Symplectic topology of integrable Hamiltonian systems, I: Arnold-Liouville with singularities

By Nguyen Tien Zung. 2001

203 citations recorded by the source corpus.

The classical Arnold-Liouville theorem describes the geometry 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.

Subfields: Mathematical Physics, Statistical and Nonlinear Physics

View Paper: https://www.semanticscholar.org/paper/621472fc066a134056b80316c59045544647ff6d

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In Mathematical Physics

1. N. Leung
2. Margaret Symington
3. A. Oshemkov
4. Alexey Bolsinov
5. Nguyen Tien Zung
6. Renato Vianna
7. 'Alvaro Pelayo
8. San Vũ Ngọc

In Statistical and Nonlinear Physics

1. A. Oshemkov
2. Alexey Bolsinov
3. Margaret Symington
4. N. Leung
5. Nguyen Tien Zung
6. San Vũ Ngọc
7. 'Alvaro Pelayo

Citations in the corpus

• Symplectic Topology of Integrable Hamiltonian Systems, II: Topological Classification (2000, methodology): “In [39] we studied nondegenerate singularities of integrable Hamiltonian systems, where we showed, among other things, that they are topologically equivalent to almost direct products of simplest singularities.”
• Almost toric symplectic four-manifolds (2003, background): “Note that this lemma was observed by Zung [45] without proof.”
• Bi-Hamiltonian structures and singularities of integrable systems (2009, background): “It is not hard to generalize this definition to the case of arbitrary rank of dΦ(x) (see, for example, [19, 20]).”
• Symplectic theory of completely integrable Hamiltonian systems (2011, unknown): “These structures have been studied in the context of integrable systems (in particular by Nguyên Tiên Zung [115]), but they also became a central concept in the works by Symington [100] and Symington and Leung [71] in the context of symplectic geometry and topology, and by Gross and Siebert [52, 53,”