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

Nguyen Tien Zung

Compositio Mathematica (1996)

  • Volume: 101, Issue: 2, page 179-215
  • ISSN: 0010-437X

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Zung, Nguyen Tien. "Symplectic topology of integrable hamiltonian systems, I : Arnold-Liouville with singularities." Compositio Mathematica 101.2 (1996): 179-215. <http://eudml.org/doc/90442>.

@article{Zung1996,
author = {Zung, Nguyen Tien},
journal = {Compositio Mathematica},
keywords = {Arnold-Liouville theorem; integrable Hamiltonian system; moment map},
language = {eng},
number = {2},
pages = {179-215},
publisher = {Kluwer Academic Publishers},
title = {Symplectic topology of integrable hamiltonian systems, I : Arnold-Liouville with singularities},
url = {http://eudml.org/doc/90442},
volume = {101},
year = {1996},
}

TY - JOUR
AU - Zung, Nguyen Tien
TI - Symplectic topology of integrable hamiltonian systems, I : Arnold-Liouville with singularities
JO - Compositio Mathematica
PY - 1996
PB - Kluwer Academic Publishers
VL - 101
IS - 2
SP - 179
EP - 215
LA - eng
KW - Arnold-Liouville theorem; integrable Hamiltonian system; moment map
UR - http://eudml.org/doc/90442
ER -

References

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  1. 1 Arnold, V.I.: Mathematical methods of classical mechanics, Springer-Verlag, 1978. Zbl0386.70001MR690288
  2. 2 Audin, M.: The topology of torus action on symplectic manifolds, Progress in Mathematics, V. 93, Birkhauser, 1991. Zbl0726.57029MR1106194
  3. 3 Audin, M.: Courbes algébriques et systèmes intégrables: géodésiques des quadriques, preprint IRMAStrasbourg1993. Zbl0843.58064MR1295705
  4. 4 Audin, M.: Toupies, A course on integrable systems, Strasbourg1994. 
  5. 5 Audin, M. and Silhol, R.: Variétés abéliennes réelles et toupie de Kowalevski, Compositio Math.87 (1993), 153-229. Zbl0774.58012MR1219634
  6. 6 Bates, L.: Monodromy in the champagne bottle, Z. Angew. Math. Phys.42 (1991) No. 6, 837-847. Zbl0755.58028MR1140696
  7. 7 Bau, Tit and Tien Zung, Nguyen: Separation of coordinates and topology of integrable Hamiltonian systems, in preparation. Zbl0869.58022
  8. 8 Birkhoff, G.D.: Dynamical systems, AMS Colloq. Publ. IX, 1927. Zbl53.0732.01
  9. 9 Bolsinov, A.V.: Methods of computation of the Fomenko-Zieschang invariant, Advances in Soviet Mathematics, V. 6 (Fomenko ed.), 1991, 147-183. Zbl0744.58029MR1141222
  10. 10 Bolsinov, A.V.: Compatible Poisson brackets on Lie algebras and completeness of families of functions in involution, Math. USSR Izvestiya38 (1992) 69-90. Zbl0744.58030MR1130028
  11. 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. Zbl0705.58025MR1069348
  12. 12 Boucetta, M. and Molino, P.: Géométrie globale des systèmes hamiltoniens complètement integrables, CRAS Paris, Ser. I, 308 (1989) 421-424. Zbl0702.58032MR992521
  13. 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. MR1040868
  14. 14 Cushman, R. and Knörrer, H.: The energy momentum mapping of the Lagrange top, Lecture Notes in Math.1139 (1985) 12-24. Zbl0615.70002MR820467
  15. 15 Dazord, P. and Delzant, T.: Le problème général des variables action-angles, J. Diff. Geom.26 (1987) No. 2, 223-251. Zbl0634.58003MR906389
  16. 16 Delzant, T.: Hamiltoniens périodiques et image convexe de l'application moment, Bull. Soc. Math. France116 (1988) 315-339. Zbl0676.58029MR984900
  17. 17 Desolneux-Moulis, N.: Singular Lagrangian foliation associated to an integrable Hamiltonian vector field, MSRI Publ., Vol. 20 (1990) 129-136. Zbl0731.58038MR1104922
  18. 18 Devaney, R.: Transversal homoclinic orbit in an integrable system, Amer. J. Math.100 (1978) 631-642. Zbl0406.58019MR494258
  19. 19 Dufour, J.-P.: Théorème de Nekhoroshev à singularités, Séminaire Gaston Darboux, Montpellier1989- 1990, 49-56. Zbl0738.53022MR1124473
  20. 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 Zbl0752.58011
  21. 21 Duistermaat, J.J.: On global action-angle variables, Comm. Pure Appl. Math.33 (1980) 687-706. Zbl0439.58014MR596430
  22. 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. Zbl0503.58016MR674406
  23. 23 Eliasson, L.H.: Normal form for Hamiltonian systems with Poisson commuting integrals- elliptic case, Comm. Math. Helv.65 (1990) 4-35. Zbl0702.58024
  24. 24 Ercolani, N.M. and Mclaughlin, D.W.: Towarda topological classification of integrable PDE's, MSRI Publ., V. 22 (1990) 111-130. Zbl0743.58020
  25. 25 Fomenko, A.T.: Symplectic geometry, Gordon and Breach, New York, 1988, and Integrability and nonintegrability in geometry and mechanics, Kluwer, Dordrecht, 1988. MR994805
  26. 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. Zbl0753.58014MR1123281
  27. 27 Gavrilov, L.: Bifurcation of invariant manifolds in the generalized Hénon-Heiles system, Physica D, 34 (1989) 223-239. Zbl0689.58014MR982389
  28. 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. Zbl0797.34042
  29. 29 Kharlamov, M.P.: Topological analysis of integrable problems in the dynamics of a rigid body, Izd. Leningrad. Univ., Leningrad (Saint-Peterbourg) 1988 (Russian). Zbl0561.58021MR948454
  30. 30 Koiller, J.: Melnikov formulae for nearly integrable Hamiltonian systems, MSRI Publ., V. 20 (1990) 183-188. Zbl0731.58037MR1104927
  31. 31 Kowalevski, S.: Sur le problème de la rotation d'un corps solide autour d'un point fixe, Acta Math.12 (1989) 177-232. JFM21.0935.01
  32. 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. Zbl0649.58017MR925264
  33. 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. 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. Zbl0819.58018MR1213368
  35. 35 Marsden, J.E. and Weinstein, A.: Reduction of symplectic manifolds with symmetry, Rep. Math. Phys.5 (1974) 121-130. Zbl0327.58005MR402819
  36. 36 Moser, J.: On the volume elements on manifolds, Trans. AMS120 (1965) 280-296. Zbl0141.19407MR182927
  37. 37 Molino, P.: Du théorème d'Arnol'd-Liouville aux formes normales de systmèmes hamiltoniens toriques: une conjecture. Séminaire Gaston Darboux, Montpellier1989-1990, 39-47. Zbl0734.70013MR1124472
  38. 38 Nekhoroshev, N.N.: Action-angle variables and their generalizations, Trans. Moscow Math. Soc.26 (1972) 180-198. Zbl0284.58009MR365629
  39. 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. Zbl0745.58028MR1141221
  40. 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. Zbl0802.58044MR1216988
  41. 41 Rüssmann, H.: Über das Verhalten analytischer Hamiltonscher Differentialgleichungen in der Nähe einer Gleichgewichtslösung, Math. Ann.154 (1964) 285-300. Zbl0124.04701MR179409
  42. 42 Veeravalli, A.: Compactification d'actions de Rn et systèmes hamiltoniens de type torique, CRAS Paris, Ser. I317 (1993) 289-293. Zbl0810.58015MR1233428
  43. 43 Vey, J.: Sur certaines systèmes dynamiques séparables, Amer. J. Math.100 (1978) 591-614. Zbl0384.58012MR501141
  44. 44 Williamson, J.: On the algebraic problem concerning the normal forms of linear dynamical systems, Amer. J. Math.58:1 (1936) 141-163. MR1507138JFM62.1795.10
  45. 45 Zou, M.: Monodromy in two degrees of freedom integrable systems, J. Geom. and Phys.10 (1992) 37-45. Zbl0776.58017MR1195671
  46. 46 Tien Zung, Nguyen: On the general position property of simple Bott integrals, Russ. Math. Surv.45 (1990) No. 4, 179-180. Zbl0724.58030MR1075400
  47. 47 Tien Zung, Nguyen: Decomposition of nondegenerate singularities of integrable Hamiltonian systems, Lett. Math. Phys.33 (1994) 187-193. Zbl0842.58032MR1321315
  48. 48 Tien Zung, Nguyen: A note on focus-focus singularities, Diff. Geom. Appl. (to appear). Zbl0887.58023
  49. 49 Tien Zung, Nguyen: Symplectic topology of integrable Hamiltonian systems, thesis, Strasbourg May/1994. 
  50. 50 Tien Zung, Nguyen: Singularities of integrable geodesic flows on multidimensional torus and sphere, J. Geometry and Physics (to appear). Zbl0849.58053MR1375166
  51. 51 Tien Zung, Nguyen: Symplectic topology of integrable Hamiltonian systems, II: Characteristic classes and integrable surgery, preprint 1995. MR1343777

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