The minimal period problem of classical Hamiltonian systems with even potentials
type: Item
instance of: scholarly article
title: The minimal period problem of classical Hamiltonian systems with even potentials
zbMATH Open document ID: 0804.58018
DOI: 10.1016/S0294-1449(16)30199-8
published in: Annales de l'Institut Henri Poincaré. Analyse Non Linéaire
publication date: 1995-01-08T00:00:00Z
full work available at URL: http://www.numdam.org/item
full work available at URL: https://eudml.org/doc/78319
review text: The author investigates the existence of non-constant periodic solutions with prescribed minimal period for a conservative (the author says ``second order Hamiltonian'') system of the form \(\ddot x + \text{grad }V = 0\) in \(\mathbb{R}^ n\), where the potential \(V: \mathbb{R}^ n \to \mathbb{R}\) is an even superquadratic autonomous function, with no convexity assumptions. He observes that the usual direct variational formulation of the system possesses a natural \(V_ 4\)-symmetry \((V_ 4 = Z_ 2 \oplus Z_ 2\) is the Kleinian Fourgroup). Using this as well as a new iteration inequality on the Morse index and Mountain-pass theorem, the author shows that for every \(T > 0\) the above system possesses a \(T\)-periodic solution with minimal period \(T\) or \(T/3\) and this solution is even about \(t = 0\), \(t = T/2\) and odd about \(t = T/4\), \(t = 3T/4\).
Mathematics Subject Classification ID: 37J99
Mathematics Subject Classification ID: 58E05
Mathematics Subject Classification ID: 34C25
zbMATH DE Number: 554341
zbMATH Keywords: even potential
zbMATH Keywords: periodic solutions
zbMATH Keywords: \(V_ 4\)-symmetry
author: Yiming Long
reviewed by: Yuri E. Gliklikh
MaRDI profile type: Publication
cites work: Q3899074
cites work: Solutions with minimal period for Hamiltonian systems in a potential well
cites work: Solutions of minimal period for a class of convex Hamiltonian systems
cites work: Hamiltonian trajectories having prescribed minimal period
cites work: Q3697938
cites work: Une théorie de Morse pour les systèmes hamiltoniens convexes
cites work: Q3728408
cites work: Q4692985
cites work: Periodic solutions with prescribed minimal period for convex autonomous Hamiltonian systems
cites work: Location, multiplicity and Morse indices of min-max critical points.
cites work: Some results on solutions of minimal period to superquadratic Hamiltonian systems
cites work: Solutions of minimal period for a class of nonconvex Hamiltonian systems and applications to the fixed energy problem
cites work: Periodic solutions of convex autonomous Hamiltonian systems with a quadratic growth at the origin and superquadratic at infinity
cites work: Dual Morse index estimates for periodic solutions of Hamiltonian systems in some nonconvex superquadratic cases
cites work: Essential critical points of linking type and solutions of minimal period to superquadratic Hamiltonian systems
cites work: A Geometric Description of the Neighbourhood of a Critical Point Given by the Mountain-Pass Theorem
cites work: Nontrivial solutions of operator equations and Morse indices of critical points of min-max type
cites work: Periodic solutions of hamiltonian systems
cites work: Periodic Solutions of Hamiltonian Systems: A Survey
cites work: Q3749777
cites work: Morse index estimates in min-max theorems
cites work: Q3715910
OpenAlex ID: W2289105526
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description: scientific article; zbMATH DE number 554341