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CHAPTER 1 SURVEY OF THE ELEMENTARY PRINCIPLES 1

1-1 Mechanics of a particle 1

1-2 Mechanics of a system of particles 5

1-3 Constraints 11

1-4 D’Alembert’s principle and Lagrange’s equations 17

1-5 Velocity-dependent potentials and the dissipation function 21

1-6 Simple applications of the Lagrangian formulation 25

CHAPTER 2 VARIATIONAL PRINCIPLES AND LAGRANGE’S EQUATIONS 35

2-1 Hamilton’s principle 35

2-2 Some techniques of the calculus of variations 37

2-3 Derivation of Lagrange’s equations from Hamilton’s principle 43

2-4 Extension of Hamilton’s principle to nonholonomic systems 45

2-5 Advantages of a variational principle formulation 51

2-6 Conservation theorems and symmetry properties 54

CHAPTER 3 THE TWO-BODY CENTRAL FORCE PROBLEM 70

3-1 Reduction to the equivalent one-body problem 70

3-2 The equations of motion and first integrals 71

3-3 The equivalent one-dimensional problem，and classification of orbits 77

3-4 The virial theorem 82

3-5 The differential equation for the orbit，and integrable power-law potentials 85

3-6 Conditions for closed orbits （Bertrand’s theorem） 90

3-7 The Kepler problem：Inverse square law of force 94

3-8 The motion in time in the Kepler problem 98

3-9 The Laplace-Runge-Lenz vector 102

3-10 Scattering in a central force field 105

3-11 Transformation of the scattering problem to laboratory coordinates 114

CHAPTER 4 THE KINEMATICS OF RIGID BODY MOTION 128

4-1 The independent coordinates of a rigid body 128

4-2 Orthogonal transformations 132

4-3 Formal properties of the transformation matrix 137

4-4 The Euler angles 143

4-5 The Cayley-Klein parameters and related quantities 148

4-6 Euler’s theorem on the motion of a rigid body 158

4-7 Finite rotations 164

4-8 Infinitesimal rotations 166

4-9 Rate of change of a vector 174

4-10 The Coriolis force 177

CHAPTER 5 THE RIGID BODY EQUATIONS OF MOTION 188

5-1 Angular momentum and kinetic energy of motion about a point 188

5-2 Tensors and dyadics 192

5-3 The inertia tensor and the moment of inertia 195

5-4 The eigenvalues of the inertia tensor and the principal axis transformation 198

5-5 Methods of solving rigid body problems and the Euler equations of motion 203

5-6 Torque-free motion of a rigid body 205

5-7 The heavy symmetrical top with one point fixed 213

5-8 Precession of the equinoxes and of satellite orbits 225

5-9 Precession of systems of charges in a magnetic field 232

CHAPTER 6 SMALL OSCILLATIONS 243

6-1 Formulation of the problem 243

6-2 The eigenvalue of equation and the principal axis transformation 246

6-3 Frequencies of free vibration，and normal coordinates 253

6-4 Free vibrations of a linear triatomic molecule 258

6-5 Forced vibrations and the effect of dissipative forces 263

CHAPTER 7 SPECIAL RELATIVITY IN CLASSICAL MECHANICS 275

7-1 The basic program of special relativity 275

7-2 The Lorentz transformation 278

7-3 Lorentz transformations in real four dimensional spaces 288

7-4 Further descriptions of the Lorentz transformation 293

7-5 Covariant four-dimensional formulations 298

7-6 The force and energy equations in relativistic mechanics 303

7-7 Relativistic kinematics of collisions and many-particle systems 309

7-8 The Lagrangian formulation of relativistic mechanics 320

7-9 Covariant Lagrangian formulations 326

CHAPTER 8 THE HAMILTON EQUATIONS OF MOTION 339

8-1 Legendre transformations and the Hamilton equations of motion 339

8-2 Cyclic coordinates and conservation theorems 347

8-3 Routh’s procedure and oscillations about steady motion 351

8-4 The Hamiltonian formulation of relativistic mechanics 356

8-5 Derivation of Hamilton’s equations from a variational principle 362

8-6 The principle of least action 365

CHAPTER 9 CANONICAL TRANSFORMATIONS 378

9-1 The equations of canonical transformation 378

9-2 Examples of canonical transformations 386

9-3 The symplectic approach to canonical transformations 391

9-4 Poisson brackets and other canonical invariants 397

9-5 Equations of motion，infinitesimal canonical transformations，and conservations theorems in the Poisson bracket formulation 405

9-6 The angular momentum Poisson bracket relations 416

9-7 Symmetry groups of mechanical systems 420

9-8 Liouville’stheorem 426

CHAPTER 10 HAMILTON-JACOBI THEORY 438

10-1 The Hamilton-Jacobi equation for Hamilton’s principal function 438

10-2 The harmonic oscillator problem as an example of the Hamilton-Jacobi method 442

10-3 The Hamilton-Jacobi equation for Hamilton’s characteristic function 445

10-4 Separation of variables in the Hamilton-Jacobi equation 449

10-5 Action-angle variables in systems of one degree of freedom 457

10-6 Action-angle variables for completely separable systems 463

10-7 The Kepler problem in action-angle variables 472

10-8 Hamilton-Jacobi theory，geometrical optics，and wave mechanics 484

CHAPTER 11 CANONICAL PERTURBATION THEORY 499

11-1 Introduction 499

11-2 Time-dependent perturbation （variation of constants） 500

11-3 Illustrations of time-dependent perturbation theory 506

11-4 Time-independent perturbation theory in first order with one degree of freedom 515

11-5 Time-independent perturbation theory to higher order 519

11-6 Specialized perturbation techniques in celestial and space mechanics 527

11-7 Adiabatic invariants 531

CHAPTER 12 INTRODUCTION TO THE LAGRANGIAN AND HAMILTONIAN FORMULATIONS FOR CONTINUOUS SYSTEMS AND FIELDS 545

12-1 The transition from a discrete to a continuous system 545

12-2 The Lagrangian formulation for continuous systems 548

12-3 The stress-energy tensor and conservation theorems 555

12-4 Hamiltonian formulation，Poisson brackets and the momentum representation 562

12-5 Relativistic field theory 570

12-6 Examples of relativistic field theories 575

12-7 Noetner’s theorem 588

APPENDIXES 601

A Proof of Bertrand’s Theorem 601

B Euler Angles in Alternate Conventions 606

C Transformatio？Properties of dΩ 611

D The Staeckel Conditions for Separability of the Hamilton-Jacobi Equation 613

E Lagrangian Formulation of the Acoustic Field in Gases 616

BIBLIOGRAPHY 621

INDEX OF SYMBQLS 631

INDEX 643