Mathematical methods of classical mechanics

Preface Preface to the second edition Part ¢ñ 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 felds 7.Angular momentum 8.Investigation of motion in a central feld 9.The motion of a point in three-space 10.Motions of a system of n points 11.The method of similarity Part ¢ò 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.Diferentiable manifolds 19.Lagrangian dynamical systems 20.E.Noether's theorem 21.D'Alembert's 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.Euler's equations. Poinsot's description of the motion iteor lo eslqonn 30.Lagrange's top 31.Sleeping tops and fast tops Part ¢ó HAMILTONIAN MECHANICS Chapter 7 Diferential 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 Poincar¨¦ Cartan 45.Applications of the integral invariant of Poincar¨¦-Cartan 46.Huygens' principle 47.The Hamilton-Jacobi method for integrating Hamilton's 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 Kolmogorov's theorem Appendix 9 Poincar¨¦'s 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