On the Group Classification of Systems of Two Linear Second-Order Ordinary Differential Equations with Constant Coefficients

TL;DR

This paper completes the group classification of two linear second-order ODE systems with constant coefficients, especially for cases where coefficient matrices do not commute, extending previous literature.

Contribution

It provides a complete classification including new cases where coefficient matrices are non-commutative, and discusses extensions to larger systems.

Findings

Complete classification of non-commutative coefficient cases

Identification of new symmetry structures

Extension to systems with more than two equations

Abstract

The completeness of the group classification of systems of two linear second-order ordinary differential equations with constant coefficients is delineated in the paper. The new cases extend what has been done in the literature. These cases correspond to the type of equations where the commutative property of the coefficient matrices with respect to the dependent variables and the first-order derivatives in the considered system does not hold. A discussion of the results as well as a note on the extension to linear systems of second-order ordinary differential equations with more than two equations are given.