Doklady Mathematics

, Volume 96, Issue 3, pp 625–627 | Cite as

Symplectic geometry of a linear transformation with a quadratic invariant

  • V. V. Kozlov


A linear transformation with an invariant being a nondegenerate quadratic form is symplectic. The geometric properties of such transformations are discussed. A complete set of quadratic invariants which are pairwise in involution is explicitly specified. The structure of the isotropic cone on which all these integrals simultaneously vanish is investigated. Applications of the general results to the problem on the stability of a fixed point of a linear transformation with a quadratic invariant are discussed.


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  1. 1.
    V. V. Kozlov, Differ. Equations 45 (4), 510–519 (2009).MathSciNetCrossRefGoogle Scholar
  2. 2.
    J. Williamson, Am. J. Math. 58 (1), 141–163 (1936).CrossRefGoogle Scholar
  3. 3.
    M. Berger, Géométrie (Cedic, Paris, 1977; Mir, Moscow, 1984), Vol. 2.zbMATHGoogle Scholar
  4. 4.
    H. Weyl, The Classical Groups: Their Invariants and Representations (Princeton Univ. Press, Princeton, N.J., 1946; Inostrannaya Literatura, Moscow, 1947).zbMATHGoogle Scholar
  5. 5.
    V. V. Kozlov, J. Appl. Math. Mech. 56 (6), 803–809 (1992).MathSciNetCrossRefGoogle Scholar
  6. 6.
    V. I. Arnold, Mathematical Methods of Classical Mechanics, 4th ed. (Editorial URSS, Moscow, 2000; Springer Science & Business Media, New York, 2013).Google Scholar

Copyright information

© Pleiades Publishing, Ltd. 2017

Authors and Affiliations

  1. 1.Steklov Mathematical InstituteRussian Academy of SciencesMoscowRussia
  2. 2.RUDN UniversityMoscowRussia

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