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Symplectic topology of integrable Hamiltonian systems. I: Arnold-Liouville with singularities. (English) Zbl 0936.37042

Summary: The classical Arnold-Liouville theorem describes the geometric of an integrable Hamiltonian system near a regular level set of the moment map. Our results describe it near a nondegenerate singular level set: a tubular neighborhood of a connected singular nondegenerate level set, after a normal finite covering, admits a non-complete system of action-angle functions (the number of action functions is equal to the rank of the moment map), and it can be decomposed topologically, together with the associated singular Lagrangian foliation, to a direct product of simplest (codimension 1 and codimension 2) singularities. These results are essential for the global topological study of integrable Hamiltonian systems.

MSC:

37J05 Relations of dynamical systems with symplectic geometry and topology (MSC2010)
37J35 Completely integrable finite-dimensional Hamiltonian systems, integration methods, integrability tests
53D35 Global theory of symplectic and contact manifolds
57R17 Symplectic and contact topology in high or arbitrary dimension
70G40 Topological and differential topological methods for problems in mechanics
70H06 Completely integrable systems and methods of integration for problems in Hamiltonian and Lagrangian mechanics
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