Variational principles

  • V. I. Arnold
Part of the Graduate Texts in Mathematics book series (GTM, volume 60)


In this chapter we show that the motions of a newtonian potential system are extremals of a variational principle, “Hamilton’s principle of least action.”


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  1. 26.
    We should specify the class of curves oh which φ is defined and the linear space which contains h. One could assume, for example, that both spaces consist of the infinitely differentiable functions.Google Scholar
  2. 27.
    Or even for any infinitely differentiable function h.Google Scholar
  3. 28.
    If it exists.Google Scholar
  4. 29.
    One can easily see that this is the theory of “Clairaut’s equation.”Google Scholar
  5. 30.
    In practice this convex function will often be a positive definite quadratic form.Google Scholar
  6. 31.
    For this it is sufficient, for example, that the level sets of H be compact.Google Scholar
  7. 32.
    Cf, for example, the book: Halmos, Lectures on Ergodic Theory, 1956 (Mathematical Society of Japan. Publications. No. 3).Google Scholar
  8. 33.
    A set A is dense in B if there is a point of A in every neighborhood of every point of B.Google Scholar
  9. 34.
    The direct product of the sets A, B,... is the set of points (a, b,...), with aA, bB,....Google Scholar

Copyright information

© Springer Science+Business Media New York 1978

Authors and Affiliations

  • V. I. Arnold
    • 1
  1. 1.Department of MathematicsUniversity of MoscowMoscowUSSR

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