Periodic perturbations with rotational symmetry of planar systems driven by a central force
type: Item
instance of: scholarly article
title: Periodic perturbations with rotational symmetry of planar systems driven by a central force
zbMATH Open document ID: 1392.34045
author: Alessandro Fonda
author: Anna Chiara Gallo
published in: Journal of Differential Equations
publication date: 2018-04-04T00:00:00Z
review text: The authors consider the planar system \[ \ddot{x}+g(|x|)x=\epsilon \nabla_x V(t,x), \;0<\epsilon \ll 1, \] where \(g:(0,+\infty)\rightarrow \mathbb{R}\) is a continuous differentiable function and \(V:\mathbb{R}\times(\mathbb{R}^2\setminus\{0\})\rightarrow \mathbb{R}\) is continuous, \(T\)-periodic in its first variable, and twice differentiable in its second variable. Under the assumption of the rotational symmetry in perturbing force i.e. \[ \nabla_xV(T,\mathcal{R}x)=\mathcal{R}\nabla_xV(t,x) \;\text{for every } t,x, \] where \(\mathcal{R}\) is the matrix of rotation, the existence of nearly circular periodic orbits is deduced. The result is the generalization, in the planar case only, of the 1989 work of \textit{A. Ambrosetti} and \textit{V. Coti Zelati} [Math. Z. 201, No. 2, 227--242 (1989; Zbl 0653.34032)] who assumed the perturbing potential to be even. It uses similar techniques and it is not extensible to higher dimensions as discussed by the authors.
Mathematics Subject Classification ID: 34C25
Mathematics Subject Classification ID: 34E10
Mathematics Subject Classification ID: 34C14
zbMATH DE Number: 6855895
zbMATH Keywords: Kepler problem
zbMATH Keywords: periodic solutions
zbMATH Keywords: rotational symmetric systems
reviewed by: Predrag Punoševac
MaRDI profile type: Publication
full work available at URL: j.jde.2018.02.009
OpenAlex ID: W2792809344
cites work: Perturbation of Hamiltonian systems with Keplerian potentials
cites work: Periodic solutions of singular Lagrangian systems
cites work: Symmetry breaking in Hamiltonian systems
cites work: Periodic solutions of a perturbed Kepler problem in the plane: from existence to stability
cites work: Periodic solutions of symmetric perturbations of the Kepler problem
cites work: On a class of dynamical systems with singular potential
cites work: Q3218836
cites work: The Kepler problem. Group theoretical aspects, regularization and quantization, with application to the study of perturbations (incl. CD-ROM)
cites work: Q4731495
cites work: Radial periodic perturbations of the Kepler problem
cites work: Periodic orbits of radially symmetric Keplerian-like systems: a topological degree approach
cites work: Periodic solutions of radially symmetric perturbations of Newtonian systems
cites work: Periodic motions in a gravitational central field with a rotating external force
cites work: Periodic, subharmonic, and quasi-periodic oscillations under the action of a central force
cites work: Critical point theory and Hamiltonian systems
cites work: Stability of critical points under small perturbations. II: Analytic theory
cites work: Noncollision solutions to some singular minimization problems with Keplerian-like potentials
cites work: Q3326480
cites work: Periodic solutions for any planar symmetric perturbation of the Kepler problem
DOI: 10.1016/J.JDE.2018.02.009
Wikidata QID: Q130192585
description: scientific article; zbMATH DE number 6855895