The integrability conditions in the inverse problem of the calculus of variations for second-order ordinary differential equations

A novel approach to a coordinate-free analysis of the multiplier question in the inverse problem of the calculus of variations, initiated in a previous publication, is completed in the following sense: under quite general circumstances, the complete set of passivity or integrability conditions is computed for systems with arbitrary dimension n. The results are applied to prove that the problem is always solvable in the case that the Jacobi endomorphism of the system is a multiple of the identity. This generalizes to arbitrary n a result derived by Douglas for n = 2