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The Arnold-Liouville theorem of and its consequences. (Le théorème d’Arnold-Liouville et ses conséquences.) (English) Zbl 1112.37043

The author studies some consequences of the following result: a compact connected, \(n\)-dimensional manifold that admits n commuting, everywhere independent vector fields is dipheomorphic to an \(n\)-dimensional torus. Such result is applied in Section 2 to Hamiltonian mechanical systems. However, no requirement for the maximal rank of the matrix representation \(J\) of the Poisson structure is made.

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
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