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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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.
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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
- Kinetic Energy
- Integrable Equation
- Integrable System
- Hamiltonian System