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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