Advertisement

Siberian Mathematical Journal

, Volume 60, Issue 4, pp 592–604 | Cite as

Lax Pairs for Linear Hamiltonian Systems

  • A. B. ZheglovEmail author
  • D. V. OsipovEmail author
Article
  • 1 Downloads

Abstract

We construct Lax pairs for linear Hamiltonian systems of differential equations. In particular, the Gröbner bases are used for computations. It is proved that the mappings in the construction of Lax pairs are Poisson. Under study are the various properties of first integrals of the system which are obtained from Lax pairs.

Keywords

Lax pairs linear Hamiltonian systems first integrals Gröbner bases 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Kozlov V. V., “Linear systems with a quadratic integral,” J. Appl. Math. Mech., vol. 56, no. 6, 803–809 (1992).MathSciNetCrossRefGoogle Scholar
  2. 2.
    Kozlov V. V., “Linear Hamiltonian systems: Quadratic integrals, singular subspaces and stability,” Regul. Chaotic Dyn., vol. 23, no. 1, 26–46 (2018).MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Arzhantsev I. V., Gröbner Bases and Systems of Algebraic Equations [Russian], MTsNMO, Moscow (2003).zbMATHGoogle Scholar
  4. 4.
    Cox D. A., Little J., and O’Shea D., Ideals, Varieties, and Algorithms: An Introduction to Computational Algebraic Geometry and Commutative Algebra, Springer-Verlag, New York (1997).CrossRefzbMATHGoogle Scholar
  5. 5.
    Williamson J., “An algebraic problem involving the involutory integrals of linear dynamical systems,” Amer. J. Math., vol. 62, 881–911 (1940).MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Kocak H., “Linear Hamiltonian systems are integrable with quadratics,” J. Math. Phys., vol. 23, no. 12, 2375–2380 (1982).MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Williamson J., “On the algebraic problem concerning the normal forms of linear dynamical systems,” Amer. J. Math., vol. 58, no. 1, 141–163 (1936).MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Arnold V. I., Mathematical Methods of Classical Mechanics, Springer-Verlag, New York (1989).CrossRefGoogle Scholar
  9. 9.
    Zheglov A. B. and Osipov D. V., “On first integrals of linear Hamiltonian systems,” Dokl. Math., vol. 98, no. 3, 616–618 (2018).CrossRefzbMATHGoogle Scholar

Copyright information

© Pleiades Publishing, Inc. 2019

Authors and Affiliations

  1. 1.Lomonosov Moscow State UniversityMoscowRussia
  2. 2.Steklov Mathematical Institute of Russian Academy of SciencesMoscowRussia
  3. 3.National Research University Higher School of EconomicsMoscowRussia
  4. 4.National University of Science and Technology “MISiS,”MoscowRussia

Personalised recommendations