating and Reflectin Waves......Page 57
1.11 Problems......Page 59
Bibliography......Page 61
2 Geometry of Mechanics, I, Linear......Page 62
2.1 Pairs of Planes as Covariant Vectors......Page 64
2.2.1 Geometric Interpretation......Page 70
2.2.2 Calculus of Differential Forms......Page 74
2.2.3 Familiar Physics Equations Expressed Using Differential Forms......Page 78
2.3.1 Vectors and Their Duals......Page 83
2.3.2 Transformation of Coordinates......Page 85
2.3.3 Transformation of Distributions......Page 89
2.3.4 Multi-index Tensors and their Contraction......Page 90
2.3.5 Representation of a Vector as a Differential Operator......Page 93
2.4.1 Euclidean Vectors......Page 96
2.4.3 Reduction of a Quadratic Form to a Sum or Difference of Squares......Page 98
2.4.4 Introduction of Covariant Components......Page 100
2.4.5 The Reciprocal Basis......Page 101
Bibliography......Page 104
3 Geometry of Mechanics, II, Curvilinear......Page 106
3.1.1 The Metric Tensor......Page 107
3.1.2 Relating Coordinate Systems at Different Points in Space......Page 109
3.1.3 The Covariant (or Absolute) Differential......Page 114
3.2 Derivation of the Lagrange Equations from the Absolute Differential......Page 119
3.2.1 Practical Evaluation of the Christoffel Symbols......Page 125
3.3 Intrinsic Derivatives and the Bilinear Covariant......Page 126
3.4.1 Lie-Dragged Coordinate Systems......Page 128
3.4.2 Lie Derivatives of Scalars and Vectors......Page 132
3.5.1 Exponential Representation of Parameterized Curves......Page 137
3.6 Identificatio of Vector Fields with Differential Operators......Page 138
3.6.1 Loop Defect......Page 139
3.7 Coordinate Congruences......Page 140
3.8 Lie-Dragged Congruences and the Lie Derivative......Page 142
3.9 Commutators of Quasi-Basis-Vectors......Page 147
Bibliography......Page 149
4.1 Generalized Euclidean Rotations and Reflection......Page 150
4.1.1 Reflecions......Page 151
4.1.2 Expressing a Rotation as a Product of Reflection......Page 152
4.1.3 The Lie Group of Rotations......Page 153
4.2.1 Volume Determined by 3- and by n-Vectors......Page 155
4.2.2 Bivectors......Page 157
4.2.3 Multivectors and Generalization to Higher Dimensionality......Page 158
4.2.4 Local Radius of Curvature of a Particle Orbit......Page 160
4.2.5 “Supplementary” Multivectors......Page 161
4.2.7 Bivectors and Infinitesima Rotations......Page 162
4.3.1 Repeated Exterior Derivatives......Page 165
4.3.2 The Gradient Formula of Vector Analysis......Page 166
4.3.3 Vector Calculus Expressed by Differential Forms......Page 168
4.3.4 Derivation of Vector Integral Formulas......Page 171
4.3.5 Generalized Divergence and Gauss’s Theorem......Page 174
4.3.6 Metric-Free Definiion of the “Divergence” of a Vector......Page 176
4.4 Spinors in Three-Dimensional Space......Page 178
4.4.2 Demonstration that a Spinor is a Euclidean Tensor......Page 179
4.4.3 Associating a 2×2 Reflection (Rotation) Matrix with a Vector (Bivector)......Page 180
4.4.5 Representations of Reflection......Page 181
4.4.6 Representations of Rotations......Page 182
4.4.7 Operations on Spinors......Page 183
4.4.9 Real Pseudo-Euclidean Space......Page 184
Bibliography......Page 185
5.1 The Poincaré Equation......Page 186
5.1.1 Some Features of the Poincaré Equations......Page 196
5.1.2 Invariance of the Poincaré Equation......Page 197
5.1.3 Translation into the Language of Forms and Vector Fields......Page 199
5.1.4 Example: Free Motion of a Rigid Body with One Point Fixed......Page 200
5.2 Variational Derivation of the Poincaré Equation......Page 203
5.3.1 Continuous Transformation Groups......Page 206
5.3.2 Use of Infiniesimal Group Parameters as Quasicoordinates......Page 210
5.3.3 Infiniesimal Group Operators......Page 212
5.3.4 Commutation Relations and Structure Constants of the Group......Page 216
5.3.5 Qualitative Aspects of Infiniesimal Generators......Page 218
5.3.6 The Poincaré Equation in Terms of Group Generators......Page 221
5.3.7 The Rigid Body Subject to Force and Torque 5.3.7.1 Infiniesimal Operators......Page 223
5.3.7.2 Description Using Body Axes......Page 225
5.3.7.3 Commutation Relations for Simultaneous Translation and Rotation......Page 229
5.3.7.4 Bowling Ball Rolling Without Slipping......Page 231
Bibliography......Page 235
6.1 Vector Mechanics 6.1.1 Vector Description in Curvilinear Coordinates......Page 236
6.1.2 The Frenet–Serret Formulas......Page 239
6.1.3 Vector Description in an Accelerating Coordinate Frame......Page 243
6.1.4 Exploiting the Fictitious Force Description......Page 249
6.1.4.1 The Reduced Three-Body Problem......Page 251
6.2 Single Particle Equations in Gauge Invariant Form......Page 255
6.2.1 Newton’s Force Equation in Gauge Invariant Form......Page 256
6.2.2 Active Interpretation of the Transformations......Page 259
6.2.3 Newton’s Torque Equation......Page 263
6.2.4 The Plumb Bob......Page 265
6.3 Gauge Invariant Description of Rigid Body Motion......Page 269
6.3.1 Space and Body Frames of Reference......Page 270
6.3.2 Review of the Association of2×2 Matrices to Vectors......Page 273
6.3.3 “Association” of 3×3 Matrices to Vectors......Page 275
6.3.4 Derivation of the Rigid Body Equations......Page 276
6.3.5 The Euler Equations for a Rigid Body......Page 278
6.4 The Foucault Pendulum......Page 279
6.4.1 Fictitious Force Solution......Page 280
6.4.2 Gauge Invariant Solution......Page 282
6.4.3 “Parallel” Translation of Coordinate Axes......Page 287
6.5 Tumblers and Divers......Page 291
Bibliography......Page 293
7 Hamiltonian Treatment of Geometric Optics......Page 294
7.1 Analogy Between Mechanics and Geometric Optics......Page 295
7.1.1 Scalar Wave Equation......Page 296
7.1.2 The Eikonal Equation......Page 298
7.1.3 Determination of Rays from Wavefronts......Page 299
7.1.4 The Ray Equation in Geometric Optics......Page 300
7.2.1 The Lagrange Integral Invariant and Snell’s Law......Page 302
7.2.2 The Principle of Least Time......Page 304
7.3 Paraxial Optics, Gaussian Optics, Matrix Optics......Page 305
7.4 Huygens’ Principle......Page 309
Bibliography......Page 311
8.1 Hamilton–Jacobi Theory Derived from Hamilton’s Principle......Page 312
8.1.1 The Geometric Picture......Page 314
8.1.2 Constant S Wavefronts......Page 315
8.2.1 Complete Integral......Page 316
8.2.2 Finding a Complete Integral by Separation of Variables......Page 317
8.2.3 Hamilton–Jacobi Analysis of Projectile Motion......Page 318
8.2.4 The Jacobi Method for Exploiting a Complete Integral......Page 319
8.2.5 Completion of Projectile Example......Page 321
8.2.6 The Time-Independent Hamilton–Jacobi Equation......Page 322
8.2.7 Hamilton–Jacobi Treatment of 1D Simple Harmonic Motion......Page 323
8.3 The Kepler Problem......Page 324
8.3.1 Coordinate Frames......Page 325
8.3.2 Orbit Elements......Page 326
8.3.3 Hamilton–Jacobi Formulation.......Page 327
8.4.1 Classical Limit of the Schrödinger Equation......Page 331
Bibliography......Page 333
9.1.1 Form Invariance......Page 334
9.1.2 World Points and Intervals......Page 335
9.1.3 Proper Time......Page 336
9.1.4 The Lorentz Transformation......Page 338
9.1.6 4-Vectors and Tensors......Page 339
9.1.8 Antisymmetric 4-Tensors......Page 342
9.1.9 The 4-Gradient, 4-Velocity, and 4-Acceleration......Page 343
9.2.1 The Relativistic Principle of Least Action......Page 344
9.2.2 Energy and Momentum......Page 345
9.2.4 Forced Motion......Page 346
9.2.5 Hamilton–Jacobi Formulation......Page 347
9.3.1 Generalization of the Action......Page 349
9.3.2 Derivation of the Lorentz Force Law......Page 351
9.3.3 Gauge Invariance......Page 352
Bibliography......Page 355
10.1 Conservation of Linear Momentum......Page 356
10.2 Rate of Change of Angular Momentum: Poincaré Approach......Page 358
10.3 Conservation of Angular Momentum: Lagrangian Approach......Page 359
10.4 Conservation of Energy......Page 360
10.5 Cyclic Coordinates and Routhian Reduction......Page 361
10.5.1 Integrability; Generalization of Cyclic Variables......Page 364
10.6 Noether’s Theorem......Page 365
10.7.1 Ignorable Coordinates and the Energy Momentum Tensor......Page 369
10.8.1 The 4-Current Density and Charge Conservation......Page 373
10.8.2 Energy and Momentum Densities......Page 377
10.9 Angular Momentum of a System of Particles......Page 379
10.10 Angular Momentum of a Field......Page 380
Bibliography......Page 381
11 Electromagnetic Theory......Page 382
11.1.1 The Lorentz Force Equation in Tensor Notation......Page 384
11.1.2 Lorentz Transformation and Invariants of the Fields......Page 386
11.2.2 The Action for the Field, Particle System......Page 387
11.2.3 The Electromagnetic Wave Equation......Page 389
11.2.4 The Inhomogeneous Pair of Maxwell Equations......Page 390
11.2.5 Energy Density, Energy Flux, and the Maxwell Stress Energy Tensor......Page 391
Bibliography......Page 394
12.1.1 Is String Theory Appropriate?......Page 396
12.1.3 Postulating a String Lagrangian......Page 398
12.2 Area Representation in Terms of the Metric......Page 400
12.3.1 A Revised Metric......Page 401
12.3.3 The Nambu–Goto Action......Page 402
12.3.4 String Tension and Mass Density......Page 404
12.4 Equations of Motion, Boundary Conditions, and Unexcited Strings......Page 406
12.5 The Action in Terms of Transverse Velocity......Page 408
12.6 Orthogonal Parameterization by Energy Content......Page 411
12.7 General Motion of a Free Open String......Page 413
12.8 A Rotating Straight String......Page 415
12.9 Conserved Momenta of a String......Page 417
12.9.1 Angular Momentum of Uniformly Rotating Straight String......Page 418
12.10 Light Cone Coordinates......Page 419
12.11 Oscillation Modes of a Relativistic String......Page 423
Bibliography......Page 425
13.1 Introduction......Page 426
13.2 Transformation to Locally Inertial Coordinates......Page 429
13.3 Parallel Transport on a Surface......Page 430
13.3.1 Geodesic Curves......Page 433
13.4 The Twin Paradox in General Relativity......Page 434
13.5 The Curvature Tensor......Page 439
13.5.1 Properties of Curvature Tensor, Ricci Tensor, and Scalar Curvature......Page 440
13.6 The Lagrangian of General Relativity and the Energy–Momentum Tensor......Page 442
13.7 “Derivation” of the Einstein Equation......Page 445
13.8 Weak, Nonrelativistic Gravity......Page 447
13.9 The Schwarzschild Metric......Page 450
13.9.1 Orbit of a Particle Subject to the Schwarzschild Metric......Page 451
13.10 Gravitational Lensing and Red Shifts......Page 454
Bibliography......Page 457
14.1.1 The Action as a Generator of Canonical Transformations......Page 458
14.2 Time-Independent Canonical Transformation......Page 463
14.3.1 The Action Variable of a Simple Harmonic Oscillator......Page 465
14.3.2 Adiabatic Invariance of the Action......Page 466
14.3.3 Action/Angle Conjugate Variables......Page 470
14.3.4 Parametrically Driven Simple Harmonic Motion......Page 472
14.4.1 Variable Length Pendulum......Page 474
14.4.2 Charged Particle in Magnetic Field......Page 476
14.4.3 Charged Particle in a Magnetic Trap......Page 478
14.5 Accuracy of Conservation of Adiabatic Invariants......Page 483
14.6 Conditionally Periodic Motion......Page 486
14.6.1 Stäckel’s Theorem......Page 487
14.6.2 Angle Variables......Page 488
14.6.3 Action/Angle Coordinates for Keplerian Satellites......Page 491
Bibliography......Page 492
15.1 Linear Hamiltonian Systems......Page 494
15.1.2 Exponentiation, Diagonalization, and Logarithm Formation of Matrices......Page 496
15.1.4 Eigensolutions......Page 498
15.2 Periodic Linear Systems......Page 501
15.2.1 Floquet’s Theorem......Page 502
15.2.3 Characteristic Multipliers, Characteristic Exponents......Page 504
15.2.4 The Variational Equations......Page 506
Bibliography......Page 507
Perturbation Theory......Page 508
16.1.1 Derivation of the Equations......Page 509
16.1.2 Relation Between Lagrange and Poisson Brackets......Page 513
16.2 Advance of Perihelion of Mercury......Page 514
16.3 Iterative Analysis of Anharmonic Oscillations......Page 519
16.4.1 First Approximation......Page 525
16.4.2 Equivalent Linearization......Page 529
16.4.3 Power Balance, Harmonic Balance......Page 531
16.4.4 Qualitative Analysis of Autonomous Oscillators......Page 532
16.4.5 Higher K–B Approximation......Page 535
16.5.1 Canonical Perturbation Theory......Page 540
16.5.2 Application to Gravity Pendulum......Page 542
16.5.3 Superconvergence......Page 544
Bibliography......Page 545
17 Symplectic Mechanics......Page 546
17.1.1 The Canonical Momentum 1-Form......Page 547
17.1.2 The Symplectic 2-Form......Page 550
17.1.3 Invariance of the Symplectic 2-Form......Page 554
17.1.4 Use of ω to Associate Vectors and 1-Forms......Page 555
17.1.5 Explicit Evaluation of Some Inner Products......Page 556
17.1.6 The Vector Field Associated with......Page 557
17.1.7 Hamilton’s Equations in Matrix Form......Page 558
17.2.1 Symplectic Products and Symplectic Bases......Page 560
17.2.2 Symplectic Transformations......Page 562
17.2.3 Properties of Symplectic Matrices......Page 563
17.3.1 The Poisson Bracket of Two Scalar Functions......Page 571
17.3.3 The Poisson Bracket and Quantum Mechanics 17.3.3.1 Commutation Relations......Page 572
17.3.3.2 Time Evolution of Expectation Values......Page 573
17.4.1 Integral Invariants in Electricity and Magnetism......Page 574
17.4.2 The Poincaré–Cartan Integral Invariant......Page 577
17.5 Invariance of the Poincaré–Cartan Integral Invariant I.I.......Page 579
17.5.1 The Extended Phase Space 2-Form and its Special Eigenvector......Page 580
17.5.2 Proof of Invariance of the Poincaré Relative Integral Invariant......Page 582
17.6 Symplectic System Evolution......Page 583
17.6.1 Liouville’s Theorem and Generalizations......Page 585
Bibliography......Page 587
Index......Page 588