Symmetries of linear systems of second-order ordinary differential equations

Abstract

In this paper, several problems concerning the Lie algebra structure of symmetries and variational symmetries of a general linear system of second-order ordinary differential equations are studied. In particular, a necessary and sufficient condition is obtained, in terms of the coefficients of the system, for the system’s symmetry algebra to be of maximal dimension (i.e., n2+4n+3) and isomorphic to sl(n+2,R), the well-known symmetry algebra of the free-particle equation x″=0. When this condition is satisfied, it is proved that the system is Lagrangian and that its variational symmetry algebra is isomorphic to a fixed, (n2+3n+6)/2- dimensional Lie algebra, whose structure (Levi–Mal’cev decomposition and realization by means of a matrix algebra) is determined. For the particular case of isotropic systems (which includes, as far as is known, all the examples treated in the literature), explicit formulas for the generators of both the symmetry algebra and the variational symmetry algebra are obtained.