Note on the Algebraic Aspect of the Integration of a System of Ordinary Linear Differential Equations

Abstract

In this note the Lie algebra generated by the coefficient matrix of a system of ordinary, linear, first‐order differential equations is considered. A systematic discussion, based on some well‐known results in the theory of Lie albegras, is given for the reduction of the problem of integration of such a system. For the purposes of this note the integration of a system of equations for which the coefficient matrix does not depend on the independent variable is regarded as ``elementary.'' It will be shown that the problem of integrating any system of linear ordinary differential equations can be reduced to the problem of integrating a set of such systems, each one of which has the property that the corresponding Lie algebra is simple, and in such a way that the sum of the dimensionalities of the Lie algebras of the reduced systems in the set does not exceed the dimensionality of the Lie algebra of the original system. The application of the reduction principle to the equations of motion in classical mechanics and in quantum mechanics is considered. It is shown that the principle in question applies to a class of Hamiltonian equations of motion not customarily regarded as describing linear systems.