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Non-commutative Integrability, Moment Map and Geodesic Flows
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abstract
The purpose of this paper is to discuss the relationship between commutative and non-commutative integrability of Hamiltonian systems and to construct new examples of integrable geodesic flows on Riemannian manifolds. In particular, we prove that the geodesic flow of the bi-invariant metric on any bi-quotient of a compact Lie group is integrable in non-commutative sense by means of polynomial integrals, and therefore, in classical commutative sense by means of $C^\infty$--smooth integrals.
Forward citations
Cited by 4 Pith papers
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Integrable systems from Poisson reductions of generalized Hamiltonian torus actions
Develops sufficient conditions for Poisson reduction of generalized Hamiltonian torus actions to preserve integrability and applies them to open problems on Lie group doubles and flat-connection moduli spaces.
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A Lax representation and integrability of homogeneous exact magnetic flows on spheres in all dimensions
The paper proves complete integrability of homogeneous exact magnetic flows on spheres in all dimensions via a Lax representation.
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Contact line bundles, foliations, and integrability
A line-bundle approach to contact integrability unifies cooriented and non-cooriented systems and covers dissipative contact Hamiltonians.
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Collective superintegrable systems from the Guillemin--Sternberg torus action
Collective Hamiltonians of any Hamiltonian action of a compact semisimple Lie group form a superintegrable system, with action variables given by the Guillemin-Sternberg torus momentum map.
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