zbMATH — the first resource for mathematics

Une théorie de Morse pour les systèmes hamiltoniens convexes. (French) Zbl 0537.58018
Author’s abstract: ”This paper deals with convex Hamiltonian systems. It is shown that, on any prescribed energy level, either the closed trajectories are infinitely many, or they fulfil a resonance condition. It follows that, generically, there are infinitely many closed trajectories on a prescribed energy level. A dual form of the least action principle, Morse theory, an iteration formula for the index, and Thom’s transversality theorems, are used to obtain these results.”
Reviewer: N.Papaghiuc

58E05 Abstract critical point theory (Morse theory, Lyusternik-Shnirel’man theory, etc.) in infinite-dimensional spaces
37J99 Dynamical aspects of finite-dimensional Hamiltonian and Lagrangian systems
Full Text: DOI Numdam EuDML
[1] R. Abraham, J. Robbin, Transversal mappings and flows. Benjamin. · Zbl 0171.44404
[2] Amann, H.; Zehnder, E., Nontrivial solutions for a class of nonresonance problems and applications to nonlinear differential equations, Ann. Sc. Norm. Sup. Pisa, t. 7, 539-603, (1980) · Zbl 0452.47077
[3] V. Arnold, Chapitres supplémentaires de la théorie des équations différentielles ordinaires. Éditions Mir, 1980 (original russe, 1978).
[4] V. Arnold, Méthodes mathématiques de la mécanique classique. Éditions Mir, 1974 (original russe, 1972).
[5] Ballmann, W.; Thorbergsson, G.; Ziller, W., Closed geodesics on positively curved manifolds, Annals of Math., t. 116, 213-247, (1982) · Zbl 0495.58010
[6] H. Berestycki, J. M. Lasry, G. Mancini, B. RUF, Existence of multiple periodic orbits on starshaped Hamiltonian surfaces. Preprint, 1983. · Zbl 0569.58027
[7] Birkhoff, G., Dynamical systems, (1927), AMS Colloquium Publications, (réédité, 1966) · JFM 53.0732.01
[8] R. Bott, Non-degenerate critical manifolds. Ann. of Math., 1954, p. 248-261. · Zbl 0058.09101
[9] Bott, R., On the iteration of closed geodesics and Sturm intersection theory, Comm. PAM, t. 9, 176-206, (1956) · Zbl 0074.17202
[10] Bott, R., Morse theory, old and new, Bull. AMS (New Series), t. 7, 331-358, (1982) · Zbl 0505.58001
[11] Clarke, F., Periodic solutions of Hamiltonian inclusions, J. Diff. Eq., t. 40, 1-6, (1980)
[12] Clarke, F.; Ekeland, I., Hamiltonian trajectories having prescribed minimal period, Comm. Pure App. Math., t. 33, 103-116, (1980) · Zbl 0403.70016
[13] C. Conley, E. Zehnder, Morse type index theory for flows and periodic solutions for Hamiltonian equations. Comm. Pure App. Math., to appear. · Zbl 0559.58019
[14] Croke, C.; Weinstein, A., Closed curves on convex hypersurfaces and periods of nonlinear oscillations, Inv. Math., t. 64, 199-202, (1981) · Zbl 0471.70020
[15] Duistermaat, J., On the Morse index in variational calculus, Advances in Math., t. 21, 173-195, (1976) · Zbl 0361.49026
[16] Ekeland, I., Periodic solutions of hamilton’s equations and a theorem of P. Rabinowitz, J. Diff. Eq., t. 34, 523-534, (1979) · Zbl 0446.70019
[17] Ekeland, I.; Lasry, J. M., On the number of closed trajectories for a Hamiltonian flow on a convex energy surface, Ann. Math., t. 112, 283-319, (1980) · Zbl 0449.70014
[18] I. Ekeland, R. Temam, Analyse convexe et problèmes variationnels. Dunod-Gauthier-Villars. · Zbl 0281.49001
[19] Gelfand, I.; Lidsky, V., On the structure of the regions of stability of linear canonical systems of differential equations with periodic coefficients, Uspekhi Math. Naouk, AMS Translation, t. 8, 143-181, (1958) · Zbl 0079.10905
[20] S. Jorna, ed., Topics is nonlinear dynamics. AIP Conference Proceedings, 1978.
[21] Klingenberg, Lectures on closed geodesics, (1981), Springer · Zbl 0397.58018
[22] M. Krasnosellskii, Topological methods in the theory of nonlinear integral equations. Pergamon Press.
[23] Krein, M., Generalisation of certain investigations of A.M. liapounov on linear differential equations with periodic coefficients, Doklady Akad. Naouk, USSR, t. 73, 445-448, (1950) · Zbl 0041.05602
[24] Meyer, W., Kritische mannigfaltigkeiten in hilbertmannigfaltigkeiten, Math. Ann., t. 170, 45-66, (1967) · Zbl 0142.21604
[25] Nemytskii, V.; Stepanov, V., Qualitative theory of differential equations, (1960), Princeton University Press · Zbl 0089.29502
[26] H. Poincaré, Les méthodes nouvelles de la mécanique céleste. Gauthier-Villars, p. 1892-1899.
[27] R. Robinson, The C^{1}closing lemma, preprint. · Zbl 0548.58012
[28] M. Struwe, On a critical point theory for minimal surfaces spanning a wire. Bonn preprint SFB n^{o} 569. · Zbl 0521.49028
[29] V. Yakubovich, V. Starzhinskii, Linear differential equations with periodic coefficients. Halsted Press, John Wiley et Sons. · Zbl 0066.33701
[30] Ekeland, I., Dualité et stabilité des systèmes hamiltoniens, CRAS Paris, t. 294, 673-676, (1982) · Zbl 0491.70021
[31] Ekeland, I., Une théorie de Morse pour LES systèmes hamiltoniens, CRAS Paris, t. 296, 117-120, (1983) · Zbl 0566.58014
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. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.