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 i n the inverse problem of the calculus of variations, initiated in a previou s publication, is completed in the following sense: under quite general cir cumstances, the complete set of passivity or integrability conditions is co mputed for systems with arbitrary dimension n. The results are applied to p rove that the problem is always solvable in the case that the Jacobi endomo rphism of the system is a multiple of the identity. This generalizes to arb itrary n a result derived by Douglas for n = 2.