Nonholonomic LR systems as Generalized Chaplygin systems with an Invariant Measure and Geodesic Flows on Homogeneous Spaces

Details Nonholonomic LR systems as Generalized Chaplygin systems with an Invariant Measure and Geodesic Flows on Homogeneous Spaces Nonholonomic LR systems as Generalized Chaplygin systems with an Invariant Measure and Geodesic Flows on Homogeneous Spaces

2003-07-08

arxiv.org/abs/math-ph/0307016v3

Journal of Nonlinear Science, Vol. 14 (2004) 341 - 381.

Physics

Mathematical Physics

39 pages, the proof of Lemma 4.3 and some references are added, to appear in Journal of Nonlinear Science

Scientific paper

10.1007/s00332-004-0603-3

We consider a class of dynamical systems on a 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 $G \to Q=G/H$, $H$ 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 $V(k,n)$ as the corresponding homogeneous spaces. For $k=1$ and a special choice of the left-invariant 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 $S^{n-1}$. 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. In this case we also explicitly reconstruct the motion on the group SO(n).

Affiliated with

Also associated with

No associations

Rating

Nonholonomic LR systems as Generalized Chaplygin systems with an Invariant Measure and Geodesic Flows on Homogeneous Spaces does not yet have a rating. At this time, there are no reviews or comments for this scientific paper.

If you have personal experience with Nonholonomic LR systems as Generalized Chaplygin systems with an Invariant Measure and Geodesic Flows on Homogeneous Spaces, we encourage you to share that experience with our LandOfFree.com community. Your opinion is very important and Nonholonomic LR systems as Generalized Chaplygin systems with an Invariant Measure and Geodesic Flows on Homogeneous Spaces will most certainly appreciate the feedback.

Rate now

Comments { 0 }

Profile ID: LFWR-SCP-O-10830