1 FUNDAMENTALS OF MECHANICS 1
1.1 Elementary Kinematics 1
1.1.1 Trajectories of Point Particles 1
1.1.2 Position,Velocity,and Acceleration 3
1.2 Principles of Dynamics 5
1.2.1 Newton's Laws 5
1.2.2 The Two Principles 6
Principle 1 7
Principle 2 7
Discussion 9
1.2.3 Consequences of Newton's Equations 10
Introduction 10
Force is a Vector 11
1.3 One-Particle Dynamical Variables 13
1.3.1 Momentum 14
1.3.2 Angular Momentum 14
1.3.3 Energy and Work 15
In Three Dimensions 15
Application to One-Dimensional Motion 18
1.4 Many-Particle Systems 22
1.4.1 Momentum and Center of Mass 22
Center of Mass 22
Momentum 24
Variable Mass 24
1.4.2 Energy 26
1.4.3 Angular Momentum 27
1.5 Examples 29
1.5.1 Velocity Phase Space and Phase Portraits 29
The Cosine Potential 29
The Kepler Problem 31
1.5.2 A System with Energy Loss 34
1.5.3 Noninertial Frames and the Equivalence Principle 38
Equivalence Principle 38
Rotating Frames 41
Problems 42
2 LAGRANGIAN FORMULATION OF MECHANICS 48
2.1 Constraints and Configuration Manifolds 49
2.1.1 Constraints 49
Constraint Equations 49
Constraints and Work 50
2.1.2 Generalized Coordinates 54
2.1.3 Examples of Configuration Manifolds 57
The Finite Line 57
The Circle 57
The Plane 57
The Two-Sphere S2 57
The Double Pendulum 60
Discussion 60
2.2 Lagrange's Equations 62
2.2.1 Derivation of Lagrange's Equations 62
2.2.2 Transformations of Lagrangians 67
Equivalent Lagrangians 67
Coordinate Independence 68
Hessian Condition 69
2.2.3 Conservation of Energy 70
2.2.4 Charged Particle in an Electromagnetic Field 72
The Lagrangian 72
A Time-Dependent Coordinate Transformation 74
2.3 Central Force Motion 77
2.3.1 The General Central Force Problem 77
Statement of the Problem;Reduced Mass 77
Reduction to Two Freedoms 78
The Equivalent One-Dimensional Problem 79
2.3.2 The Kepler Problem 84
2.3.3 Bertrand's Theorem 88
2.4 The Tangent Bundle TQ 92
2.4.1 Dynamics on TQ 92
Velocities Do Not Lie in Q 92
Tangent Spaces and the Tangent Bundle 93
Lagrange's Equations and Trajectories on TQ 95
2.4.2 TQ as a Differential Manifold 97
Differential Manifolds 97
Tangent Spaces and Tangent Bundles 100
Application to Lagrange's Equations 102
Problems 103
3 TOPICS IN LAGRANGIAN DYNAMICS 108
3.1 The Variational Principle and Lagrange's Equations 108
3.1.1 Derivation 108
The Action 108
Hamilton's Principle 110
Discussion 112
3.1.2 Inclusion of Constraints 114
3.2 Symmetry and Conservation 118
3.2.1 Cyclic Coordinates 118
Invariant Submanifolds and Conservation of Momentum 118
Transformations,Passive and Active 119
Three Examples 123
3.2.2 Noether's Theorem 124
Point Transformations 124
The Theorem 125
3.3 Nonpotential Forces 128
3.3.1 Dissipative Forces in the Lagrangian Formalism 129
Rewriting the EL Equations 129
The Dissipative and Rayleigh Functions 129
3.3.2 The Damped Harmonic Oscillator 131
3.3.3 Comment on Time-Dependent Forces 134
3.4 A Digression on Geometry 134
3.4.1 Some Geometry 134
Vector Fields 134
One-Forms 135
The Lie Derivative 136
3.4.2 The Euler-Lagrange Equations 138
3.4.3 Noether's Theorem 139
One-Parameter Groups 139
The Theorem 140
Problems 143
4 SCATTERING AND LINEAR OSCILLATIONS 147
4.1 Scattering 147
4.1.1 Scattering by Central Forces 147
General Considerations 147
The Rutherford Cross Section 153
4.1.2 Tne Inverse Scattering Problem 154
General Treatment 154
Example:Coulomb Scattering 156
4.1.3 Chaotic Scattering,Cantor Sets,and Fractal Dimension 157
Two Disks 158
Three Disks,Cantor Sets 162
Fractal Dimension and Lyapunov Exponent 166
Some Further Results 169
4.1.4 Scattering of a Charge by a Magnetic Dipole 170
The St？rmer Problem 170
The Equatorial Limit 171
The General Case 174
4.2 Linear Oscillations 178
4.2.1 Linear Approximation:Small Vibrations 178
Linearization 178
Normal Modes 180
4.2.2 Commensurate and Incommensurate Frequencies 183
The Invariant Torus T 183
The Poincaré Map 185
4.2.3 A Chain of Coupled Oscillators 187
General Solution 187
The Finite Chain 189
4.2.4 Forced and Damped Oscillators 192
Forced Undamped Oscillator 192
Foreed Damped Oscillator 193
Problems 197
5 HAMILTONIAN FORMULATION OF MECHANICS 201
5.1 Hamilton's Canonical Equations 202
5.1.1 Local Considerations 202
From the Lagrangian to the Hamiltonian 202
A Brief Review of Special Relativity 207
The Relativistic Kepler Problem 211
5.1.2 The Legendre Transform 212
5.1.3 Unified Coordinates on T*Q and Poisson Brackets 215
The ξ Notation 215
Variational Derivation of Hamilton's Equations 217
Poisson Brackets 218
Poisson Brackets and Hamiltonian Dynamics 222
5.2 Symplectic Geometry 224
5.2.1 The Cotangent Manifold 224
5.2.2 Two-Forms 225
5.2.3 The Symplectic Form ω 226
5.3 Canonical Transformations 231
5.3.1 Local Considerations 231
Reduction on T*Q by Constants of the Motion 231
Definition of Canonical Transformations 232
Changes Induced by Canonical Transformations 234
Two Examples 236
5.3.2 Intrinsic Approach 239
5.3.3 Generating Functions of Canonical Transformations 240
Generating Functions 240
The Generating Functions Gives the New Hamiltonian 242
Generating Functions of Type 244
5.3.4 One-Parameter Groups of Canonical Transformations 248
Infinitesimal Generators of One-Parameter Groups;Hamiltonian Flows 249
The Hamiltonian Noether Theorem 251
Flows and Poisson Brackets 252
5.4 Two Theorems:Liouville and Darboux 253
5.4.1 Liouville's Volume Theorem 253
Volume 253
Integration on T*Q;The Liouville Theorem 257
Poincaré Invariants 260
Density of States 261
5.4.2 Darboux's Theorem 268
The Theorem 269
Reduction 270
Problems 275
Canonicity Implies PB Preservation 280
6 TOPICS IN HAMILTONIAN DYNAMICS 284
6.1 The Hamilton-Jacobi Method 284
6.1.1 The Hamilton-Jacobi Equation 285
Derivation 285
Properties of Solutions 286
Relation to the Action 288
6.1.2 Separation of Variables 290
The Method of Separation 291
Example:Charged Particle in a Magnetic Field 294
6.1.3 Geometry and the HJ Equation 301
6.1.4 The Analogy Between Optics and the HJ Method 303
6.2 Completely Integrable Systems 307
6.2.1 Action-Angle Variables 307
Invariant Tori 307
The φαand Jα 309
The Canonical Transformation to AA Variables 311
Example:A Particle on a Vertical Cylinder 314
6.2.2 Liouville's Integrability Theorem 320
Complete Integrability 320
The Tori 321
The Jα 323
Example:the Neumann Problem 324
6.2.3 Motion on the Tori 328
Rational and Irrational Winding Lines 328
Fourier Series 331
6.3 Perturbation Theory 332
6.3.1 Example:The Quartic Oscillator;Secular Perturbation Theory 332
6.3.2 Hamiltonian Perturbation Theory 336
Perturbation via Canonical Transformations 337
Averaging 339
Canonical Perturbation Theory in One Freedom 340
Canonical Perturbation Theory in Many Freedoms 346
The Lie Transformation Method 351
Example:The Quartic Oscillator 357
6.4 Adiabatic Invariance 359
6.4.1 The Adiabatic Theorem 360
Oscillator with Time-Dependent Frequency 360
The Theorem 361
Remarks on N＞1 363
6.4.2 Higher Approximations 364
6.4.3 The Hannay Angle 365
6.4.4 Motion of a Charged Particle in a Magnetic Field 371
The Action Integral 371
Three Magnetic Adiabatic Invariants 374
Problems 377
7 NONLINEAR DYNAMICS 382
7.1 Nonlinear Oscillators 383
7.1.1 A Model System 383
7.1.2 Driven Quartic Oscillator 386
Damped Driven Quartic Oscillator;Harmonic Analysis 387
Undamped Driven Quartic Oscillator 390
7.1.3 Example:The van der Pol Oscillator 391
7.2 Stability of Solutions 396
7.2.1 Stability of Autonomous Systems 397
Definitions 397
The Poincaré-Bendixon Theorem 399
Linearization 400
7.2.2 Stability of Nonautonomous Systems 410
The Poincaré Map 410
Linearization of Discrete Maps 413
Example:The Linearized Hénon Map 417
7.3 Parametric Oscillators 418
7.3.1 Floquet Theory 419
The Floquet Operator R 419
Standard Basis 420
Eigenvalues of R and Stability 421
Dependence on G 424
7.3.2 The Vertically Driven Pendulum 424
The Mathieu Equation 424
Stability of the Pendulum 426
The Inverted Pendulum 427
Damping 429
7.4 Discrete Maps;Chaos 431
7.4.1 The Logistic Map 431
Definition 432
Fixed Points 432
Period Doubling 434
Universality 442
Further Remarks 444
7.4.2 The Circle Map 445
The Damped Driven Pendulum 445
The Standard Sine Circle Map 446
Rotation Number and the Devil's Staircase 447
Fixed Points of the Circle Map 450
7.5 Chaos in Hamiltonian Systems and the KAM Theorem 452
7.5.1 The Kicked Rotator 453
The Dynamical System 453
The Standard Map 454
Poincaré Map of the Perturbed System 455
7.5.2 The Hénon Map 460
7.5.3 Chaos in Hamiltonian Systems 463
Poincaré-Birkhoff Theorem 464
The Twist Map 466
Numbers and Properties of the Fixed Points 467
The Homoclinic Tangle 468
The Transition to Chaos 472
7.5.4 The KAM Theorem 474
Background 474
Two Conditions:Hessian and Diophantine 475
The Theorem 477
A Brief Description of the Proof of KAM 480
Problems 483
Number Theory 486
The Unit Interval 486
A Diophantine Condition 487
The Circle and the Plane 488
KAM and Continued Fractions 489
8 RIGID BODIES 492
8.1 Introduction 492
8.1.1 Rigidity and Kinematics 492
Definition 492
The Angular Velocity Vector ω 493
8.1.2 Kinetic Energy and Angular Momentum 495
Kinetic Energy 495
Angular Momentum 498
8.1.3 Dynamics 499
Space and Body Systems 499
Dynamical Equations 500
Example:The Gyrocompass 503
Motion of the Angular Momentum J 505
Fixed Points and Stability 506
The Poinsot Construction 508
8.2 The Lagrangian and Hamiltonian Formulations 510
8.2.1 The Configuration Manifold QR 510
Inertial,Space,and Body Systems 510
The Dimension of QR 511
The Structure of QR 512
8.2.2 The Lagrangian 514
Kinetic Energy 514
The Constraints 515
8.2.3 The Euler-Lagrange Equations 516
Derivation 516
The Angular Velocity Matrix Ω 518
8.2.4 The Hamiltonian Formalism 519
8.2.5 Equivalence to Euler's Equations 520
Antisymmetric Matrix-Vector Correspondence 520
The Torque 521
The Angular Velocity Pseudovector and Kinematics 522
Transformations of Velocities 523
Hamilton's Canonical Equations 524
8.2.6 Discussion 525
8.3 Euler Angles and Spinning Tops 526
8.3.1 Euler Angles 526
Definition 526
R in Terms of the Euler Angles 527
Angular Velocities 529
Discussion 531
8.3.2 Geometric Phase for a Rigid Body 533
8.3.3 Spinning Tops 535
The Lagrangian and Hamiltonian 536
The Motion of the Top 537
Nutation and Precession 539
Quadratic Potential;the Neumann Problem 542
8.4 Cayley-Klein Parameters 543
8.4.1 2×2 Matrix Representation of 3-Vectors and Rotations 543
3-Vectors 543
Rotations 544
8.4.2 The Pauli Matrices and CK Parameters 544
Definitions 544
Finding RU 545
Axis and Angle in terms of the CK Parameters 546
8.4.3 Relation Between SU(2)and SO(3) 547
Problems 549
9 CONTINUUM DYNAMICS 553
9.1 Lagrangian Formulation of Continuum Dynamics 553
9.1.1 Passing to the Continuum Limit 553
The Sine-Gordon Equation 553
The Wave and Klein-Gordon Equations 556
9.1.2 The Variational Principle 557
Introduction 557
Variational Derivation of the EL Equations 557
The Functional Derivative 560
Discussion 560
9.1.3 Maxwell's Equations 561
Some Special Relativity 561
Electromagnetic Fields 562
The Lagrangian and the EL Equations 564
9.2 Noether's Theorem and Relativistic Fields 565
9.2.1 Noether's Theorem 565
The Theorem 565
Conserved Currents 566
Energy and Momentum in the Field 567
Example:The Electromagnetic Energy-Momentum Tensor 569
9.2.2 Relativistic Fields 571
Lorentz Transformations 571
Lorentz Invariant L and Conservation 572
Free Klein-Gordon Fields 576
Complex K-G Field and Interaction with the Maxwell Field 577
Discussion of the Coupled Field Equations 579
9.2.3 Spinors 580
Spinor Fields 580
A Spinor Field Equation 582
9.3 The Hamiltonian Formalism 583
9.3.1 The Hamiltonian Formalism for Fields 583
Definitions 583
The Canonical Equations 584
Poisson Brackets 586
9.3.2 Expansion in Orthonormal Functions 588
Orthonormal Functions 589
Particle-like Equations 590
Example:Klein-Gordon 591
9.4 Nonlinear Field Theory 594
9.4.1 The Sine-Gordon Equation 594
Soliton Solutions 595
Properties of sG Solitons 597
Multiple-Soliton Solutions 599
Generating Soliton Solutions 601
Nonsoliton Solutions 605
Josephson Junctions 608
9.4.2 The Nonlinear K-G Equation 608
The Lagrangian and the EL Equation 608
Kinks 609
9.5 Fluid Dynamics 610
9.5.1 The Euler and Navier-Stokes Equations 611
Substantial Derivative and Mass Conservation 611
Euler's Equation 612
Viscosity and Incompressibility 614
The Navier-Stokes Equations 615
Turbulence 616
9.5.2 The Burgers Equation 618
The Equation 618
Asymptotic Solution 620
9.5.3 Surface Waves 622
Equations for the Waves 622
Linear Gravity Waves 624
Nonlinear Shallow Water Waves:the KdV Equation 626
Single KdV Solitons 629
Multiple KdV Solitons 631
9.6 Hamiltonian Formalism for Nonlinear Field Theory 632
9.6.1 The Field Theory Analog of Particle Dynamics 633
From Particles to Fields 633
Dynamical Variables and Equations of Motion 634
9.6.2 The Hamiltonian Formalism 634
The Gradient 634
The Symplectic Form 636
The Condition for Canonicity 636
Poisson Brackets 636
9.6.3 The kdV Equation 637
KdV as a Hamiltonian Field 637
Constants of the Motion 638
Generating the Constants of the Motion 639
More on Constants of the Motion 640
9.6.4 The Sine-Gordon Equation 642
Two-Component Field Variables 642
sG as a Hamiltonian Field 643
Problems 646
EPILOGUE 648
APPENDIX:VECTOR SPACES 649
General Vector Spaces 649
Linear Operators 651
Inverses and Eigenvalues 652
Inner Products and Hermitian Operators 653
BIBLIOGRAPHY 656
INDEX 663