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Integrable equations, related with the Poisson algebra

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Abstract

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.

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Literature cited

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    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).

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Additional information

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

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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

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Keywords

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