《Mathematical Methods of Classical Mechanics》

Part I NEWTONIAN MECHANICS1

Chapter 1 Experimental facts3

1. The principles of relativity and determinacy3

2. The galilean group and Newton's equations4

3. Examples of mechanical systems11

Chapter 2 Investigation of the equations of motion15

4. Systems with one degree of freedom15

5. Systems with two degrees of freedom22

6. Conservative force fields28

7. Angular momentum30

8. Investigation of motion in a central field33

9. The motion of a point in three-space42

10. Motions of a system of n pomts44

11. The method of similarity50

Part Ⅱ LAGRANGIAN MECHANICS53

Chapter 3 Variational principles55

12. Calculus of variations55

13. Lagrange's equations59

14. Legendre transformations61

15. Hamilton's equations65

16. Liouville's theorem68

Chapter 4 Lagrangian mechanics on manifolds75

17. Holonomic constraints75

18. Differentiable manifolds77

19. Lagrangian dynamical systems83

20. E. Noether's theorem88

21. D'Alembert's principle91

Chapter 5 Oscillations98

22. Linearization98

23. Small oscillations103

24. Behavior of characteristic frequencies110

25. Parametric resonance113

Chapter 6 Rigid Bodies123

26. Motion in a moving coordinate system123

27. Inertial forces and the Coriolis force129

28. Rigid bodies133

29. Euler's equations. Poinsot's description of the motion142

30. Lagrange's top148

31. Sleeping tops and fast tops154

Part Ⅲ HAMILTONIAN MECHANICS161

Chapter 7 Differential forms163

32. Exterior forms163

33. Exterior multiplication170

34. Differential forms174

35. Integration of differential forms181

36. Exterior differentiation188

Chapter 8 Symplectic manifolds201

37. Symplectic structures on manifolds201

38. Hamiltonian phase flows and their integral invariants204

39. The Lie algebra of vector fields208

40. The Lie algebra of hamiltonian functions214

41. Symplectic geometry219

42. Parametric resonance in systems with many degrees of freedom225

43. A symplectic atlas229

Chapter 9 Canonical formalism233

44. The integral invariant of Poincare-Cartan233

45. Applications of the integral invariant of Poincare-Cartan240

46. Huygens' principle248

47. The Hamilton-Jacobi method for integrating Hamilton's canonical equations258

48. Generating functions266

Chapter 10 Introduction to perturbation theory271

49. Integrable systems271

50. Action-angle variables279

51. Averaging285

52. Averaging of perturbations291

Appendix 1 Riemannian curvature301

Appendix 2 Geodesics of left-invariant metrics on Lie groups and the hydrodynamics of ideal fluids318

Appendix 3 Symplectic structures on algebraic manifolds343

Appendix 4 Contact structures349

Appendix 5 Dynamical systems with symmetries371

Appendix 6 Normal forms of quadratic hamiltonians381

Appendix 7 Normal forms of hamiltonian systems near stationary points and closed trajectories385

Appendix 8 Theory of perturbations of conditionally periodic motion, and Kolmogorov's theorem399

Appendix 9 Poincare's geometric theorem, its generalizations and applications416

Appendix 10 Multiplicities of characteristic frequencies, and ellipsoids depending on parameters425

Appendix 11 Short wave asymptotics438

Appendix 12 Lagrangian singularities446

Appendix 13 The Korteweg-de Vries equation453

Appendix 14 Poisson structures456

Appendix 15 On elliptic coordinates469

Appendix 16 Singularities of ray systems480

Index503