Invariant measures of modified LR and L+R systems

Abstract

We introduce a class of dynamical systems having an invariant measure, the modifications of well-known systems on Lie groups: LR and L+R systems. As an example, we study a modified Veselova nonholonomic rigid body problem, considered as a dynamical system on the product of the Lie algebra so(n) with the Stiefel variety Vn,r, as well as the associated єL+R system on so(n) × Vn,r. In the 3-dimensional case, these systems model the nonholonomic problems of motion of a ball and a rubber ball over a fixed sphere.