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Periodic solutions of a class of Hamiltonian systems with singularities

Published online by Cambridge University Press:  14 November 2011

Antonio Ambrosetti  and
Ivar Ekeland
Antonio Ambrosetti
Affiliation:
Scuola Normale Superiore, Pisa, 56100, Italy
Ivar Ekeland
Affiliation:
CERCEEMADE, Université Paris Dauphine, UA CNRS n. 749, France
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Synopsis

This paper deals with a class of time-periodic Hamiltonian systems obtained by a time-dependent perturbation from an autonomous system with a singularity at q = 0 in configuration space. It is shown that, T being the period of the perturbation, nondegenerate families of T-periodic orbits in the unperturbed problem branch off into a certain number of T-periodic orbits for the perturbed problem.

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 114 , Issue 1-2 , 1990 , pp. 1 - 13
Copyright
Copyright © Royal Society of Edinburgh 1990

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References

1Ambrosetti, A. and zelati, V. Coti. Critical points with lack of compactness and singular dynamical systems. Ann. Mat. Pura Appl. 149 (1987), 237–259.CrossRefGoogle Scholar
2Ambrosetti, A. and Zelati, V. Coti. Solutions periodiques sans collision pour une classe de potentiels de type Keplerien. C.R. Acad. Sri. Paris 305 (1987), 813–815.Google Scholar
3Ambrosetti, A. and Zelati, V. Coti. Perturbation of Hamiltonian systems with Keplerian potentials. Math. Zeit. (to appear).Google Scholar
4Ambrosetti, A., Zelati, V. Coti and Ekeland, I.. Symmetry breaking in Hamiltonian systems. J. Differential Equations 67 (1987), 165–184.CrossRefGoogle Scholar
5Ambrosetti, A. and Ekeland, I.. Perturbation results for a class of singular Hamiltonian Systems. Atti Ace. Naz. Lincei (to appear).Google Scholar
6Bahri, A. and Rabinowitz, P. H.. A minimax method for a class of Hamiltonian systems with singular potentials (preprint).Google Scholar
7Clarke, F.. Periodic solutions of Hamiltonian inclusions. J. Differential Equations 40 (1981), 1–6.CrossRefGoogle Scholar
8Degiovanni, M. and Giannoni, F.. Dynamical systems with Newtonian type potentials. Ann. Scuola Norm. Pisa (to appear).Google Scholar
9Degiovanni, M., Giannoni, F. and Marino, A.. Periodic solutions of dynamical systems with Newtonian type potential. Atti Ace. Naz. Lincei 81 (1987), 271–278.Google Scholar
10Ekeland, I., A perturbation theory near convex Hamiltonian systems. J. Differential Equations 50 (1983), 407–440.CrossRefGoogle Scholar
11Greco, C.. Periodic solutions of a class of singular Hamiltonian systems. Nonlinear Anal. T.M.A. 12 (1988), 259–270.CrossRefGoogle Scholar
12Kowalski, J.. Introduction to Celestial Mechanics (Dordrecht: Reidel, 1967).Google Scholar
13Meyer, K.. Periodic solutions of the N-body problem. J. Differential Equations 39 (1981), 2–38.CrossRefGoogle Scholar
14Moser, J.. Regularization of Kepler's problem and the averaging method on a manifold. Comm. Pure Appl. Math. 23 (1970), 609–636.CrossRefGoogle Scholar
15Poincaré, H.. Les methodes nouvelles de la Mecanique Celeste (Paris: Libr. Albert Blanchard, 1987).Google Scholar
16Sternberg, S.. Celestial Mechanics—Part II (New York: W. A. Benjamin, 1969).Google Scholar