Periodic first integrals for Hamiltonian systems of Lie type
type: Item
instance of: scholarly article
zbMATH Open document ID: 1246.34002
DOI: 10.1142/S0219887811005634
author: Ruben Flores Espinoza
published in: International Journal of Geometric Methods in Modern Physics
publication date: 2012-06-12T00:00:00Z
full work available at URL: https://arxiv.org/abs/1004.1132
Mathematics Subject Classification ID: 34A05
Mathematics Subject Classification ID: 34C14
Mathematics Subject Classification ID: 37J15
Mathematics Subject Classification ID: 53Z05
Mathematics Subject Classification ID: 70H33
zbMATH DE Number: 6045599
zbMATH Keywords: periodic Hamiltonian systems
zbMATH Keywords: Lie system
zbMATH Keywords: Euler system
zbMATH Keywords: periodic first integrals
zbMATH Keywords: Floquet theory
zbMATH Keywords: Milne-Pinney equation
MaRDI profile type: Publication
arXiv ID: 1004.1132
cites work: Superposition rules, Lie theorem, and partial differential equations
cites work: Lie-Scheffers systems: a geometric approach
cites work: Ermakov systems, exact solution, and gemetrical angles and phases
cites work: A Remark on Rational Isochronous Potentials
cites work: Class of Exact Invariants for Classical and Quantum Time-Dependent Harmonic Oscillators
title: Periodic first integrals for Hamiltonian systems of Lie type
review text: Let \((P, \omega )\) be a symplectic manifold and \(X\) a vector field on \(\mathbb{R}\times P\). Then \(X\) is called a \textit{Hamiltonian vector field of Lie type} if \(X(t, x)=\sum _{\alpha =1}^n b_{\alpha }(t)X_{\alpha }(x), x\in P\), with \((X_{\alpha })\) Hamiltonian vector fields on \(P\) spanning an \(n\)-dimensional real Lie algebra and \((b_{\alpha })\) smooth real functions. If \((b_{\alpha })\) are all periodic with the same period then \(X\) is called periodic. The present paper provides a general criterion for the existence of periodic first integrals for periodic Hamiltonian vector fields of Lie type. The Floquet theory is used to derive general conditions for the existence of a Poisson algebra of periodic first integrals. A very interesting application is considered for the Milne-Pinney system that describes the time evolution of an isotonic oscillator.
reviewed by: Mircea Crasmareanu
description: scientific article; zbMATH DE number 6045599