Abstract

We study integrable discretizations of geodesic flows of Euclidean metrics on the cotangent bundles of the Stiefel manifolds \(V_{n,r}\). In particular, for \(n=3\) and \(r=2\), after the identification \(V_{3,2}\cong\mathrm{SO}(3)\), we obtain a discrete analog of the Euler case of the rigid body motion corresponding to the inertia operator \(I=(1,1,2)\). In addition, billiard-type mappings are considered; one of them turns out to be the “square root” of the discrete Neumann system on \(V_{n,r}\).

中文翻译：

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Stiefel流形上的离散测地流

摘要

我们研究了Stiefel流形\（V_ {n，r} \）的余切束上的欧几里得度量测地线流的可积离散化。特别是，对于\（n = 3 \）和\（r = 2 \），在标识\（V_ {3,2} \ cong \ mathrm {SO}（3）\）之后，我们获得了的离散模拟对应于惯性算子\（I =（1,1,2）\）的刚体运动的Euler情况。另外，还考虑了台球类型的映射。其中之一是\（V_ {n，r} \）上离散Neumann系统的“平方根” 。