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Multiple periodic trajectories in a relativistic gravitational field. (English) Zbl 0725.34038

Variational methods, Proc. Conf., Paris/Fr. 1988, Prog. Nonlinear Differ. Equ. Appl. 4, 373-381 (1990).
[For the entire collection see Zbl 0713.00009.]
The paper deals with a class of conservative systems with singular potentials. The existence of n closed, geometrically distinct trajectories with prescribed energy is proved. The result applies, for example, to potentials arising in the Post-Newtonian Celestial Mechanics, when the Kepler potential \(-| x|^{-1}\) is substituted by \(- | x|^{-1}-k| x|^{-2}\) where \(k\approx 1/c^ 2\), c \(=\) speed of light. The paper is related to other ones dealing with conservative systems with singular potentials: see, for example, the first author and V. Coti Zelati [Arch. Ration. Mech. Anal. 112, No.4, 339-362 (1990)] and references therein.
Reviewer: A.Ambrosetti

MSC:

34C25 Periodic solutions to ordinary differential equations
83C40 Gravitational energy and conservation laws; groups of motions
70F15 Celestial mechanics

Citations:

Zbl 0713.00009