Part I NEWTONIAN MECHANICS 1
Chapter 1 Experimental facts 3
1. The principles of relativity and determinacy 3
2. The galilean group and Newton's equations 4
3. Examples of mechanical systems 11
Chapter 2 Investigation of the equations of motion 15
4. Systems with one degree of freedom 15
5. Systems with two degrees of freedom 22
6. Conservative force fields 28
7. Angular momentum 30
8. Investigation of motion in a central field 33
9. The motion of a point in three-space 42
10. Motions of a system of n pomts 44
11. The method of similarity 50
Part Ⅱ LAGRANGIAN MECHANICS 53
Chapter 3 Variational principles 55
12. Calculus of variations 55
13. Lagrange's equations 59
14. Legendre transformations 61
15. Hamilton's equations 65
16. Liouville's theorem 68
Chapter 4 Lagrangian mechanics on manifolds 75
17. Holonomic constraints 75
18. Differentiable manifolds 77
19. Lagrangian dynamical systems 83
20. E. Noether's theorem 88
21. D'Alembert's principle 91
Chapter 5 Oscillations 98
22. Linearization 98
23. Small oscillations 103
24. Behavior of characteristic frequencies 110
25. Parametric resonance 113
Chapter 6 Rigid Bodies 123
26. Motion in a moving coordinate system 123
27. Inertial forces and the Coriolis force 129
28. Rigid bodies 133
29. Euler's equations. Poinsot's description of the motion 142
30. Lagrange's top 148
31. Sleeping tops and fast tops 154
Part Ⅲ HAMILTONIAN MECHANICS 161
Chapter 7 Differential forms 163
32. Exterior forms 163
33. Exterior multiplication 170
34. Differential forms 174
35. Integration of differential forms 181
36. Exterior differentiation 188
Chapter 8 Symplectic manifolds 201
37. Symplectic structures on manifolds 201
38. Hamiltonian phase flows and their integral invariants 204
39. The Lie algebra of vector fields 208
40. The Lie algebra of hamiltonian functions 214
41. Symplectic geometry 219
42. Parametric resonance in systems with many degrees of freedom 225
43. A symplectic atlas 229
Chapter 9 Canonical formalism 233
44. The integral invariant of Poincare-Cartan 233
45. Applications of the integral invariant of Poincare-Cartan 240
46. Huygens' principle 248
47. The Hamilton-Jacobi method for integrating Hamilton's canonical equations 258
48. Generating functions 266
Chapter 10 Introduction to perturbation theory 271
49. Integrable systems 271
50. Action-angle variables 279
51. Averaging 285
52. Averaging of perturbations 291
Appendix 1 Riemannian curvature 301
Appendix 2 Geodesics of left-invariant metrics on Lie groups and the hydrodynamics of ideal fluids 318
Appendix 3 Symplectic structures on algebraic manifolds 343
Appendix 4 Contact structures 349
Appendix 5 Dynamical systems with symmetries 371
Appendix 6 Normal forms of quadratic hamiltonians 381
Appendix 7 Normal forms of hamiltonian systems near stationary points and closed trajectories 385
Appendix 8 Theory of perturbations of conditionally periodic motion, and Kolmogorov's theorem 399
Appendix 9 Poincare's geometric theorem, its generalizations and applications 416
Appendix 10 Multiplicities of characteristic frequencies, and ellipsoids depending on parameters 425
Appendix 11 Short wave asymptotics 438
Appendix 12 Lagrangian singularities 446
Appendix 13 The Korteweg-de Vries equation 453
Appendix 14 Poisson structures 456
Appendix 15 On elliptic coordinates 469
Appendix 16 Singularities of ray systems 480
Index 503