On the existence of an integral invariant of a smooth dynamic system

Abstract

The existence of an integral invariant with a smooth density for a dynamic system in a cylindrical phase space is considered. The well-known Krylov-Bogolyubov theorem guarantees the existence of an invariant measure for any system in a compact space (for a discussion of these topics see /1, 2/). But this measure is often concentrated in invariant sets of small dimensionality and in general is not an integral invariant with a summable density. For useful applications of ergodic theory, and in the theory of the Euler-Jacobi integrating factor, an invariant measure in the form of an integral invariant with smooth density is useful. Effective criteria for the existence of such measures in smooth dynamic systems are described. The general results are illustrated by examples from non-holonomic mechanics.