Periodic solutions for a class of planar, singular dynamical systems
type: Item
instance of: scholarly article
title: Periodic solutions for a class of planar, singular dynamical systems
zbMATH Open document ID: 0633.34034
author: Vittorio Coti Zelati
published in: Journal de Mathématiques Pures et Appliquées. Neuvième Série
publication date: 1989-01-01T00:00:00Z
review text: 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 [\textit{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.
Mathematics Subject Classification ID: 34C25
Mathematics Subject Classification ID: 37-XX
Mathematics Subject Classification ID: 70F05
zbMATH DE Number: 4030005
zbMATH Keywords: singular dynamical systems
zbMATH Keywords: estimates
zbMATH Keywords: angular momentum conservation
zbMATH Keywords: critical points
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 4030005