×

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

References:

[1] Arnold, V.I. : Mathematical methods of classical mechanics , Springer-Verlag, 1978. · Zbl 0386.70001
[2] Audin, M. : The topology of torus action on symplectic manifolds, Progress in Mathematics , V. 93, Birkhauser, 1991. · Zbl 0726.57029
[3] Audin, M. : Courbes algébriques et systèmes intégrables: géodésiques des quadriques , preprint IRMA Strasbourg 1993. · Zbl 0843.58064
[4] Audin, M. : Toupies, A course on integrable systems , Strasbourg 1994.
[5] Audin, M. and Silhol, R. : Variétés abéliennes réelles et toupie de Kowalevski , Compositio Math. 87 (1993), 153-229. · Zbl 0774.58012
[6] Bates, L. : Monodromy in the champagne bottle , Z. Angew. Math. Phys. 42 (1991) No. 6, 837-847. · Zbl 0755.58028 · doi:10.1007/BF00944566
[7] Bau, Tit and Tien Zung, Nguyen : Separation of coordinates and topology of integrable Hamiltonian systems , in preparation. · Zbl 0869.58022
[8] Birkhoff, G.D. : Dynamical systems , AMS Colloq. Publ. IX, 1927. · JFM 53.0732.01
[9] Bolsinov, A.V. : Methods of computation of the Fomenko-Zieschang invariant, Advances in Soviet Mathematics , V. 6 (Fomenko ed.), 1991, 147-183. · Zbl 0744.58029
[10] Bolsinov, A.V. : Compatible Poisson brackets on Lie algebras and completeness of families of functions in involution , Math. USSR Izvestiya 38 (1992) 69-90. · Zbl 0744.58030 · doi:10.1070/IM1992v038n01ABEH002187
[11] Bolsinov, A.V. , Matveev, S.V. and Fomenko, A.T. : Topological classification of integrable Hamiltonian systems with two degrees of freedom. List of all systems of small complexity , Russ. Math. Surv. 45 (1990) No. 2, 59-99. · Zbl 0705.58025 · doi:10.1070/RM1990v045n02ABEH002344
[12] Boucetta, M. and Molino, P. : Géométrie globale des systèmes hamiltoniens complètement integrables , CRAS Paris, Ser. I, 308 (1989) 421-424. · Zbl 0702.58032
[13] Condevaux, M. , Dazord, P. and Molino, P. : Géométrie du moment, Séminaire Sud-Phodanien , Publications du départment de math., Univ. Claude Bernard - Lion I, 1988.
[14] Cushman, R. and Knörrer, H. : The energy momentum mapping of the Lagrange top , Lecture Notes in Math. 1139 (1985) 12-24. · Zbl 0615.70002
[15] Dazord, P. and Delzant, T. : Le problème général des variables action-angles , J. Diff. Geom. 26 (1987) No. 2, 223-251. · Zbl 0634.58003 · doi:10.4310/jdg/1214441368
[16] Delzant, T. : Hamiltoniens périodiques et image convexe de l’application moment , Bull. Soc. Math. France 116 (1988) 315-339. · Zbl 0676.58029 · doi:10.24033/bsmf.2100
[17] Desolneux-Moulis, N. : Singular Lagrangian foliation associated to an integrable Hamiltonian vector field , MSRI Publ., Vol. 20 (1990) 129-136. · Zbl 0731.58038
[18] Devaney, R. : Transversal homoclinic orbit in an integrable system , Amer. J. Math. 100 (1978) 631-642. · Zbl 0406.58019 · doi:10.2307/2373844
[19] Dufour, J.-P. : Théorème de Nekhoroshev à singularités, Séminaire Gaston Darboux , Montpellier 1989- 1990, 49-56. · Zbl 0738.53022
[20] Dufour, J.-P. and Molino, P. : Compactification d’action de Rn et variables action-angle avec singularités , MSRI Publ., Vol. 20 (1990) (Séminaire Sud-Rhodanien de Géométrie à Berkeley, 1989, P. Dazord and A. Weinstein eds.) 151-167 · Zbl 0752.58011
[21] Duistermaat, J.J. : On global action-angle variables , Comm. Pure Appl. Math. 33 (1980) 687-706. · Zbl 0439.58014 · doi:10.1002/cpa.3160330602
[22] Duistermaat, J.J. and Heckman, G.J. : On the variation in the cohomology of the symplectic form of the reduced phase space and Addendum , Invent. Math. 69 (1982) 259-269 and 72 (1983) 153-158. · Zbl 0503.58016 · doi:10.1007/BF01389132
[23] Eliasson, L.H. : Normal form for Hamiltonian systems with Poisson commuting integrals- elliptic case , Comm. Math. Helv. 65 (1990) 4-35. · Zbl 0702.58024 · doi:10.1007/BF02566590
[24] Ercolani, N.M. and Mclaughlin, D.W. : Towarda topological classification of integrable PDE’s , MSRI Publ., V. 22 (1990) 111-130. · Zbl 0743.58020
[25] Fomenko, A.T. : Symplectic geometry , Gordon and Breach, New York, 1988, and Integrability and nonintegrability in geometry and mechanics , Kluwer, Dordrecht, 1988.
[26] Fomenko, A.T. : Topological classification of all integrable Hamiltonian systems of general types with two degrees of freedom , MSRI Publ., Vol. 22 (1991) 131-340. · Zbl 0753.58014
[27] Gavrilov, L. : Bifurcation of invariant manifolds in the generalized Hénon-Heiles system , Physica D, 34 (1989) 223-239. · Zbl 0689.58014 · doi:10.1016/0167-2789(89)90236-4
[28] Gavrilov, L. , Ouazzani-Jamil, M. and Caboz, R. : Bifurcation diagrams and Fomenko’s surgery on Liouville tori of the Kolossoff potential U = p + (1/\rho ) - k cos \Phi , Ann. Sci. Ecole Norm. Sup., 4e serie, 26 (1993) 545-564. · Zbl 0797.34042 · doi:10.24033/asens.1680
[29] Kharlamov, M.P. : Topological analysis of integrable problems in the dynamics of a rigid body , Izd. Leningrad. Univ., Leningrad (Saint-Peterbourg) 1988 (Russian). · Zbl 0561.58021
[30] Koiller, J. : Melnikov formulae for nearly integrable Hamiltonian systems , MSRI Publ., V. 20 (1990) 183-188. · Zbl 0731.58037
[31] Kowalevski, S. : Sur le problème de la rotation d’un corps solide autour d’un point fixe , Acta Math. 12 (1989) 177-232. · JFM 21.0935.01
[32] Lerman, L. and Umanskii, Ya. : Structure of the Poisson action of R 2 on a four-dimensional symplectic manifold, I and II , Selecta Math. Sovietica Vol. 6, No. 4 (1987) 365-396, and Vol. 7, No. 1 (1988) 39-48. · Zbl 0649.58017
[33] 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, 67-76.
[34] Lerman, L. and Umanskii, Ya. : Classification of four-dimensional integrable hamiltonian systems and Poisson actions of R2 in extended neighborhoods of simple singular points, I and II , Russian Math. Sb. 77 (1994) 511-542 and 78 (1994) 479-506. · Zbl 0819.58018 · doi:10.1070/SM1994v078n02ABEH003481
[35] Marsden, J.E. and Weinstein, A. : Reduction of symplectic manifolds with symmetry , Rep. Math. Phys. 5 (1974) 121-130. · Zbl 0327.58005 · doi:10.1016/0034-4877(74)90021-4
[36] Moser, J. : On the volume elements on manifolds , Trans. AMS 120 (1965) 280-296. · Zbl 0141.19407 · doi:10.2307/1994022
[37] Molino, P. : Du théorème d’Arnol’d-Liouville aux formes normales de systmèmes hamiltoniens toriques: une conjecture. Séminaire Gaston Darboux , Montpellier 1989-1990, 39-47. · Zbl 0734.70013
[38] Nekhoroshev, N.N. : Action-angle variables and their generalizations , Trans. Moscow Math. Soc. 26 (1972) 180-198. · Zbl 0284.58009
[39] Oshemkov, A.A. : Fomenko invariants for the main integrable cases of the rigid body motion equations , Advances in Soviet Mathematics, V. 6 (1991) A. T. Fomenko ed., 67-146. · Zbl 0745.58028
[40] Polyakova, L. and Tien Zung, Nguyen : A topological classification of integrable geodesic flows on the two-dimensional sphere with an additional integral quadratic in the momenta , J. Nonlinear Sci. 3 (1993) No.1, 85-108. · Zbl 0802.58044 · doi:10.1007/BF02429860
[41] Rüssmann, H. : Über das Verhalten analytischer Hamiltonscher Differentialgleichungen in der Nähe einer Gleichgewichtslösung , Math. Ann. 154 (1964) 285-300. · Zbl 0124.04701 · doi:10.1007/BF01362565
[42] Veeravalli, A. : Compactification d’actions de Rn et systèmes hamiltoniens de type torique , CRAS Paris, Ser. I 317 (1993) 289-293. · Zbl 0810.58015
[43] Vey, J. : Sur certaines systèmes dynamiques séparables , Amer. J. Math. 100 (1978) 591-614. · Zbl 0384.58012 · doi:10.2307/2373841
[44] Williamson, J. : On the algebraic problem concerning the normal forms of linear dynamical systems , Amer. J. Math. 58:1 (1936) 141-163. · Zbl 0013.28401 · doi:10.2307/2371062
[45] Zou, M. : Monodromy in two degrees of freedom integrable systems , J. Geom. and Phys. 10 (1992) 37-45. · Zbl 0776.58017 · doi:10.1016/0393-0440(92)90006-M
[46] Tien Zung, Nguyen : On the general position property of simple Bott integrals , Russ. Math. Surv. 45 (1990) No. 4, 179-180. · Zbl 0724.58030 · doi:10.1070/RM1990v045n04ABEH002370
[47] Tien Zung , Nguyen: Decomposition of nondegenerate singularities of integrable Hamiltonian systems , Lett. Math. Phys. 33 (1994) 187-193. · Zbl 0842.58032 · doi:10.1007/BF00749620
[48] Tien Zung, Nguyen : A note on focus-focus singularities , Diff. Geom. Appl. (to appear). · Zbl 0887.58023 · doi:10.1016/S0926-2245(96)00042-3
[49] Tien Zung, Nguyen : Symplectic topology of integrable Hamiltonian systems , thesis, Strasbourg May/1994. · Zbl 0995.37040
[50] Tien Zung, Nguyen : Singularities of integrable geodesic flows on multidimensional torus and sphere , J. Geometry and Physics (to appear). · Zbl 0849.58053 · doi:10.1016/0393-0440(95)00008-9
[51] Tien Zung, Nguyen : Symplectic topology of integrable Hamiltonian systems, II: Characteristic classes and integrable surgery , preprint 1995. · Zbl 1127.53308 · doi:10.1023/A:1026133814607
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.