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Mathematical Methods of Classical Mechanics

Mathematical Methods of Classical Mechanics (title of the original in Russian: Математические методы классической механики) is a 1974 textbook by mathematician Vladimir I. Arnold. Originally written in Russian, an English translation was produced in 1978 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."

See also

References

Bibliography