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Decomposition of nondegenerate singularities of integrable Hamiltonian systems. (English) Zbl 0842.58032
The main purpose of the paper is to give a topological classification of stable nondegenerate singularities of smooth integrable Hamiltonian systems. Namely, the author shows that all such singularities can be decomposed diffeomorphically, after a finite covering, to the direct product of simplest (codimension 1 and codimension 2 focus-focus) singularities.
Reviewer: Y.Kozai (Tokyo)

MSC:
37J05 Relations of dynamical systems with symplectic geometry and topology (MSC2010)
37J35 Completely integrable finite-dimensional Hamiltonian systems, integration methods, integrability tests
58K45 Singularities of vector fields, topological aspects
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[1] Arnold, V. I.,Mathematical Methods of Classical Mechanics, Mir, Moscow, 1978. · Zbl 0386.70001
[2] Bolsinov, A. V.,Methods of computing Fomenko-Zieschang invariant, Advances in Soviet Mathematics, 6, 1991 (Fomenko ed.). · Zbl 0744.58029
[3] Dufour, J. P. and Molino, P.,Compactification d’action de R n et variables action-angle avec singularités, MSRI Publ. 20, Springer-Verlag, New York, 1990, pp. 151-167. · Zbl 0752.58011
[4] Eliasson, L. H., Normal form for Hamiltonian systems with Poisson commuting integrals - elliptic case,Comment Math. Helv. 65, 4-35, (1990). · Zbl 0702.58024 · doi:10.1007/BF02566590
[5] Fomenko, A. T.,Symplectic Geometry, Gordon and Breach, New York, 1988, andIntegrability and Nonintegrability in Geometry and Mechanics, Kluwer, Dordrecht, 1988.
[6] Lerman, L. and Umanskii, Ya., Structure of the Poisson action ofR 2 on a four dimensional symplectic manifold, I and II,Selecta Math. Sovietica 6(4), 365-396 (1987),7(1), 39-48 (1988). · Zbl 0635.58009
[7] Lerman, L. and Umanskii, Ya., Classification of four-dimensional integrable Hamiltonian systems in extended neighborhoods of simple singular points,Methods of Qualitative Theory of Bifurcations, Izdat. Gorkov. Univ., Gorki, 1988, pp. 67-76.
[8] Vey, J., Sur certaines systèmes dynamiques séparables,Amer. J. Math. 100, 591-614 (1978). · Zbl 0384.58012 · doi:10.2307/2373841
[9] Williamson, J. On the algebraic problem concerning the normal forms of linear dynamical systems,Amer. J. Math. 58(1), 141-163 (1936). · Zbl 0013.28401 · doi:10.2307/2371062
[10] Nguyen Tien Zung, Singularities of integrable geodesic flows on multi-dimensional torus and sphere, preprint 5/1994. · Zbl 0995.37040
[11] Nguyen Tien Zung, Symplectic topology of integrable Hamiltonian systems, Ph D thesis, Strasbourg, 1994. · Zbl 0995.37040
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