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• W1978992889 abstract "We study integrable geodesic flows on Stiefel varieties $V_{n,r}=SO(n)/SO(n-r)$ given by the Euclidean, normal (standard), Manakov-type, and Einstein metrics. We also consider natural generalizations of the Neumann systems on $V_{n,r}$ with the above metrics and proves their integrability in the non-commutative sense by presenting compatible Poisson brackets on $(T^*V_{n,r})/SO(r)$. Various reductions of the latter systems are described, in particular, the generalized Neumann system on an oriented Grassmannian $G_{n,r}$ and on a sphere $S^{n-1}$ in presence of Yang-Mills fields or a magnetic monopole field. Apart from the known Lax pair for generalized Neumann systems, an alternative (dual) Lax pair is presented, which enables one to formulate a generalization of the Chasles theorem relating the trajectories of the systems and common linear spaces tangent to confocal quadrics. Additionally, several extensions are considered: the generalized Neumann system on the complex Stiefel variety $W_{n,r}=U(n)/U(n-r)$, the matrix analogs of the double and coupled Neumann systems." @default.
• W1978992889 created "2016-06-24" @default.
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• W1978992889 date "2010-12-24" @default.
• W1978992889 modified "2023-10-17" @default.
• W1978992889 title "Geodesic flows and Neumann systems on Stiefel varieties: geometry and integrability" @default.
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• W1978992889 doi "https://doi.org/10.1007/s00209-010-0818-y" @default.
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