Time-dependent linear systems derivable from a variational principle

This paper presents a study of time-varying linear systems of second order ordinary differential equations, which can be derived from a Lagrangian after multiplication by a suitable matrix. It concerns a generalization of previous studies on systems with constant coefficients. After a simplification of the Helmholtz conditions, it is shown that the problem is reduced to a purely algebraic one, provided one can solve a matrix differential equation which produces the transformation to canonical form of the given system. This further leads to a theoretical characterization of all systems admitting a multiplier. Various algebraic relations are derived, involving constant matrices only, which can help to detect, prior to any integration procedure, whether or not: a multiplier exists. They are referred to as the generalized commutativity conditions. The first of these, which is sufficient for the existence of a Lagrangian, is shown to allow also a simple construction of a quadratic first integral, and to have some other interesting features. The paper ends with an example.