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.

• 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

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.

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."

• List of textbooks in classical and quantum mechanics

• Arnold, Vladimir I. (16 May 1989) [First published in 1974]. Mathematical Methods of Classical Mechanics Математические методы классической механики. Graduate Texts in Mathematics. Vol. 60. Translated by Vogtmann, Karen; Weinstein, Alan D. (2nd ed.). New York: Springer-Verlag. ISBN 978-0-387-96890-2. OCLC 18681352.