Analytical mini-max methods for Hamiltonian brake orbits of prescribed energy
type: Item
instance of: scholarly article
title: Analytical mini-max methods for Hamiltonian brake orbits of prescribed energy
zbMATH Open document ID: 0665.70022
DOI: 10.1016/0022-247X(88)90039-X
published in: Journal of Mathematical Analysis and Applications
publication date: 1988-01-01T00:00:00Z
review text: Brake orbits of a conservative system \(\ddot q=-V'(q)\) \((q(t)\in {\mathbb{R}}^ N)\) are non-constant solutions, satisfying \(\dot q(0)=\dot q(T)=0\) for \(T>0\). They give rise to a 2T-periodic solution. The author presents a constructive formulation of brake orbits with prescribed energy \(E>0\) via an analytical mini-max procedure. The potential V is assumed to be strictly convex, with global minimum at zero. The starting point is a kind of modified principle of least action for fixed energy orbits: after a rescaling of the equations of motion, the author constructs a function \(J_ E\) on the Sobolev space \(X=W_{1,2}([0,1],{\mathbb{R}}^ N)\), whose critical points are the brake orbits of energy E. It is argued that X is the union of halfspaces \(M_{\xi}\) in \({\mathbb{R}}^{N+1}\), where \(\xi\) runs over the space of normalized functions with zero mean value. Restricted to each \(M_{\xi}\), \(J_ E\) is shown to have a unique critical point (a positive maximum) in the interior. Letting \(N_ E\) denote the union of these maxima, the problem is posed to minimize \(J_ E\) over \(N_ E\). It is shown that this problem has at least one solution x, with \(J_ E(x)>0\) and that the corresponding brake orbit is one with minimal period. In the final section, the author calls on Lyusternik-Schnirelmann theory to derive a multiplicity result concerning the number of distinct brake orbits with energy E.
Mathematics Subject Classification ID: 70H30
Mathematics Subject Classification ID: 34C25
Mathematics Subject Classification ID: 49J35
zbMATH DE Number: 4089002
zbMATH Keywords: Brake orbits
zbMATH Keywords: conservative system
zbMATH Keywords: periodic solution
zbMATH Keywords: analytical mini-max procedure
zbMATH Keywords: fixed energy orbits
zbMATH Keywords: Sobolev space
zbMATH Keywords: Lyusternik-Schnirelmann theory
author: Embrecht W. C. Van Groesen
reviewed by: W. Sarlet
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 4089002