Symplectic topology of integrable hamiltonian systems, I : Arnold-Liouville with singularities
Zung, Nguyen Tien
Compositio Mathematica, Tome 104 (1996), p. 179-215 / Harvested from Numdam
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     author = {Zung, Nguyen Tien},
     title = {Symplectic topology of integrable hamiltonian systems, I : Arnold-Liouville with singularities},
     journal = {Compositio Mathematica},
     volume = {104},
     year = {1996},
     pages = {179-215},
     mrnumber = {1389366},
     zbl = {0936.37042},
     language = {en},
     url = {http://dml.mathdoc.fr/item/CM_1996__101_2_179_0}
}

Zung, Nguyen Tien. Symplectic topology of integrable hamiltonian systems, I : Arnold-Liouville with singularities. Compositio Mathematica, Tome 104 (1996) pp. 179-215. http://gdmltest.u-ga.fr/item/CM_1996__101_2_179_0/

1 Arnold, V.I.: Mathematical methods of classical mechanics, Springer-Verlag, 1978. | MR 690288 | Zbl 0386.70001

2 Audin, M.: The topology of torus action on symplectic manifolds, Progress in Mathematics, V. 93, Birkhauser, 1991. | MR 1106194 | Zbl 0726.57029

3 Audin, M.: Courbes algébriques et systèmes intégrables: géodésiques des quadriques, preprint IRMA Strasbourg 1993. | MR 1295705 | 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. | Numdam | MR 1219634 | Zbl 0774.58012

6 Bates, L.: Monodromy in the champagne bottle, Z. Angew. Math. Phys. 42 (1991) No. 6, 837-847. | MR 1140696 | Zbl 0755.58028

7 Bau, Tit and Tien Zung, Nguyen: Separation of coordinates and topology of integrable Hamiltonian systems, in preparation.

8 Birkhoff, G.D.: Dynamical systems, AMS Colloq. Publ. IX, 1927.

9 Bolsinov, A.V.: Methods of computation of the Fomenko-Zieschang invariant, Advances in Soviet Mathematics, V. 6 (Fomenko ed.), 1991, 147-183. | MR 1141222 | 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. | MR 1130028 | Zbl 0744.58030

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. | MR 1069348 | Zbl 0705.58025

12 Boucetta, M. and Molino, P.: Géométrie globale des systèmes hamiltoniens complètement integrables, CRAS Paris, Ser. I, 308 (1989) 421-424. | MR 992521 | 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. | MR 1040868

14 Cushman, R. and Knörrer, H.: The energy momentum mapping of the Lagrange top, Lecture Notes in Math. 1139 (1985) 12-24. | MR 820467 | 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. | MR 906389 | Zbl 0634.58003

16 Delzant, T.: Hamiltoniens périodiques et image convexe de l'application moment, Bull. Soc. Math. France 116 (1988) 315-339. | Numdam | MR 984900 | Zbl 0676.58029

17 Desolneux-Moulis, N.: Singular Lagrangian foliation associated to an integrable Hamiltonian vector field, MSRI Publ., Vol. 20 (1990) 129-136. | MR 1104922 | Zbl 0731.58038

18 Devaney, R.: Transversal homoclinic orbit in an integrable system, Amer. J. Math. 100 (1978) 631-642. | MR 494258 | Zbl 0406.58019

19 Dufour, J.-P.: Théorème de Nekhoroshev à singularités, Séminaire Gaston Darboux, Montpellier 1989- 1990, 49-56. | MR 1124473 | 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. | MR 596430 | Zbl 0439.58014

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. | MR 674406 | Zbl 0503.58016

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

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. | MR 994805

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. | MR 1123281 | Zbl 0753.58014

27 Gavrilov, L.: Bifurcation of invariant manifolds in the generalized Hénon-Heiles system, Physica D, 34 (1989) 223-239. | MR 982389 | Zbl 0689.58014

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/ρ) - k cos Φ, Ann. Sci. Ecole Norm. Sup., 4e serie, 26 (1993) 545-564. | Numdam | Zbl 0797.34042

29 Kharlamov, M.P.: Topological analysis of integrable problems in the dynamics of a rigid body, Izd. Leningrad. Univ., Leningrad (Saint-Peterbourg) 1988 (Russian). | MR 948454

30 Koiller, J.: Melnikov formulae for nearly integrable Hamiltonian systems, MSRI Publ., V. 20 (1990) 183-188. | MR 1104927 | 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. | MR 925264 | 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. | MR 1213368 | Zbl 0819.58018

35 Marsden, J.E. and Weinstein, A.: Reduction of symplectic manifolds with symmetry, Rep. Math. Phys. 5 (1974) 121-130. | MR 402819 | Zbl 0327.58005

36 Moser, J.: On the volume elements on manifolds, Trans. AMS 120 (1965) 280-296. | MR 182927 | Zbl 0141.19407

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. | MR 1124472 | Zbl 0734.70013

38 Nekhoroshev, N.N.: Action-angle variables and their generalizations, Trans. Moscow Math. Soc. 26 (1972) 180-198. | MR 365629 | 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. | MR 1141221 | 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. | MR 1216988 | Zbl 0802.58044

41 Rüssmann, H.: Über das Verhalten analytischer Hamiltonscher Differentialgleichungen in der Nähe einer Gleichgewichtslösung, Math. Ann. 154 (1964) 285-300. | MR 179409 | Zbl 0124.04701

42 Veeravalli, A.: Compactification d'actions de Rn et systèmes hamiltoniens de type torique, CRAS Paris, Ser. I 317 (1993) 289-293. | MR 1233428 | Zbl 0810.58015

43 Vey, J.: Sur certaines systèmes dynamiques séparables, Amer. J. Math. 100 (1978) 591-614. | MR 501141 | Zbl 0384.58012

44 Williamson, J.: On the algebraic problem concerning the normal forms of linear dynamical systems, Amer. J. Math. 58:1 (1936) 141-163. | JFM 62.1795.10 | MR 1507138

45 Zou, M.: Monodromy in two degrees of freedom integrable systems, J. Geom. and Phys. 10 (1992) 37-45. | MR 1195671 | Zbl 0776.58017

46 Tien Zung, Nguyen: On the general position property of simple Bott integrals, Russ. Math. Surv. 45 (1990) No. 4, 179-180. | MR 1075400 | Zbl 0724.58030

47 Tien Zung, Nguyen: Decomposition of nondegenerate singularities of integrable Hamiltonian systems, Lett. Math. Phys. 33 (1994) 187-193. | MR 1321315 | Zbl 0842.58032

48 Tien Zung, Nguyen: A note on focus-focus singularities, Diff. Geom. Appl. (to appear). | Zbl 0887.58023

49 Tien Zung, Nguyen: Symplectic topology of integrable Hamiltonian systems, thesis, Strasbourg May/1994.

50 Tien Zung, Nguyen: Singularities of integrable geodesic flows on multidimensional torus and sphere, J. Geometry and Physics (to appear). | MR 1375166 | Zbl 0849.58053

51 Tien Zung, Nguyen: Symplectic topology of integrable Hamiltonian systems, II: Characteristic classes and integrable surgery, preprint 1995. | MR 1343777