On the dynamics of mechanical systems with the homogeneous polynomial potential \(V = ax ^{4} + cx ^{2} y ^{2}\)
type: Item
instance of: scholarly article
title: On the dynamics of mechanical systems with the homogeneous polynomial potential \(V = ax ^{4} + cx ^{2} y ^{2}\)
zbMATH Open document ID: 1196.37097
author: Claudio Vidal
author: Manuel Falconi
author: Ernesto A. Lacomba
published in: Journal of Dynamics and Differential Equations
publication date: 2009-10-14T00:00:00Z
review text: The authors consider a planar mechanical system \(\ddot{q} =\nabla V(q)\), \(q = (x, y) \in \mathbb R^{2}\), where the potential \(V\) is a homogeneous polynomial of degree four in two variables, \(V(x, y) = ax^{4} + cx^{2}y^{2}\), \(a, c\in \mathbb R\). The associated Hamiltonian system is \(\dot{x} = p_1\), \(\dot{y}=p_2\), with \(\dot{p_1} = 2x[2ax^{2} + cy^{2}]\) and \(\dot{p_2} = 2cx^{2}y\). The authors give relevant information about the dynamics of the Hamiltonian system as a function of the parameters \(a, c\in \mathbb R\). Also they study the Hill region as a function of the energy level and the parameters \(a\) and \(c\). The authors characterize explicitly some particular solutions of the mechanical system, namely equilibria and homothetic solutions. In general the paper is devoted to analyze certain dynamical aspects of the planar mechanical system associated to the potential \(V\).
reviewed by: Patricia Domínguez
Mathematics Subject Classification ID: 37J30
Mathematics Subject Classification ID: 70H99
zbMATH DE Number: 5615002
zbMATH Keywords: potential
zbMATH Keywords: polynomial, Hamiltonian system
MaRDI profile type: Publication
full work available at URL: s10884-009-9144-7
OpenAlex ID: W2005164933
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DOI: 10.1007/S10884-009-9144-7
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description: scientific article; zbMATH DE number 5615002