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The Liouville equation as a Hamiltonian system. (English. Russian original) Zbl 1455.37054
Math. Notes 108, No. 3, 339-343 (2020); translation from Mat. Zametki 108, No. 3, 360-365 (2020).
The author considers smooth dynamical systems on closed manifolds with invariant measure. The evolution of the density of a nonstationary invariant measure is described by the well-known Liouville equation and for ergodic dynamical systems, the latter is expressed in Hamiltonian form. An infinite collection of quadratic invariants that are pairwise in involution with respect to the Poisson bracket generated by the Hamiltonian structure is indicated. Some remarks and open questions are mentioned at the end of the paper.

##### MSC:
 37K10 Completely integrable infinite-dimensional Hamiltonian and Lagrangian systems, integration methods, integrability tests, integrable hierarchies (KdV, KP, Toda, etc.) 37K06 General theory of infinite-dimensional Hamiltonian and Lagrangian systems, Hamiltonian and Lagrangian structures, symmetries, conservation laws
##### Keywords:
Liouville equation; Hamiltonian systems; integrability
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##### References:
 [1] Kozlov, V. V., Phenomena of nonintegrability in Hamiltonian systems, Proceedings of the International Congress of Mathematicians, 0, 1161-1170 (1987) · Zbl 0675.58015 [2] Moshchevitin, N. G., Existence and smoothness of the integral of a Hamiltonian system of a certain form, Math. Notes, 49, 5, 498-501 (1991) · Zbl 0741.70013 [3] Kozlov, V. V., Linear systems with quadratic integral and complete integrability of the Schrödinger equation, Russian Math. Surveys, 74, 5-449, 959-961 (2019) · Zbl 1442.37079 [4] Kozlov, V. V., Linear systems with a quadratic integral, J. Appl. Math. Mech., 56, 6, 803-809 (1992) · Zbl 0792.70014 [5] Treshchev, D. V.; Shkalikov, A. A., On the Hamiltonian property of linear dynamical systems in Hilbert space, Math. Notes, 101, 6, 1033-1039 (2017) · Zbl 06769031 [6] Kozlov, V. V., Multi-Hamiltonian property of a linear system with quadratic invariant, St. Petersburg Mathematical Journal, 30, 5, 877-883 (2019) · Zbl 07090628 [7] Williamson, J., An algebraic problem involving the involutory integrals of linear dynamical systems, Amer. J. Math., 62, 881-911 (1940) · Zbl 0024.13301 [8] Kocak, H., Linear Hamiltonian Systems are Integrable with Quadratics, J. Math. Phys., 23, 12, 2375-2380 (1982) · Zbl 0507.70015 [9] Kozlov, V. V., Linear Hamiltonian systems: quadratic integrals, singular subspaces and stability, Regul. Chaotic Dyn., 23, 1, 26-46 (2018) · Zbl 1400.37061 [10] Zheglov, A. B.; Osipov, D. V., On first integrals of linear Hamiltonian systems, Dokl. Math., 98, 3, 616-618 (2018) · Zbl 1408.37099 [11] Zheglov, A. B.; Osipov, D. V., Lax pairs for linear Hamiltonian systems, Siberian Math. J., 60, 4, 592-604 (2019) · Zbl 1427.37046
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