Some perturbation results for nonlinear problems
type: Item
instance of: scholarly article
title: Some perturbation results for nonlinear problems
zbMATH Open document ID: 0799.58063
author: Carlo Carminati
published in: Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Serie IX. Rendiconti Lincei. Matematica e Applicazioni
publication date: 1994-07-13T00:00:00Z
full work available at URL: https://eudml.org/doc/244265
review text: A Hamiltonian system is considered whose Hamiltonian is a combination of a quadratic polynomial corresponding to the linear equations of motion, and a small higher-order part which is treated as a small perturbation. For this system, a brake orbit is defined as an orbit on which the velocity vanishes twice (at two different moments of time). Once the brake orbit is found, one can construct from it a periodic orbit. In this work, a theorem is proven which gives a minimum number of the brake orbits on a surface with a fixed value of energy. Generally speaking, the minimum number is equal to the number of degrees of freedom of the system. The theorem is proven by reducing it to a theorem giving a minimum number of closed geodesics on a sphere with a slightly perturbed metric; in turn, the latter theorem is proven by means of variational methods.
Mathematics Subject Classification ID: 37G99
Mathematics Subject Classification ID: 37J40
Mathematics Subject Classification ID: 37D40
Mathematics Subject Classification ID: 53D25
Mathematics Subject Classification ID: 58E05
zbMATH DE Number: 591103
zbMATH Keywords: periodic solutions
zbMATH Keywords: geodesics
reviewed by: B. A. Malomed
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 591103