Periodic solutions of Hamiltonian systems of 3-body type

  • A. Bahri

    École Nationale d’Ingénieurs de Tunis Campus Universitaire Le Belvédère, Tunis, Tunisie, Mathematics Department. Rutgers University, New Brunswick, NJ 08903, U.S.A.
  • P.H. Rabinowitz

    Mathematics Department and Center for the Mathematical Sciences, University of Wisconsin, Madison, WI 53706, U.S.A.


We study the question of the existence of periodic solutions of Hamiltonian systems of the form:

\[ q\limits^{¨} + \mathrm{V}_{q}\left(t,\:q\right) = 0\:\mathrm{where}\:\mathrm{V} = \sum \limits_{1\:≦\:i\: \neq \:j}^{3}\mathrm{V}_{ij}\left(t,\:q_{i}−q_{j}\right) \]

with V(t, ξ) T-periodic in t and singular at ξ = 0. Under hypotheses on V of 3-body type, we prove that the functional corresponding to (✶) has an unbounded sequence of critical points provided that the singularity of V at 0 is strong enough.

Cite this article

A. Bahri, P.H. Rabinowitz, Periodic solutions of Hamiltonian systems of 3-body type. Ann. Inst. H. Poincaré Anal. Non Linéaire 8 (1991), no. 6, pp. 561–649

DOI 10.1016/S0294-1449(16)30252-9