Mathematical Methods of Classical Mechanics

Mathematical Methods of Classical Mechanics is a textbook by mathematician Vladimir I. Arnold. It was originally written in Russian, and later translated into English by A. Weinstein and K. Vogtmann. It is aimed at graduate students.

Contents

 * Part I: Newtonian Mechanics
 * Chapter 1: Experimental Facts
 * Chapter 2: Investigation of the Equations of Motion
 * Part II: Lagrangian Mechanics
 * Chapter 3: Variational Principles
 * Chapter 4: Lagrangian Mechanics on Manifolds
 * Chapter 5: Oscillations
 * Chapter 6: Rigid Bodies
 * Part III: Hamiltonian Mechanics
 * Chapter 7: Differential forms
 * Chapter 8: Symplectic Manifolds
 * Chapter 9: Canonical Formalism
 * Chapter 10: Introduction to Perturbation Theory
 * Appendices
 * Riemannian curvature
 * Geodesics of left-invariant metrics on Lie groups and the hydrodynamics of ideal fluids
 * Symplectic structures on algebraic manifolds
 * Contact structures
 * Dynamical systems with symmetries
 * Normal forms of quadratic Hamiltonians
 * Normal forms of Hamiltonian systems near stationary points and closed trajectories
 * Theory of perturbations of conditionally period motion and Kolmogorov's theorem
 * Poincaré's geometric theorem, its generalizations and applications
 * Multiplicities of characteristic frequencies, and ellipsoids depending on parameters
 * Short wave asymptotics
 * Lagrangian singularities
 * The Kortweg-de Vries equation
 * Poisson structures
 * On elliptic coordinates
 * Singularities of ray systems

Russian original and translations
The original Russian first edition Математические методы классической механики was published in 1974 by Наука. A second edition was published in 1979, and a third in 1989. The book has since been translated into a number of other languages, including French, German, Japanese and Mandarin.

Reviews
The Bulletin of the American Mathematical Society said, "The [book] under review [...] written by a distinguished mathematician [...is one of] the first textbooks [to] successfully to present to students of mathematics and physics, [sic] classical mechanics in a modern setting."

A book review in the journal Celestial Mechanics said, "In summary, the author has succeeded in producing a mathematical synthesis of the science of dynamics. The book is well presented and beautifully translated [...] Arnold's book is pure poetry; one does not simply read it, one enjoys it."