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.

中文翻译：

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作为哈密顿系统的刘维尔方程

考虑了具有不变测度的封闭流形上的光滑动力系统。非平稳不变测度的密度演化由众所周知的 Liouville 方程描述。对于遍历动力系统，Liouville 方程以哈密顿形式表示。指示了关于由哈密顿结构生成的泊松括号成对对合的二次不变量的无限集合。