Springer Nature is making SARS-CoV-2 and COVID-19 research free. View research | View latest news | Sign up for updates

Integrable equations, related with the Poisson algebra

  • 34 Accesses

  • 4 Citations


The general r-matrix scheme for the construction of integrable Hamiltonian systems is applied to a Poisson algebra, i.e., the algebra of functions on a symplectic manifold, the commutator in which is defined by the Poisson bracket. Integrable systems of two types are constructed: Hamiltonian systems on the group of symplectic diffeomorphisms, whose Hamiltonians are sums of a leftinvariant kinetic energy and a potential, and first-order systems of equations for two functions of one variable.

This is a preview of subscription content, log in to check access.

Literature cited

  1. 1.

    V. I. Arnol'd, Mathematical Methods of Classical Mechanics, Springer, New York (1978).

  2. 2.

    A. G. Reiman and M. A. Semenov-Tyan-Shanskii, “Group theoretic methods in the theory of integrable systems,” in: Current Problems in Mathematics. Fundamental Directions, Vol. 16 [in Russian], VINITI, Moscow (1987), pp. 119–193.

  3. 3.

    A. G. Reiman, “Integrable Hamiltonian systems connected with graded Lie algebras,” Zap. Nauchn. Semin. Leningr. Otd. Mat. Inst.,95, 3–54 (1980).

Download references

Additional information

Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 169, pp. 44–50, 1988.

Rights and permissions

Reprints and Permissions

About this article

Cite this article

Golenishcheva-Kutuzova, M.I., Reiman, A.G. Integrable equations, related with the Poisson algebra. J Math Sci 54, 890–894 (1991). https://doi.org/10.1007/BF01101116

Download citation


  • Manifold
  • Kinetic Energy
  • Integrable Equation
  • Integrable System
  • Hamiltonian System