|Affiliations:||Mathematical Institute of the Serbian Academy of Sciences and Arts||Title:||Nonholonomic LR systems as generalized chaplygin systems with an invariant measure and flows on homogeneous spaces||Journal:||Journal of Nonlinear Science||Volume:||14||Issue:||4||First page:||341||Last page:||381||Issue Date:||1-Aug-2004||Rank:||M21a||ISSN:||0938-8974||DOI:||10.1007/s00332-004-0603-3||Abstract:||
We consider a class of dynamical systems on a compact Lie group G with a left-invariant 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 ultidimensional 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) as the corresponding homogeneous spaces. For k = 1 and a special choice of the left-invariant metric on SO(n), we prove that after a time substitution the reduced system becomes an integrable Hamiltonian system describing a geodesic flow on the unit sphere S n-1. 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 for any dimension. In this case we also explicitly reconstruct the motion on the group SO(n).
|Publisher:||Springer Link||Project:||Russian Federation program of academic groups No. 00-15-96146 and INTAS project 00-221
Serbian Ministry of Science and Technology, Project 1643 (Geometry and Topology of Manifolds and Integrable Dynamical Systems)
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