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1 FUNDAMENTALS OF MECHANICS1

1.1 Elementary Kinematics1

1.1.1 Trajectories of Point Particles1

1.1.2 Position,Velocity,and Acceleration3

1.2 Principles of Dynamics5

1.2.1 Newton's Laws5

1.2.2 The Two Principles6

Principle 17

Principle 27

Discussion9

1.2.3 Consequences of Newton's Equations10

Introduction10

Force is a Vector11

1.3 One-Particle Dynamical Variables13

1.3.1 Momentum14

1.3.2 Angular Momentum14

1.3.3 Energy and Work15

In Three Dimensions15

Application to One-Dimensional Motion18

1.4 Many-Particle Systems22

1.4.1 Momentum and Center of Mass22

Center of Mass22

Momentum24

Variable Mass24

1.4.2 Energy26

1.4.3 Angular Momentum27

1.5 Examples29

1.5.1 Velocity Phase Space and Phase Portraits29

The Cosine Potential29

The Kepler Problem31

1.5.2 A System with Energy Loss34

1.5.3 Noninertial Frames and the Equivalence Principle38

Equivalence Principle38

Rotating Frames41

Problems42

2 LAGRANGIAN FORMULATION OF MECHANICS48

2.1 Constraints and Configuration Manifolds49

2.1.1 Constraints49

Constraint Equations49

Constraints and Work50

2.1.2 Generalized Coordinates54

2.1.3 Examples of Configuration Manifolds57

The Finite Line57

The Circle57

The Plane57

The Two-Sphere S257

The Double Pendulum60

Discussion60

2.2 Lagrange's Equations62

2.2.1 Derivation of Lagrange's Equations62

2.2.2 Transformations of Lagrangians67

Equivalent Lagrangians67

Coordinate Independence68

Hessian Condition69

2.2.3 Conservation of Energy70

2.2.4 Charged Particle in an Electromagnetic Field72

The Lagrangian72

A Time-Dependent Coordinate Transformation74

2.3 Central Force Motion77

2.3.1 The General Central Force Problem77

Statement of the Problem;Reduced Mass77

Reduction to Two Freedoms78

The Equivalent One-Dimensional Problem79

2.3.2 The Kepler Problem84

2.3.3 Bertrand's Theorem88

2.4 The Tangent Bundle TQ92

2.4.1 Dynamics on TQ92

Velocities Do Not Lie in Q92

Tangent Spaces and the Tangent Bundle93

Lagrange's Equations and Trajectories on TQ95

2.4.2 TQ as a Differential Manifold97

Differential Manifolds97

Tangent Spaces and Tangent Bundles100

Application to Lagrange's Equations102

Problems103

3 TOPICS IN LAGRANGIAN DYNAMICS108

3.1 The Variational Principle and Lagrange's Equations108

3.1.1 Derivation108

The Action108

Hamilton's Principle110

Discussion112

3.1.2 Inclusion of Constraints114

3.2 Symmetry and Conservation118

3.2.1 Cyclic Coordinates118

Invariant Submanifolds and Conservation of Momentum118

Transformations,Passive and Active119

Three Examples123

3.2.2 Noether's Theorem124

Point Transformations124

The Theorem125

3.3 Nonpotential Forces128

3.3.1 Dissipative Forces in the Lagrangian Formalism129

Rewriting the EL Equations129

The Dissipative and Rayleigh Functions129

3.3.2 The Damped Harmonic Oscillator131

3.3.3 Comment on Time-Dependent Forces134

3.4 A Digression on Geometry134

3.4.1 Some Geometry134

Vector Fields134

One-Forms135

The Lie Derivative136

3.4.2 The Euler-Lagrange Equations138

3.4.3 Noether's Theorem139

One-Parameter Groups139

The Theorem140

Problems143

4 SCATTERING AND LINEAR OSCILLATIONS147

4.1 Scattering147

4.1.1 Scattering by Central Forces147

General Considerations147

The Rutherford Cross Section153

4.1.2 Tne Inverse Scattering Problem154

General Treatment154

Example:Coulomb Scattering156

4.1.3 Chaotic Scattering,Cantor Sets,and Fractal Dimension157

Two Disks158

Three Disks,Cantor Sets162

Fractal Dimension and Lyapunov Exponent166

Some Further Results169

4.1.4 Scattering of a Charge by a Magnetic Dipole170

The St？rmer Problem170

The Equatorial Limit171

The General Case174

4.2 Linear Oscillations178

4.2.1 Linear Approximation:Small Vibrations178

Linearization178

Normal Modes180

4.2.2 Commensurate and Incommensurate Frequencies183

The Invariant Torus T183

The Poincaré Map185

4.2.3 A Chain of Coupled Oscillators187

General Solution187

The Finite Chain189

4.2.4 Forced and Damped Oscillators192

Forced Undamped Oscillator192

Foreed Damped Oscillator193

Problems197

5 HAMILTONIAN FORMULATION OF MECHANICS201

5.1 Hamilton's Canonical Equations202

5.1.1 Local Considerations202

From the Lagrangian to the Hamiltonian202

A Brief Review of Special Relativity207

The Relativistic Kepler Problem211

5.1.2 The Legendre Transform212

5.1.3 Unified Coordinates on T*Q and Poisson Brackets215

The ξ Notation215

Variational Derivation of Hamilton's Equations217

Poisson Brackets218

Poisson Brackets and Hamiltonian Dynamics222

5.2 Symplectic Geometry224

5.2.1 The Cotangent Manifold224

5.2.2 Two-Forms225

5.2.3 The Symplectic Form ω226

5.3 Canonical Transformations231

5.3.1 Local Considerations231

Reduction on T*Q by Constants of the Motion231

Definition of Canonical Transformations232

Changes Induced by Canonical Transformations234

Two Examples236

5.3.2 Intrinsic Approach239

5.3.3 Generating Functions of Canonical Transformations240

Generating Functions240

The Generating Functions Gives the New Hamiltonian242

Generating Functions of Type244

5.3.4 One-Parameter Groups of Canonical Transformations248

Infinitesimal Generators of One-Parameter Groups;Hamiltonian Flows249

The Hamiltonian Noether Theorem251

Flows and Poisson Brackets252

5.4 Two Theorems:Liouville and Darboux253

5.4.1 Liouville's Volume Theorem253

Volume253

Integration on T*Q;The Liouville Theorem257

Poincaré Invariants260

Density of States261

5.4.2 Darboux's Theorem268

The Theorem269

Reduction270

Problems275

Canonicity Implies PB Preservation280

6 TOPICS IN HAMILTONIAN DYNAMICS284

6.1 The Hamilton-Jacobi Method284

6.1.1 The Hamilton-Jacobi Equation285

Derivation285

Properties of Solutions286

Relation to the Action288

6.1.2 Separation of Variables290

The Method of Separation291

Example:Charged Particle in a Magnetic Field294

6.1.3 Geometry and the HJ Equation301

6.1.4 The Analogy Between Optics and the HJ Method303

6.2 Completely Integrable Systems307

6.2.1 Action-Angle Variables307

Invariant Tori307

The φαand Jα309

The Canonical Transformation to AA Variables311

Example:A Particle on a Vertical Cylinder314

6.2.2 Liouville's Integrability Theorem320

Complete Integrability320

The Tori321

The Jα323

Example:the Neumann Problem324

6.2.3 Motion on the Tori328

Rational and Irrational Winding Lines328

Fourier Series331

6.3 Perturbation Theory332

6.3.1 Example:The Quartic Oscillator;Secular Perturbation Theory332

6.3.2 Hamiltonian Perturbation Theory336

Perturbation via Canonical Transformations337

Averaging339

Canonical Perturbation Theory in One Freedom340

Canonical Perturbation Theory in Many Freedoms346

The Lie Transformation Method351

Example:The Quartic Oscillator357

6.4 Adiabatic Invariance359

6.4.1 The Adiabatic Theorem360

Oscillator with Time-Dependent Frequency360

The Theorem361

Remarks on N＞1363

6.4.2 Higher Approximations364

6.4.3 The Hannay Angle365

6.4.4 Motion of a Charged Particle in a Magnetic Field371

The Action Integral371

Three Magnetic Adiabatic Invariants374

Problems377

7 NONLINEAR DYNAMICS382

7.1 Nonlinear Oscillators383

7.1.1 A Model System383

7.1.2 Driven Quartic Oscillator386

Damped Driven Quartic Oscillator;Harmonic Analysis387

Undamped Driven Quartic Oscillator390

7.1.3 Example:The van der Pol Oscillator391

7.2 Stability of Solutions396

7.2.1 Stability of Autonomous Systems397

Definitions397

The Poincaré-Bendixon Theorem399

Linearization400

7.2.2 Stability of Nonautonomous Systems410

The Poincaré Map410

Linearization of Discrete Maps413

Example:The Linearized Hénon Map417

7.3 Parametric Oscillators418

7.3.1 Floquet Theory419

The Floquet Operator R419

Standard Basis420

Eigenvalues of R and Stability421

Dependence on G424

7.3.2 The Vertically Driven Pendulum424

The Mathieu Equation424

Stability of the Pendulum426

The Inverted Pendulum427

Damping429

7.4 Discrete Maps;Chaos431

7.4.1 The Logistic Map431

Definition432

Fixed Points432

Period Doubling434

Universality442

Further Remarks444

7.4.2 The Circle Map445

The Damped Driven Pendulum445

The Standard Sine Circle Map446

Rotation Number and the Devil's Staircase447

Fixed Points of the Circle Map450

7.5 Chaos in Hamiltonian Systems and the KAM Theorem452

7.5.1 The Kicked Rotator453

The Dynamical System453

The Standard Map454

Poincaré Map of the Perturbed System455

7.5.2 The Hénon Map460

7.5.3 Chaos in Hamiltonian Systems463

Poincaré-Birkhoff Theorem464

The Twist Map466

Numbers and Properties of the Fixed Points467

The Homoclinic Tangle468

The Transition to Chaos472

7.5.4 The KAM Theorem474

Background474

Two Conditions:Hessian and Diophantine475

The Theorem477

A Brief Description of the Proof of KAM480

Problems483

Number Theory486

The Unit Interval486

A Diophantine Condition487

The Circle and the Plane488

KAM and Continued Fractions489

8 RIGID BODIES492

8.1 Introduction492

8.1.1 Rigidity and Kinematics492

Definition492

The Angular Velocity Vector ω493

8.1.2 Kinetic Energy and Angular Momentum495

Kinetic Energy495

Angular Momentum498

8.1.3 Dynamics499

Space and Body Systems499

Dynamical Equations500

Example:The Gyrocompass503

Motion of the Angular Momentum J505

Fixed Points and Stability506

The Poinsot Construction508

8.2 The Lagrangian and Hamiltonian Formulations510

8.2.1 The Configuration Manifold QR510

Inertial,Space,and Body Systems510

The Dimension of QR511

The Structure of QR512

8.2.2 The Lagrangian514

Kinetic Energy514

The Constraints515

8.2.3 The Euler-Lagrange Equations516

Derivation516

The Angular Velocity Matrix Ω518

8.2.4 The Hamiltonian Formalism519

8.2.5 Equivalence to Euler's Equations520

Antisymmetric Matrix-Vector Correspondence520

The Torque521

The Angular Velocity Pseudovector and Kinematics522

Transformations of Velocities523

Hamilton's Canonical Equations524

8.2.6 Discussion525

8.3 Euler Angles and Spinning Tops526

8.3.1 Euler Angles526

Definition526

R in Terms of the Euler Angles527

Angular Velocities529

Discussion531

8.3.2 Geometric Phase for a Rigid Body533

8.3.3 Spinning Tops535

The Lagrangian and Hamiltonian536

The Motion of the Top537

Nutation and Precession539

Quadratic Potential;the Neumann Problem542

8.4 Cayley-Klein Parameters543

8.4.1 2×2 Matrix Representation of 3-Vectors and Rotations543

3-Vectors543

Rotations544

8.4.2 The Pauli Matrices and CK Parameters544

Definitions544

Finding RU545

Axis and Angle in terms of the CK Parameters546

8.4.3 Relation Between SU(2)and SO(3)547

Problems549

9 CONTINUUM DYNAMICS553

9.1 Lagrangian Formulation of Continuum Dynamics553

9.1.1 Passing to the Continuum Limit553

The Sine-Gordon Equation553

The Wave and Klein-Gordon Equations556

9.1.2 The Variational Principle557

Introduction557

Variational Derivation of the EL Equations557

The Functional Derivative560

Discussion560

9.1.3 Maxwell's Equations561

Some Special Relativity561

Electromagnetic Fields562

The Lagrangian and the EL Equations564

9.2 Noether's Theorem and Relativistic Fields565

9.2.1 Noether's Theorem565

The Theorem565

Conserved Currents566

Energy and Momentum in the Field567

Example:The Electromagnetic Energy-Momentum Tensor569

9.2.2 Relativistic Fields571

Lorentz Transformations571

Lorentz Invariant L and Conservation572

Free Klein-Gordon Fields576

Complex K-G Field and Interaction with the Maxwell Field577

Discussion of the Coupled Field Equations579

9.2.3 Spinors580

Spinor Fields580

A Spinor Field Equation582

9.3 The Hamiltonian Formalism583

9.3.1 The Hamiltonian Formalism for Fields583

Definitions583

The Canonical Equations584

Poisson Brackets586

9.3.2 Expansion in Orthonormal Functions588

Orthonormal Functions589

Particle-like Equations590

Example:Klein-Gordon591

9.4 Nonlinear Field Theory594

9.4.1 The Sine-Gordon Equation594

Soliton Solutions595

Properties of sG Solitons597

Multiple-Soliton Solutions599

Generating Soliton Solutions601

Nonsoliton Solutions605

Josephson Junctions608

9.4.2 The Nonlinear K-G Equation608

The Lagrangian and the EL Equation608

Kinks609

9.5 Fluid Dynamics610

9.5.1 The Euler and Navier-Stokes Equations611

Substantial Derivative and Mass Conservation611

Euler's Equation612

Viscosity and Incompressibility614

The Navier-Stokes Equations615

Turbulence616

9.5.2 The Burgers Equation618

The Equation618

Asymptotic Solution620

9.5.3 Surface Waves622

Equations for the Waves622

Linear Gravity Waves624

Nonlinear Shallow Water Waves:the KdV Equation626

Single KdV Solitons629

Multiple KdV Solitons631

9.6 Hamiltonian Formalism for Nonlinear Field Theory632

9.6.1 The Field Theory Analog of Particle Dynamics633

From Particles to Fields633

Dynamical Variables and Equations of Motion634

9.6.2 The Hamiltonian Formalism634

The Gradient634

The Symplectic Form636

The Condition for Canonicity636

Poisson Brackets636

9.6.3 The kdV Equation637

KdV as a Hamiltonian Field637

Constants of the Motion638

Generating the Constants of the Motion639

More on Constants of the Motion640

9.6.4 The Sine-Gordon Equation642

Two-Component Field Variables642

sG as a Hamiltonian Field643

Problems646

EPILOGUE648

APPENDIX:VECTOR SPACES649

General Vector Spaces649

Linear Operators651

Inverses and Eigenvalues652

Inner Products and Hermitian Operators653

BIBLIOGRAPHY656

INDEX663