Periodic solutions for a class of planar, singular dynamical systems. (English) Zbl 0633.34034

Coti Zelati, Vittorio

J. Math. Pures Appl., IX. Sér. 68, No. 1, 109-119 (1989).

It is proved that the existence of at least one T-periodic solution of minimal period T for a system of differential equations of the form - ÿ\(=\nabla V(y)\) where \(y\in {\mathbb{R}}^ 2\), \(V: {\mathbb{R}}^ 2\setminus \{0\}\to {\mathbb{R}}\) and \(V(y)\equiv -| y| ^{-\alpha}-U(y)\) with \(\alpha >1\) and U, spherically symmetric in a ball of radius c small, is such that \(U(y)| y| ^{\alpha}\to 0\) as \(| y| \to 0\). Moreover such a solution is not all contained in the little ball of radius c. The problem of finding T-periodic solutions for singular potentials has been studied by many authors. See [A. Ambrosetti and the author, Periodic solutions of singular dynamical systems, in Periodic solutions of Hamiltonian systems and related topics, Proc. NATO Adv. Res. Workshop, Il Ciocco/Italy 1986, NATO ASI Ser., Ser. C 209, 1-10] for a survey of the results of this field. In particular, little is known on the problem when V(y) behaves, close to the singularities set, as \(| y| ^{-\alpha}\), \(1\leq \alpha <2\). In fact, in such a situation, one loses the control on the behavior of the action functional (whose critical points are the T-periodic solutions of (V)) on the functions passing through the origin. In the present paper estimates are found using the angular momentum conservation which allows one to find critical points of the action functional “far” from the singularity set.

Reviewer: V.Coti Zelati

MSC:

34C25	Periodic solutions to ordinary differential equations
37-XX	Dynamical systems and ergodic theory
70F05	Two-body problems

Keywords:

singular dynamical systems; estimates; angular momentum conservation; critical points