Dynamics of a System in the Presence of Invariant Relationships

The possibility of the existence of an invariant measure with smooth density is discussed in two cases related to invariant sets, namely, at the levels of partial integrals and at the joint invariant level of two or more functions. A version of Jacobi’s last multiplier theorem is presented, which supplements similar results of S.A. Chaplygin and V.V. Kozlov. Conditions are investigated under which the invariant sets represent a two-dimensional torus with an invariant measure of smooth density defined on it. This means that Kolmogorov’s theorem is applicable, which implies that, after making an appropriate change of coordinates, the motion becomes conditionally periodic.