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1．Elementary Newtonian Mechanics 1

1.1 Newton's Laws（1687）and Their Interpretation 1

1.2 Uniform Rectilinear Motion and Inertial Systems 4

1.3 Inertial Frames in Relative Motion 6

1.4 Momentum and Force 6

1.5 Typical Forces.A Remark About Units 8

1.6 Space，Time，and Forces 10

1.7 The Two-Body System with Internal Forces 11

1.7.1 Center-of-Mass and Relative Motion 11

1.7.2 Example:The Gravitational Force Between Two Celestial Bodies（Kepler's Problem） 13

1.7.3 Center-of-Mass and Relative Momentum in the Two-Body System 19

1.8 Systems of Finitely Many Particles 20

1.9 The Principle of Center-of-Mass Motion 21

1.10 The Principle of Angular-Momentum Conservation 21

1.11 The Principle of Energy Conservation 22

1.12 The Closed n-Particle System 23

1.13 Galilei Transformations 24

1.14 Space and Time with Galilei Invariance 27

1.15 Conservative Force Fields 29

1.16 One-Dimensional Motion of a Point Particle 32

1.17 Examples of Motion in One Dimension 34

1.17.1 The Harmonic Oscillator 34

1.17.2 The Planar Mathematical Pendulum 36

1.18 Phase Space for the n-Particle System（in R3） 37

1.19 Existence and Uniqueness of the Solutions of？＝？（？，t） 38

1.20 Physical Consequences of the Existence and Uniqueness Theorem 40

1.21 Linear Systems 42

1.21.1 Linear，Homogeneous Systems 42

1.21.2 Linear，Inhomogeneous Systems 43

1.22 Integrating One-Dimensional Equations of Motion 43

1.23 Example:The Planar Pendulum for Arbitrary Deviations from the Vertical 45

1.24 Example:The Two-Body System with a Central Force 48

1.25 Rotating Reference Systems:Coriolis and Centrifugal Forces 55

1.26 Examples of Rotating Reference Systems 56

1.27 Scattering of Two Particles that Interact via a Central Force:Kinematics 64

1.28 Two-Particle Scattering with a Central Force:Dynamics 68

1.29 Example:Coulomb Scattering of Two Particles with Equal Mass and Charge 72

1.30 Mechanical Bodies of Finite Extension 76

1.31 Time Averages and the Virial Theorem 80

Appendix:Practical Examples 82

2．The Principles of Canonical Mechanics 89

2.1 Constraints and Generalized Coordinates 89

2.1.1 Definition of Constraints 89

2.1.2 Generalized Coordinates 91

2.2 D'Alembert's Principle 91

2.2.1 Definition of Virtual Displacements 91

2.2.2 The Static Case 92

2.2.3 The Dynamical Case 92

2.3 Lagrange's Equations 94

2.4 Examples of the Use of Lagrange's Equations 95

2.5 A Digression on Variational Principles 97

2.6 Hamilton's Variational Principle（1834） 100

2.7 The Euler-Lagrange Equations 100

2.8 Further Examples of the Use of Lagrange's Equations 101

2.9 A Remark About Nonuniqueness of the Lagrangian Function 103

2.10 Gauge Transformations of the Lagrangian Function 104

2.11 Admissible Transformations of the Generalized Coordinates 105

2.12 The Hamiltonian Function and Its Relation to the Lagrangian Function L 106

2.13 The Legendre Transformation for the Case of One Variable 107

2.14 The Legendre Transformation for the Case of Several Variables 109

2.15 Canonical Systems 110

2.16 Examples of Canonical Systems 111

2.17 The Variational Principle Applied to the Hamiltonian Function 113

2.18 Symmetries and Conservation Laws 114

2.19 Noether's Theorem 115

2.20 The Generator for Infinitesimal Rotations About an Axis 117

2.21 More About the Rotation Group 119

2.22 Infinitesimal Rotations and Their Generators 121

2.23 Canonical Transformations 123

2.24 Examples of Canonical Transformations 127

2.25 The Structure of the Canonical Equations 128

2.26 Example:Linear Autonomous Systems in One Dimension 129

2.27 Canonical Transformations in Compact Notation 131

2.28 On the Symplectic Structure of Phase Space 133

2.29 Liouville's Theorem 136

2.29.1 The Local Form 137

2.29.2 The Global Form 138

2.30 Examples for the Use of Liouville's Theorem 139

2.31 Poisson Brackets 142

2.32 Properties of Poisson Brackets 145

2.33 Infinitesimal Canonical Transformations 147

2.34 Integrals of the Motion 148

2.35 The Hamilton-Jacobi Differential Equation 151

2.36 Examples for the Use of the Hamilton-Jacobi Equation 152

2.37 The Hamilton-Jacobi Equation and Integrable Systems 156

2.37.1 Local Rectification of Hamiltonian Systems 156

2.37.2 Integrable Systems 160

2.37.3 Angle and Action Variables 165

2.38 Perturbing Quasiperiodic Hamiltonian Systems 166

2.39 Autonomous，Nondegenerate Hamiltonian Systems in the Neighborhood of Integrable Systems 169

2.40 Examples.The Averaging Principle 170

2.40.1 The Anharmonic Oscillator 170

2.40.2 Averaging of Perturbations 172

2.41 Generalized Theorem of Noether 174

Appendix:Practical Examples 182

3．The Mechanics of Rigid Bodies 187

3.1 Definition of Rigid Body 187

3.2 Infinitesimal Displacement of a Rigid Body 189

3.3 Kinetic Energy and the Inertia Tensor 191

3.4 Properties of the Inertia Tensor 193

3.5 Steiner's Theorem 197

3.6 Examples of the Use of Steiner's Theorem 198

3.7 Angular Momentum of a Rigid Body 203

3.8 Force-Free Motion of Rigid Bodies 205

3.9 Another Parametrization of Rotations:The Euler Angles 207

3.10 Definition of Eulerian Angles 209

3.11 Equations of Motion of Rigid Bodies 210

3.12 Euler's Equations of Motion 213

3.13 Euler's Equations Applied to a Force-Free Top 216

3.14 The Motion of a Free Top and Geometric Constructions 220

3.15 The Rigid Body in the Framework of Canonical Mechanics 223

3.16 Example:The Symmetric Children's Top in a Gravitational Field 227

3.17 More About the Spinning Top 229

3.18 Spherical Top with Friction:The"Tippe Top" 231

3.18.1 Conservation Law and Energy Considerations 232

3.18.2 Equations of Motion and Solutions with Constant Energy 234

Appendix:Practical Examples 238

4．Relativistic Mechanics 241

4.1 Failures of Nonrelativistic Mechanics 242

4.2 Constancy of the Speed of Light 245

4.3 The Lorentz Transformations 246

4.4 Analysis of Lorentz and Poincaré Transformations 252

4.4.1 Rotations and Special Lorentz Tranformations（"Boosts"） 254

4.4.2 Interpretation of Special Lorentz Transformations 258

4.5 Decomposition of Lorentz Transformations into Their Components 259

4.5.1 Proposition on Orthochronous，Proper Lorentz Transformations 259

4.5.2 Corollary of the Decomposition Theorem and Some Consequences 261

4.6 Addition of Relativistic Velocities 264

4.7 Galilean and Lorentzian Space-Time Manifolds 266

4.8 Orbital Curves and Proper Time 270

4.9 Relativistic Dynamics 272

4.9.1 Newton's Equation 272

4.9.2 The Energy-Momentum Vector 274

4.9.3 The Lorentz Force 277

4.10 Time Dilatation and Scale Contraction 279

4.11 More About the Motion of Free Particles 281

4.12 The Conformal Group 284

5．Geometric Aspects of Mechanics 285

5.1 Manifolds of Generalized Coordinates 286

5.2 Differentiable Manifolds 289

5.2.1 The Euclidean Space Rn 289

5.2.2 Smooth or Differentiable Manifolds 291

5.2.3 Examples of Smooth Manifolds 293

5.3 Geometrical Objects on Manifolds 297

5.3.1 Functions and Curves on Manifolds 298

5.3.2 Tangent Vectors on a Smooth Manifold 300

5.3.3 The Tangent Bundle of a Manifold 302

5.3.4 Vector Fields on Smooth Manifolds 303

5.3.5 Exterior Forms 307

5.4 Calculus on Manifolds 309

5.4.1 Differentiable Mappings of Manifolds 309

5.4.2 Integral Curves of Vector Fields 311

5.4.3 Exterior Product of One-Forms 313

5.4.4 The Exterior Derivative 315

5.4.5 Exterior Derivative and Vectors in R3 317

5.5 Hamilton-Jacobi and Lagrangian Mechanics 319

5.5.1 Coordinate Manifold Q，Velocity Space TQ，and Phase Space TQ 319

5.5.2 The Canonical One-Form on Phase Space 323

5.5.3 The Canonical，Symplectic Two-Form on M 326

5.5.4 Symplectic Two-Form and Darboux's Theorem 328

5.5.5 The Canonical Equations 331

5.5.6 The Poisson Bracket 334

5.5.7 Time-Dependent Hamiltonian Systems 337

5.6 Lagrangian Mechanics and Lagrange Equations 339

5.6.1 The Relation Between the Two Formulations of Mechanics 339

5.6.2 The Lagrangian Two-Form 341

5.6.3 Energy Function on TQ and Lagrangian Vector Field 342

5.6.4 Vector Fields on Velocity Space TQ and Lagrange Equations 344

5.6.5 The Legendre Transformation and the Correspondence of Lagrangian and Hamiltonian Functions 346

5.7 Riemannian Manifolds in Mechanics 349

5.7.1 Affine Connection and Parallel Transport 350

5.7.2 Parallel Vector Fields and Geodesics 352

5.7.3 Geodesics as Solutions of Euler-Lagrange Equations 353

5.7.4 Example:Force-Free Asymmetric Top 354

6．Stability and Chaos 357

6.1 Qualitative Dynamics 357

6.2 Vector Fields as Dynamical Systems 358

6.2.1 Some Definitions of Vector Fields and Their Integral Curves 360

6.2.2 Equilibrium Positions and Linearization of Vector Fields 362

6.2.3 Stability of Equilibrium Positions 365

6.2.4 Critical Points of Hamiltonian Vector Fields 369

6.2.5 Stability and Instability of the Free Top 371

6.3 Long-Term Behavior of Dynamical Flows and Dependence on External Parameters 373

6.3.1 Flows in Phase Space 374

6.3.2 More General Criteria for Stability 375

6.3.3 Attractors 378

6.3.4 The Poincaré Mapping 382

6.3.5 Bifurcations of Flows at Critical Points 386

6.3.6 Bifurcations of Periodic Orbits 390

6.4 Deterministic Chaos 392

6.4.1 Iterative Mappings in One Dimension 392

6.4.2 Qualitative Definitions of Deterministic Chaos 394

6.4.3 An Example:The Logistic Equation 398

6.5 Quantitative Measures of Deterministic Chaos 403

6.5.1 Routes to Chaos 403

6.5.2 Liapunov Characteristic Exponents 407

6.5.3 Strange Attractors 409

6.6 Chaotic Motions in Celestial Mechanics 411

6.6.1 Rotational Dynamics of Planetary Satellites 411

6.6.2 Orbital Dynamics of Asteroids with Chaotic Behavior 417

7．Continuous Systems 421

7.1 Discrete and Continuous Systems 421

7.2 Transition to the Continuous System 425

7.3 Hamilton's Variational Principle for Continuous Systems 427

7.4 Canonically Conjugate Momentum and Hamiltonian Density 429

7.5 Example:The Pendulum Chain 430

7.6 Comments and Outlook 434

Exercises 439

Chapter 1:Elementary Newtonian Mechanics 439

Chapter 2:The Principles of Canonical Mechanics 446

Chapter 3:The Mechanics of Rigid Bodies 454

Chapter 4:Relativistic Mechanics 457

Chapter 5:Geometric Aspects of Mechanics 460

Chapter 6:Stability and Chaos 463

Solution of Exercises 467

Chapter 1:Elementary Newtonian Mechanics 467

Chapter 2:The Principles of Canonical Mechanics 483

Chapter 3:The Mechanics of Rigid Bodies 503

Chapter 4:Relativistic Mechanics 511

Chapter 5:Geometric Aspects of Mechanics 523

Chapter 6:Stability and Chaos 528

Appendix 537

A．Some Mathematical Notions 537

B．Historical Notes 540

Bibliography 547

Index 549