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The Liouville Equation as a Hamiltonian System

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Abstract

Smooth dynamical systems on closed manifolds with invariant measure are considered. The evolution of the density of a nonstationary invariant measure is described by the well-known Liouville equation. For ergodic dynamical systems, the Liouville equation 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.

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Funding

This research was supported by the Russian Science Foundation under grant 19-71-30012.

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  1. Steklov Mathematical Institute of Russian Academy of Sciences, 119991, Moscow, Russia

    V. V. Kozlov

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  1. V. V. Kozlov
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Correspondence to V. V. Kozlov.

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Kozlov, V.V. The Liouville Equation as a Hamiltonian System. Math Notes 108, 339–343 (2020). https://doi.org/10.1134/S0001434620090035

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  • DOI: https://doi.org/10.1134/S0001434620090035

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