×

Orbites périodiques des systèmes hamiltoniens du type de celui des trois corps. (Periodic orbits of Hamiltonian systems of the three-body type). (French) Zbl 0731.70007

The paper announces some interesting new results concerning periodic orbits of so-called three-body like problems (which are presented in full detail in a paper with the same title, in English). The general setting is about three bodies interacting via a T-periodic potential, satisfying a number of natural conditions at infinity and also Gordon’s “hypothesis of strong interaction”. The terminology in this respect is a bit misleading as within the “family of gravitational potentials”, the true gravitational potential does not satisfy that hypothesis. Using a variational approach on a Hilbert space of orbits with a suitable norm, the main theorems state the existence of a double infinity of T-periodic solutions. There is a brief discussion of the autonomous case and of the complications which arise when the strong interaction hypothesis is dropped: collisions then cannot be excluded and require passing to generalized solutions. Much of the attention goes to the main complication in the variational approach for three bodies, namely the violation of the Palais-Smale condition. The problem is being dealt with by an extension of the classical variational techniques via a generalization of the idea of Morse’s lemma, leading to a result which the authors qualify as a Morse lemma at infinity.
Reviewer: W.Sarlet (Gent)

MSC:

70F07 Three-body problems
37G99 Local and nonlocal bifurcation theory for dynamical systems
34C25 Periodic solutions to ordinary differential equations
PDFBibTeX XMLCite
Full Text: DOI Numdam EuDML

References:

[1] Poincaré, H.) .- Les méthodes nouvelles de la Mécanique Céleste, Librairie Albert Blanchard, Paris1987 · JFM 30.0834.08
[2] Alekseev, V.M.) .- On the capture orbits for the three-body problem for negative energy constant, Uspekhi Mat. Nauk, 24 (1969) pp. 185-186 · Zbl 0188.29101
[3] Alekseev, V.M.) . - Sur l’allure finale du mouvement dans le problème des trois corps, Actes du Congrès Int. des Math.1970, 2, pp. 893-907, Gauthier-Villars, Paris1971 · Zbl 0266.70005
[4] Gordon, W.B.) . - Conservative dynamical systems involving strong forces, Trans. Amer. Math. Soc., 204 (1975) pp. 113-135 · Zbl 0276.58005
[5] Ambroseti, A.) and Coti-Zelati, V.) .- Critical points with lack of compactness and applications to singular Hamiltonian systems, to appear
[6] Degiovanni, M.), Giannoni, F.) and Marino, A.) .- Periodic solutions of dynamical systems with Newtonian type potentials, in ”Periodic Solutions of Hamiltonian Systems and Related Topics (P.H. Rabinowitz, et al Eds) 29, pp. 111-115, , Reidel, Dordrecht1987 · Zbl 0632.34038
[7] Greco, C.) . - Periodic solutions of a class of singular hamiltonian systems, Nonlinear Analysis : TMA, 12 (1988) pp. 259-270 · Zbl 0648.34048
[8] Coti-Zelati, V.) - Morse Theory and periodic solutions of Hamiltonian systems, Preprint, SISSATrieste · Zbl 0652.34053
[9] Sullivan, D.) and Vigué-Poirrier, M.) .- The homology theory of the closed geodesic problem, J. Diff. Geom., 11 (1976) pp. 633-644 · Zbl 0361.53058
[10] Bahri, A.) and Rabinowitz, P.H.) .- A minimax method for a class of Hamiltonian systems with singular potentials, J. Funct. Anal., 82 (1983) pp. 412-428 · Zbl 0681.70018
[11] Bahri, A.) and Rabinowitz, P.H.) .- Periodic solutions of Hamiltonian systemsof 3-body type, to appear · Zbl 0745.34034
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. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.