ar X iv : m at h - ph / 0 30 70 16 v 1 8 J ul 2 00 3 Nonholonomic LR systems as Generalized Chaplygin systems with an Invariant Measure and Geodesic Flows on Homogeneous Spaces ∗

We consider a class of dynamical systems on a Lie group G with a leftinvariant metric and right-invariant nonholonomic constraints (so called LR systems) and show that, under a generic condition on the constraints, such systems can be regarded as generalized Chaplygin systems on the principle bundle G → Q = G/H , H being a Lie subgroup. In contrast to generic Chaplygin systems, the reductions of our LR systems onto the homogeneous space Q always possess an invariant measure. We study the case G = SO(n), when LR systems are multidimensional generalizations of the Veselova problem of a nonholonomic rigid body motion, which admit a reduction to systems with an invariant measure on the (co)tangent bundle of Stiefel varieties V (k, n). For k = 1 and a special choice of the leftinvariant metric on SO(n), we prove that under a change of time, the reduced system becomes an integrable Hamiltonian system describing a geodesic flow on the unit sphere S. This provides a first example of a nonholonomic system with more than two degrees of freedom for which the celebrated Chaplygin reducibility theorem is applicable.