2 1 2 A ug 2 00 3 Nonholonomic LR systems as Generalized Chaplygin systems with an Invariant Measure and Geodesic Flows on Homogeneous Spaces ∗
2 1 2 A ug 2 00 3 Nonholonomic LR systems as Generalized Chaplygin systems with an Invariant Measure and Geodesic Flows on Homogeneous Spaces ∗
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2 1 2 A ug 2 00 3 非完整 LR 系统作为具有不变测度和均匀空间上测地流的广义 Chaplygin 系统 *
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发表时间：
2003
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通讯作者：
Yuri N. Fedorov
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作者：
Yuri N. Fedorov
考虑紧致李群G上具有左不变度量和右不变非完整约束的一类动力系统（称为LR系统），并证明了在一般的约束条件下，这类系统可被看作是主丛G → Q = G/H上的广义Chaubergin系统，H是李群.与一般的Chaubergin系统相比，我们的LR系统在齐次空间Q上的约化总是具有不变测度。我们研究的情况G = SO（n），LR系统是多维推广的Veselova问题的非完整刚体运动，这允许减少到系统的不变测度的（余）切丛的Stiefel品种V（k，n）作为相应的齐性空间。对于k = 1和SO（n）上左不变度量的特殊选择，我们证明了在时间的变化下，约化系统成为描述单位球面S上测地线流的可积Hamilton系统.这提供了一个具有两个以上自由度的非完整系统的第一个例子，著名的Chaubergin约化定理是适用的。在这种情况下，我们也明确重建群SO（n）上的运动。CRAAMS主题分类37 J60、37 J35、70 H45
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 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. In this case we also explicitly reconstruct the motion on the group SO(n). ∗AMS Subject Classification 37J60, 37J35, 70H45