Helmholtz conditions,covariance, and invariance identities

This paper is concerned with the problem of finding a multiplier matrixg which converts a prescribed system of second-order ordinary differential equations to the Euler-Lagrange form. Sufficient conditions for the existence of a multiplier matrix are given in the form of an infinite system of linear algebraic equations, provided the entries ofg may be regarded as components of a (0, 2) symmetric tensor field. As an application, conditions for the local existence of a metric tensor compatible with a given torsion-free connection are deduced.