Perturbation of Hamiltonian systems with Keplerian potentials
type: Item
instance of: scholarly article
title: Perturbation of Hamiltonian systems with Keplerian potentials
zbMATH Open document ID: 0653.34032
DOI: 10.1007/BF01160679
author: A. Ambrosetti
author: Vittorio Coti Zelati
published in: Mathematische Zeitschrift
publication date: 1989-01-01T00:00:00Z
full work available at URL: https://eudml.org/doc/174049
review text: The paper deals with the existence of periodic solutions for a second order Hamiltonian system of the form \[ (P_{\epsilon})\quad q''+\alpha q| q|^{-\alpha -2}+\epsilon V_ q(t,q)=0 \] for \(\epsilon\) small, where \(\alpha >0\) and V is T-periodic in t. It is shown that \((P_{\epsilon})\) has ``several'' T-periodic solutions bifurcating from the circular orbits of the unperturbed system \((P_ 0)\quad q''+\alpha q| q|^{-\alpha -2}=0\) without any assumption on V if \(\alpha\neq 1\), and under the assumption V even in q and T/2 periodic in t if \(\alpha =1\). In the case that V does not depend on time, a change of variable allows to prove the same existence result (for any given \(T>0)\) also for \(\epsilon =1\), provided V grows less than \(| q|^{-\alpha}\) in the origin.
Mathematics Subject Classification ID: 34C25
Mathematics Subject Classification ID: 70H05
zbMATH DE Number: 4065463
zbMATH Keywords: Keplerian potentials
zbMATH Keywords: existence
zbMATH Keywords: periodic solutions
zbMATH Keywords: second order Hamiltonian system
zbMATH Keywords: circular orbits
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 4065463