Mathematical Methods of Classical Mechanics

Mathematical physics book by V.I. Arnold

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.

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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

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.