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Global action-angle variables for non-commutative integrable systems. (English) Zbl 1435.37080

The aim of this paper is to extend to Poisson manifolds the results on the existence of global action-angle variables for commutative and non-commutative Liouville integrable systems on symplectic manifolds.
The authors analyze the obstructions to the existence of global action-angle variables for regular non-commutative integrable systems (regular NCI systems) on Poisson manifolds. The notion of a regular NCI system is reformulated geometrically as an \(r\)-dimensional foliation whose tangent bundle is spanned by Hamiltonian vector fields associated to the local first integrals of the foliations (i.e., locally defined functions that are constant along the leaves).
Several new concepts as the action bundle, the action lattice bundle of NCI systems and the action, angle and transverse foliations are introduced and exploited in the paper. The obstructions, often of cohomological nature, to the existence of the mentioned foliations are also presented. Many interesting examples are provided. Some of them come from classical mechanics while other examples are artificially constructed to illustrate the non-triviality of the obstructions.

MSC:

37J30 Obstructions to integrability for finite-dimensional Hamiltonian and Lagrangian systems (nonintegrability criteria)
37J39 Relations of finite-dimensional Hamiltonian and Lagrangian systems with topology, geometry and differential geometry (symplectic geometry, Poisson geometry, etc.)
37J35 Completely integrable finite-dimensional Hamiltonian systems, integration methods, integrability tests
37J06 General theory of finite-dimensional Hamiltonian and Lagrangian systems, Hamiltonian and Lagrangian structures, symmetries, invariants
70G45 Differential geometric methods (tensors, connections, symplectic, Poisson, contact, Riemannian, nonholonomic, etc.) for problems in mechanics
53D17 Poisson manifolds; Poisson groupoids and algebroids
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