This book constructs the mathematical apparatus of classical mechanics from the beginning, examining basic problems in dynamics like the theory of oscillations and the Hamiltonian formalism. Discussion includes qualitative methods of the theory of dynamical systems and of asymptotic methods like averaging and adiabatic invariance. Preface v Preface to the Second edition vii Part I NEWTONIAN MECHANICS 1(52) Experimental facts 3(12) The principles of relativity and determinacy 3(1) The galilean group and Newton's equations 4(7) Examples of mechanical systems 11(4) Investigation of the equations of motion 15(38) Systems with one degree of freedom 15(7) Systems with two degrees of freedom 22(6) Conservative force fields 28(2) Angular momentum 30(3) Investigation of motion in a central field 33(9) The motion of a point in three-space 42(2) Motions of a system of n points 44(6) The method of similarity 50(3) Part II LAGRAGIAN MECHANICS 53(108) Variational principles 55(20) Calculus of variations 55(4) Lagrange's equations 59(2) Legendre transformations 61(4) Hamilton's equations 65(3) Liouville's theorem 68(7) Lagrangian mechanics on manifolds 75(23) Holonomic constraints 75(2) Differentiable manifolds 77(6) Lagragian dynamical systems 83(5) E. Noether's theorem 88(3) D'Alembert's principle 91(7) Oscillations 98(25) Linearization 98(5) Small oscillations 103(7) Behavior of characteristic frequencies 110(3) Parametric resonance 113(10) Rigid bodies 123(38) Motion in a moving coordinate system 123(6) Inertial forces and the Coriolis force 129(4) Rigid bodies 133(9) Euler's equations. Poinsot's description of the motion 142(6) Language's top 148(6) Sleeping tops and fast tops 154(7) Part III HAMILTONIAN MECHANICS 161(140) Differential forms 163(38) Exterior forms 163(7) Exterior multiplication 170(4) Differential forms 174(7) Integration of differential forms 181(7) Exterior differentiation 188(13) Symplectic manifolds 201(32) Symplectic structures on manifolds 201(3) Hamiltonian phase flows and their integral invariants 204(4) The Lie algebra of vector fields 208(6) The Lie algebra of hamiltonian functions 214(5) Symplectic geometry 219(6) Parametric resonance in systems with many degrees of freedom 225(4) A symplectic atlas 229(4) Canonical formalism 233(38) The intergral invariant of Poincare-Cartan 233(7) Applications of the integral invariant of Poincare-Cartan 240(8) Huygens' principle 248(10) The Hamilton- Jacobi method for integrating Hamilton's canonical equations 258(8) Generating functions 266(5) Introduction to perturbation theory 271(30) Integrable systems 271(8) Action- angle variables 279(6) Averaging 285(6) Averaging of perturbations 291(10) Appendix 1 Riemannian curvature 301(17) Appendix 2 Geodesics of left-invariant metrics on Lie groups and the hydrodynamics of ideal fluids 318(25) Appendix 3 Symplectic structures on algebraic manifolds 343(6) Appendix 4 Contact structures 349(22) Appendix 5 Dynamical systems with symmetries 371(10) Appendix 6 Normal forms of quadratic hamiltonians 381(4) Appendix 7 Normal forms of hamiltonian systems near stationary points and closed trajectories 385(14) Appendix 8 Theory of perturbations of conditionally periodic motion, and Kolmogorov's theorem 399(17) Appendix 9 Poincare's geometric theorem, its generalizations and applications 416(9) Appendix 10 Multiplicities of characteristic frequencies, and ellipsoids depending on parameters 425(13) Appendix 11 Short wave asymptotics 438(8) Appendix 12 Lagrangian singularities 446(7) Appendix 13 The Korteweg-de Vries equation 453(3) Appendix 14 Poisson structures 456(13) Appendix 15 On elliptic coordinates 469(11) Appendix 16 Singularities of ray systems 480(31) Index 511