The inverse problem of the calculus of variations: Separable systems

This paper deals with the inverse problem of the calculus of variations for systems of second-order ordinary differential equations. The case of the p roblem which Douglas, in his classification of pairs of such equations, cal led the 'separated case' is generalized to arbitrary dimension. After ident ifying the conditions which should specify such a case for n equations in a coordinate-free way, two proofs of its variationality are presented. The f irst one follows the line of approach introduced by some of the authors in previous work, and is close in spirit, though being coordinate independent, to the Riquier analysis applied by Douglas for n = 2. The second proof is more direct and leads to the discovery that belonging to the 'separated cas e' has an intrinsic meaning for the given second-order differential equatio ns: the system is separable in the sense that it can be decoupled into n pa irs of first-order equations.