Geodesic flows and Neumann systems on Stiefel varieties: geometry and integrability
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- Fedorov, Y.N. & Jovanović, B. Math. Z. (2012) 270: 659. doi:10.1007/s00209-010-0818-y
We study integrable geodesic flows on Stiefel varieties Vn,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 Vn,r with the above metrics and proves their integrability in the non-commutative sense by presenting compatible Poisson brackets on (T*Vn,r)/SO(r). Various reductions of the latter systems are described, in particular, the generalized Neumann system on an oriented Grassmannian Gn,r and on a sphere Sn−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 Wn,r = U(n)/U(n−r), the matrix analogs of the double and coupled Neumann systems.
Mathematics Subject Classification (2000)17B80 53D25 70H06 70H33 70H45
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