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Noncommutative integrability, moment map and geodesic flows. (English) Zbl 1022.37038

This paper brings interesting new insights into the nature of so-called noncommutative integrability of Hamiltonian systems. After a brief sketch of the history of the subject and having recalled what noncommutative integrability, as introduced by Mishchenko and Fomenko, means, the authors directly come to their main result. They prove that, under mild assumptions, noncommutative integrability in fact always implies standard integrability (through commuting integrals in the category of smooth functions). In Section 2, in the context of the action of a Lie group on a connected symplectic manifold and the associated momentum map, a generalization is proved of a result of Guillemin and Sternberg on integrability of collective motion. Another interesting application is the proof of complete integrability of geodesic flows on Riemannian manifolds all of whose geodesics are closed. In the final section, it is shown that geodesic flows of bi-invariant metrics on so-called bi-quotients of compact Lie groups are always integrable.

MSC:

37J35 Completely integrable finite-dimensional Hamiltonian systems, integration methods, integrability tests
70H06 Completely integrable systems and methods of integration for problems in Hamiltonian and Lagrangian mechanics
37J15 Symmetries, invariants, invariant manifolds, momentum maps, reduction (MSC2010)
53D20 Momentum maps; symplectic reduction
53D25 Geodesic flows in symplectic geometry and contact geometry
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