Periodic solutions of Lagrangian systems on a compact manifold
type: Item
instance of: scholarly article
title: Periodic solutions of Lagrangian systems on a compact manifold
zbMATH Open document ID: 0605.58034
DOI: 10.1016/0022-0396(86)90045-8
published in: Journal of Differential Equations
publication date: 1986-01-01T00:00:00Z
review text: The author studies periodic solutions of prescribed period when the Lagrangian system is constrained to a compact manifold. This fact allows him to use many techniques developed in the theory of closed geodesics on Riemannian compact manifolds [cf. the reviewer, Foundations of Global Nonlinear Analysis, Teubner-Texte zur Mathematik, Vol. 86, Leipzig (1986)]. He considers M to be a smooth n-dimensional manifold with TM its tangent bundle. He then considers a time periodic Lagrangian of periodic T, \({\mathcal L}_ t: TM\to R\), and he searches for T-periodic solutions of the Lagrange equations, which in local coordinates are \[ (1)\quad \frac{d}{dt}\frac{\partial {\mathcal L}}{\partial \dot q}(t,q,\dot q)- \frac{\partial {\mathcal L}}{\partial q}(t,q,\dot q)=0,\quad i=1,2,...,n. \] The author proves that if the fundamental group of M is finite, then (1) has infinitely many T-periodic solutions, if \({\mathcal L}_ t\) satisfies certain physically reasonable assumptions. The author makes extensive use of the Palais-Smale condition (C).
Mathematics Subject Classification ID: 37G99
Mathematics Subject Classification ID: 37J99
zbMATH DE Number: 3977991
zbMATH Keywords: Sobolev space
zbMATH Keywords: Lie group
zbMATH Keywords: Abelian group
zbMATH Keywords: periodic solutions
zbMATH Keywords: Lagrangian system
zbMATH Keywords: compact manifold
zbMATH Keywords: Palais-Smale condition
author: Vieri Benci
reviewed by: Michael Th. Rassias
MaRDI profile type: Publication
full work available at URL: 0022-0396(86)90045-8
OpenAlex ID: W2138309527
cites work: THE TOPOLOGY OF FUNCTIONAL MANIFOLDS AND THE CALCULUS OF VARIATIONS IN THE LARGE
cites work: Q4181002
cites work: Morse theory on Hilbert manifolds
cites work: Periodic Solutions of Hamiltonian Systems: A Survey
cites work: The homology theory of the closed geodesic problem
description: scientific article; zbMATH DE number 3977991