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CHAPTER 1 SURVEY OF THE ELEMENTARY PRINCIPLES1

1-1 Mechanics of a particle1

1-2 Mechanics of a system of particles5

1-3 Constraints11

1-4 D’Alembert’s principle and Lagrange’s equations17

1-5 Velocity-dependent potentials and the dissipation function21

1-6 Simple applications of the Lagrangian formulation25

CHAPTER 2 VARIATIONAL PRINCIPLES AND LAGRANGE’S EQUATIONS35

2-1 Hamilton’s principle35

2-2 Some techniques of the calculus of variations37

2-3 Derivation of Lagrange’s equations from Hamilton’s principle43

2-4 Extension of Hamilton’s principle to nonholonomic systems45

2-5 Advantages of a variational principle formulation51

2-6 Conservation theorems and symmetry properties54

CHAPTER 3 THE TWO-BODY CENTRAL FORCE PROBLEM70

3-1 Reduction to the equivalent one-body problem70

3-2 The equations of motion and first integrals71

3-3 The equivalent one-dimensional problem，and classification of orbits77

3-4 The virial theorem82

3-5 The differential equation for the orbit，and integrable power-law potentials85

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

3-7 The Kepler problem：Inverse square law of force94

3-8 The motion in time in the Kepler problem98

3-9 The Laplace-Runge-Lenz vector102

3-10 Scattering in a central force field105

3-11 Transformation of the scattering problem to laboratory coordinates114

CHAPTER 4 THE KINEMATICS OF RIGID BODY MOTION128

4-1 The independent coordinates of a rigid body128

4-2 Orthogonal transformations132

4-3 Formal properties of the transformation matrix137

4-4 The Euler angles143

4-5 The Cayley-Klein parameters and related quantities148

4-6 Euler’s theorem on the motion of a rigid body158

4-7 Finite rotations164

4-8 Infinitesimal rotations166

4-9 Rate of change of a vector174

4-10 The Coriolis force177

CHAPTER 5 THE RIGID BODY EQUATIONS OF MOTION188

5-1 Angular momentum and kinetic energy of motion about a point188

5-2 Tensors and dyadics192

5-3 The inertia tensor and the moment of inertia195

5-4 The eigenvalues of the inertia tensor and the principal axis transformation198

5-5 Methods of solving rigid body problems and the Euler equations of motion203

5-6 Torque-free motion of a rigid body205

5-7 The heavy symmetrical top with one point fixed213

5-8 Precession of the equinoxes and of satellite orbits225

5-9 Precession of systems of charges in a magnetic field232

CHAPTER 6 SMALL OSCILLATIONS243

6-1 Formulation of the problem243

6-2 The eigenvalue of equation and the principal axis transformation246

6-3 Frequencies of free vibration，and normal coordinates253

6-4 Free vibrations of a linear triatomic molecule258

6-5 Forced vibrations and the effect of dissipative forces263

CHAPTER 7 SPECIAL RELATIVITY IN CLASSICAL MECHANICS275

7-1 The basic program of special relativity275

7-2 The Lorentz transformation278

7-3 Lorentz transformations in real four dimensional spaces288

7-4 Further descriptions of the Lorentz transformation293

7-5 Covariant four-dimensional formulations298

7-6 The force and energy equations in relativistic mechanics303

7-7 Relativistic kinematics of collisions and many-particle systems309

7-8 The Lagrangian formulation of relativistic mechanics320

7-9 Covariant Lagrangian formulations326

CHAPTER 8 THE HAMILTON EQUATIONS OF MOTION339

8-1 Legendre transformations and the Hamilton equations of motion339

8-2 Cyclic coordinates and conservation theorems347

8-3 Routh’s procedure and oscillations about steady motion351

8-4 The Hamiltonian formulation of relativistic mechanics356

8-5 Derivation of Hamilton’s equations from a variational principle362

8-6 The principle of least action365

CHAPTER 9 CANONICAL TRANSFORMATIONS378

9-1 The equations of canonical transformation378

9-2 Examples of canonical transformations386

9-3 The symplectic approach to canonical transformations391

9-4 Poisson brackets and other canonical invariants397

9-5 Equations of motion，infinitesimal canonical transformations，and conservations theorems in the Poisson bracket formulation405

9-6 The angular momentum Poisson bracket relations416

9-7 Symmetry groups of mechanical systems420

9-8 Liouville’stheorem426

CHAPTER 10 HAMILTON-JACOBI THEORY438

10-1 The Hamilton-Jacobi equation for Hamilton’s principal function438

10-2 The harmonic oscillator problem as an example of the Hamilton-Jacobi method442

10-3 The Hamilton-Jacobi equation for Hamilton’s characteristic function445

10-4 Separation of variables in the Hamilton-Jacobi equation449

10-5 Action-angle variables in systems of one degree of freedom457

10-6 Action-angle variables for completely separable systems463

10-7 The Kepler problem in action-angle variables472

10-8 Hamilton-Jacobi theory，geometrical optics，and wave mechanics484

CHAPTER 11 CANONICAL PERTURBATION THEORY499

11-1 Introduction499

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

11-3 Illustrations of time-dependent perturbation theory506

11-4 Time-independent perturbation theory in first order with one degree of freedom515

11-5 Time-independent perturbation theory to higher order519

11-6 Specialized perturbation techniques in celestial and space mechanics527

11-7 Adiabatic invariants531

CHAPTER 12 INTRODUCTION TO THE LAGRANGIAN AND HAMILTONIAN FORMULATIONS FOR CONTINUOUS SYSTEMS AND FIELDS545

12-1 The transition from a discrete to a continuous system545

12-2 The Lagrangian formulation for continuous systems548

12-3 The stress-energy tensor and conservation theorems555

12-4 Hamiltonian formulation，Poisson brackets and the momentum representation562

12-5 Relativistic field theory570

12-6 Examples of relativistic field theories575

12-7 Noetner’s theorem588

APPENDIXES601

A Proof of Bertrand’s Theorem601

B Euler Angles in Alternate Conventions606

C Transformatio？Properties of dΩ611

D The Staeckel Conditions for Separability of the Hamilton-Jacobi Equation613

E Lagrangian Formulation of the Acoustic Field in Gases616

BIBLIOGRAPHY621

INDEX OF SYMBQLS631

INDEX643