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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