經(jīng)典力學(xué)的數(shù)學(xué)方法

內(nèi)容概要

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.

書籍目錄

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 Newton‘s 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. Lagrange's equations　　14. Legendre transformations　　15. Hamilton's equations　　16. Liouville's theorem　Chapter 4　Lagrangian mechanics on manifolds　　17. Holonomic constraints　　18. Differentiable manifolds　　19. Lagrangian dynamical systems　　20. E. Noether's theorem　　21. D'Alembert's principle　Chapter 5　scillations　　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. Euler's equations. Poinsot's description of the motion　　30. Lagrange's top　　31. Sleeping tops and fast topsPart 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 invariants6　　39. The Lie algebra of vector fields　　40. The Lie algebra of hamiltonian functions    ……  Chapter 9 Canonical formalism  Chapter 10 Introduction to perturbation theoryAppendixIndex

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