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Intersection de sous-variétés lagrangiennes, fonctionnelles d’action et indice des systèmes hamiltoniens. (Intersection of Lagrangian submanifolds, action functionals and indices of Hamiltonian systems). (French) Zbl 0639.58018
Most of the advances in modern symplectic topology to a proof of Arnold’s conjectures on fixed points and Lagrangian intersections are connected beginning from Conley-Zehnder theorem with investigation of action functionals on path space in a symplectic manifold. A purely geometrical interpretation for the relative Morse index of critical points of the functional in Maslov index terms is discovered in the work and an expression of the index through conjugate points of appropriate linear Hamiltonian flows is given.
Reviewer: A.Givental’

MSC:
37J99 Dynamical aspects of finite-dimensional Hamiltonian and Lagrangian systems
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[1] ARNOLD (V. I.) . Appendice au livre de Maslov , V.P. : Théorie des perturbations et méthodes asymptotiques (traduit du russe), Dunod, Gauthier-Villars, Paris, 1972 .
[2] ARTIN (E.) . Geometric Algebra , Interscience, New York, 1957 . MR 18,553e | Zbl 0077.02101 · Zbl 0077.02101
[3] AMANN (H.) et ZEHNDER (E.) . Periodic solutions of asymptotically linear hamiltonian systems , Manus. Math., t. 32, 1980 , p. 149-189. MR 82i:58026 | Zbl 0443.70019 · Zbl 0443.70019 · doi:10.1007/BF01298187 · eudml:154723
[4] BERESTICKY (H.) , LASRY (J. M.) , MANCINI (G.) et RUF (B.) . Existence of multiple periodic orbits on starshaped hamiltonian surfaces , Comm. Pure Appl. Math., t. 38, 1985 , p. 253-289. Zbl 0569.58027 · Zbl 0569.58027 · doi:10.1002/cpa.3160380302
[5] BOREL (A.) . Cohomologie des espaces localement compacts d’après J. Leray , Lecture Notes in Math. n^\circ 2, Springer-Verlag, Berlin, Heidelberg, New York. 1984 . MR 30 #4252 | Zbl 0126.38803 · Zbl 0126.38803 · doi:10.1007/BFb0097851
[6] BROUSSEAU (V.) . Thèse de 3e cycle , Université de Paris-IX, 1985 .
[7] CHAPERON (M.) . Une idée du type géodésiques brisées pour les systèmes hamiltoniens , C.R. Acad. Sc., Paris, t. 298, 1984 , p. 293-296. MR 86f:58049 | Zbl 0576.58010 · Zbl 0576.58010
[8] CLARKE (F. H.) . Periodic solutions to Hamiltonian inclusions , J. of Diff. Eq., t. 40, 1980 , p. 1-6. MR 83a:58035 | Zbl 0461.34030 · Zbl 0461.34030 · doi:10.1016/0022-0396(81)90007-3
[9] CLARKE (F. H.) et EKELAND (I.) . Hamiltonian trajectories having prescribed minimal period , Comm. Pure Appl. Math., t. 33, 1980 , p. 103-116. MR 81e:70017 | Zbl 0403.70016 · Zbl 0403.70016 · doi:10.1002/cpa.3160330202
[10] CONLEY (C.) et ZEHNDER (E.) . Morse type index theory for flows and periodic solutions for Hamiltonian equations , Comm. Pure Appl. Math., t. 37, 1984 , p. 207-253. MR 86b:58021 | Zbl 0559.58019 · Zbl 0559.58019 · doi:10.1002/cpa.3160370204
[11] DUISTERMAAT (J. J.) . On the Morse index in variational calculus , Advances in Math., t. 21, 1976 , p. 173-195. MR 58 #31190 | Zbl 0361.49026 · Zbl 0361.49026 · doi:10.1016/0001-8708(76)90074-8
[12] EKELAND (I.) . Une théorie de Morse pour les systèmes hamiltoniens convexes , Annales de l’I.H.P., Analyse non linéaire, vol. 1.1, 1984 , p. 19-78. Numdam | MR 85f:58023 | Zbl 0537.58018 · Zbl 0537.58018 · numdam:AIHPC_1984__1_1_19_0 · eudml:78065
[13] EKELAND (I.) . Index theory for periodic solutions of convex Hamiltonian systems , Cahiers de M.D. n^\circ 85003, Université de Paris-Dauphine, 1985 . · Zbl 0596.34023
[14] GUILLEMIN (V.) et STERNBERG (S.) . Geometric Asymptotics , Math. Surveys, n^\circ 14, A.M.S. Providence R.I., 1977 . MR 58 #24404 | Zbl 0364.53011 · Zbl 0364.53011 · www.ams.org
[15] HUSEMOLLER (D.) . Fibre Bundles , 2nd éd., Springer-Verlag, Berlin, New York, 1975 . MR 51 #6805 | Zbl 0307.55015 · Zbl 0307.55015
[16] LAUDENBACH (F.) et SIKORAV (J. C.) . Persistance d’intersection avec la section nulle au cours d’une isotopie hamiltonienne dans un fibre cotangent , Inventiones Math., vol. 82, Fasc. 2, p. 349-357, 1985 . MR 87c:58042 | Zbl 0592.58023 · Zbl 0592.58023 · doi:10.1007/BF01388807 · eudml:143290
[17] MORSE (M.) . The calculus of Variations in the large , A.M.S. Coll. Publ., vol. 18, A.M.S., New York, 1934 . MR 98f:58070 | Zbl 0011.02802 | JFM 60.0450.01 · Zbl 0011.02802 · www.emis.de
[18] MILNOR (J. W.) and STASHEFF (J. D.) . Characteristic classes , Annals of Math. Studies, n^\circ 76, Princeton Univ. Press, 1974 , Princeton N.J. MR 55 #13428 | Zbl 0298.57008 · Zbl 0298.57008
[19] SIKORAV (J. C.) . Sur les immersions lagrangiennes dans un fibré cotangent admettant une phase génératrice globale , C.R. Acad. Sc. Paris, t. 302, série 1, 1986 , p. 119-122. MR 87h:58070 | Zbl 0602.58019 · Zbl 0602.58019
[20] STEENROD (N.) . The topology of fibre bundles , Princeton University Press, 1951 , Princeton N.J. MR 12,522b | Zbl 0054.07103 · Zbl 0054.07103
[21] VITERBO (C.) . Thèse de 3e cycle , Université de Paris-IX, 1985 .
[22] WEINSTEIN (A.) . Lectures on symplectic manifolds , C.B.M.S. regional conference series in Math., n^\circ 29, A.M.S., 1979 , Providence R.I. MR 82b:58039 | Zbl 0406.53031 · Zbl 0406.53031
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