Systems of second-order linear ODE's with constant coefficients and their symmetries
type: Item
instance of: scholarly article
title: Systems of second-order linear ODE's with constant coefficients and their symmetries
zbMATH Open document ID: 1235.34108
published in: Communications in Nonlinear Science and Numerical Simulation
publication date: 2011-07-13T00:00:00Z
review text: This well-written paper concerns the study of symmetry algebras (of Lie point symmetries) for systems \[ \ddot y=M_1\dot y+M_2 y+M_3(t), \] where \( y, M_3(t)\in\mathbb{R}^n\) and \(M_1,M_2\) are \(n\times n\) real matrices. Through appropriate changes of variables, this problem is reduced to the investigation of symmetry algebras of systems \[ \ddot x =J x, \] where \(J\) is an \(n\times n\) real matrix in Jordan form and \({ x}\in\mathbb{R}^n\). Among other achievements, the author proves the following main results. For \(n=3\), there was a symmetry algebra that was left out in the analysis performed in \textit{C. Wafo Soh} [Commun. Nonlinear Sci. Numer. Simul. 15, 139--143 (2010; Zbl 1221.34092)]. For \(n=4\), the dimension of the symmetry algebras may be 13, 15, 17, 19 or 35 (when \(J\) is a diagonal matrix) and ranges from 13 to 20 otherwise. Additionally, generators of these Lie algebras are explicitly detailed. For general \(n\) and diagonal \(J\), with \(J\neq \lambda Id\) and \(\lambda\in\mathbb{R}\), the generators and the dimension of the symmetry algebras are provided in terms of the eigenvalues of \(J\) and their multiplicities. Some results about the algebraic structure of such Lie algebras are also obtained.
Mathematics Subject Classification ID: 34C14
Mathematics Subject Classification ID: 34A30
Mathematics Subject Classification ID: 17B60
zbMATH DE Number: 5920353
zbMATH Keywords: symmetry algebra
zbMATH Keywords: Lie point symmetry
zbMATH Keywords: second-order ODE's with constant coefficients
zbMATH Keywords: linearization
zbMATH Keywords: Levi factor
author: Rutwig Campoamor Stursberg
reviewed by: J. de Lucas
MaRDI profile type: Publication
full work available at URL: j.cnsns.2010.10.033
OpenAlex ID: W1985227160
Wikidata QID: Q58324962
cites work: Symmetry Lie algebras of \(n\)th order ordinary differential equations
cites work: Linearization criteria for a system of second-order ordinary differential equations
cites work: Symmetries of linear systems of second-order ordinary differential equations
cites work: Symmetry breaking of systems of linear second-order ordinary differential equations with constant coefficients
cites work: Lie point symmetries for systems of second order linear ordinary differential equations
cites work: Matrix Analysis
cites work: Q3924413
DOI: 10.1016/J.CNSNS.2010.10.033
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description: scientific article; zbMATH DE number 5920353