Homoclinics and heteroclinics for a Lagrangian system
type: Item
instance of: scholarly article
title: Homoclinics and heteroclinics for a Lagrangian system
zbMATH Open document ID: 0936.37036
author: Paul H. Rabinowitz
published in: Ricerche di Matematica
publication date: 2000-05-22T00:00:00Z
review text: The paper considers a family of Lagrangian systems on \(\mathbb{T}^{2}\), described by the Lagrangian \[ L(q)=\sum_{i,j=1}^{2}a_{ij}(q)\dot q_{i}\dot q_{j}-V(q), \] where \(V\) and \(a_{i,j}\) are \(C^{2}\)-differentiable and \(V(0)=0>V(x)\), \(x\in \mathbb{T}^{2}\). In recent years some papers [see e.g. \textit{S. Bolotin} and \textit{P. Negrini}, A variational criterion for nonintegrability, Russ. J. Math. Phys. 5, 415-436 (1998)] have established the existence of heteroclinic orbits of prescribed homotopy type for such families. The results required a geometrical assumption that implies the minimum of an associated variational problem is a periodic orbit of the Lagrangian system rather than a homoclinic on \(\mathbb{T}^{2}\). In this paper the author shows that some stronger geometrical condition together with the variational framework developed by the author leads to much richer families of homoclinic and heteroclinic solutions of the system.
Mathematics Subject Classification ID: 37J45
Mathematics Subject Classification ID: 34C37
Mathematics Subject Classification ID: 70H03
Mathematics Subject Classification ID: 70K44
zbMATH DE Number: 1370122
zbMATH Keywords: homoclinic trajectory
zbMATH Keywords: heteroclinic trajectory
zbMATH Keywords: Lagrangian system
zbMATH Keywords: heteroclinic orbits
zbMATH Keywords: homoclinic solutions
zbMATH Keywords: heteroclinic solutions
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 1370122