Systems of second-order linear ODE's with constant coefficients and their symmetries
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.
second-order ODE's with constant coefficients
scientific article; zbMATH DE number 5920353
scientific article