经典力学的数学方法(英文版)

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书名:经典力学的数学方法(英文版)
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出版社:世界图书出版公司北京公司
价格:87元

简介

Many different mathematical methods and concepts are used in classical mechanics: differential equations and phase flows, smooth mappings and manifolds, Lie groups and Lie algebras, symplectic geometry and ergodic theory. Many modern mathematical theories arose from problems in mechanics and only later acquired that axiomatic-abstract form which makes them so hard to study.

Preface
Preface to the second edition
Part I
NEWTONIAN MECHANICS
Chapter 1
Experimental facts
1. The principles of relativity and determinacy
2. The galilean group and Newtons equations
3. Examples of mechanical systems
Chapter 2
Investigation of the equations of motion
4. Systems with one degree of freedom
5. Systems with two degrees of freedom
6. Conservative force fields
7. Angular momentum
8. Investigation of motion in a central field
9. The motion of a point in three-space
10. Motions of a system of n points
11. The method of similarity
Part II
LAGRANGIAN MECHANICS
Chapter 3
Variational principles
12. Calculus of variations
13. Lagranges equations
14. Legendre transformations
15. Hamiltons equations
16. Liouvilles theorem
Chapter 4
Lagrangian mechanics on manifolds
17. Holonomic constraints
18. Differentiable manifolds
19. Lagrangian dynamical systems
20. E. Noethers theorem
21. DAlemberts principle
Chapter 5
Oscillations
22. Linearization
23. Small oscillations
24. Behavior of characteristic frequencies
25. Parametric resonance
Chapter 6
Rigid bodies
26. Motion in a moving coordinate system
27. Inertial forces and the Coriolis force
28. Rigid bodies
29. Eulers equations. Poinsots deScription of the motion
30. Lagranges top
31. Sleeping tops and fast tops
Part III
HAMILTONIAN MECHANICS
Chapter 7
Differential forms
32. Exterior forms
33. Exterior multiplication
34. Differential forms
35. Integration of differential forms
36. Exterior differentiation
Chapter 8
Symplectic manifolds
37. Symplectic structures on manifolds
38. Hamiltonian phase flows and their integral invariants
39. The Lie algebra of vector fields
40. The Lie algebra of hamiltonian functions
41. Symplectic geometry
42. Parametric resonance in systems with many degrees of freedom
43. A symplectic atlas
Chapter 9
Canonical formalism
44. The integral invariant of Poincare-Cartan
45. Applications of the integral invariant of Poincare-Cartan
46. Huygens principle
47. The Hamilton-Jacobi method for integrating Hamiltons canonical equations
48. Generating functions
Chapter 10
Introduction to perturbation theory
49. Integrable systems
50. Action-angle variables
51. Averaging
52. Averaging of perturbations
Appendix 1
Riemannian curvature
Appendix 2
Geodesics of left-invariant metrics on Lie groups and the hydrodynamics of ideal fluids
Appendix 3
Symplectic structures on algebraic manifolds
Appendix 4
Contact structures
Appendix 5
Dynamical systems with symmetries
Appendix 6
Normal forms of quadratic hamiltonians
Appendix 7
Normal forms of hamiltonian systems near stationary points and closed trajectories
Appendix 8
Theory of perturbations of conditionally periodic motion, and Kolmogorovs theorem
Appendix 9
Poincares geometric theorem, its generalizations and applications
Appendix 10
Multiplicities of characteristic frequencies, and ellipsoids depending on parameters
Appendix 11
Short wave asymptotics
Appendix 12
Lagrangian singularities
Appendix 13
The Korteweg-de Vries equation
Appendix 14
Poisson structures
Appendix 15
On elliptic coordinates
Appendix 16
Singularities of ray systems
Index

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