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.^[1] It is aimed at graduate students.

Mathematical Methods of Classical Mechanics

First English edition
Author	Vladimir I. Arnol'd
Original title	Matematicheskie metody klassicheskoi mekhaniki
Language	Russian
Subjects	Mathematical physics Classical mechanics
Genre	Non-fiction
Published	1974
Publication place	Russia
Published in English	1978
Pages	xvi + 516
ISBN	0387968903

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."^[2]

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."^[3]

See also

• List of textbooks in classical and quantum mechanics

References

1. Mathematical Lives: Protagonists of the Twentieth Century From Hilbert to Wiles. Springer Science & Business Media. 2010. p. 211. ISBN 9783642136061.
2. Sneddon, Ian N. (March 1980). "Book Review of Mathematical methods of classical mechanics and A course in mathematical physics, vol. 1: Classical dynamical systems". Bulletin of the American Mathematical Society. 2 (2): 346–352. doi:10.1090/S0273-0979-1980-14755-2 – via Project Euclid.
3. Broucke, R (1982). "Book-Review - Mathematical Methods of Classical Mechanics". Celestial Mechanics. 28: 345. Bibcode:1982CeMec..28..345A. doi:10.1007/bf01243742. S2CID 189830621 – via SAO/NASA ADS.

Bibliography

• 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.