Nonholonomic LR systems as Generalized Chaplygin systems with an invariant measure and geodesic flows on homogeneous spaces
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hdl:2117/873
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Data publicació2003
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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 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 Sn−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).
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