EQUIVALENCE OF HIGHER-ORDER LAGRANGIANS .1. FORMULATION AND REDUCTION

It is shown that the general equivalence problem for an r-th order variational problem (with or without the addition of a divergence) can always be formulated as a Cartan equivalence problem on the jet bundle J(r). Moreover, equivalence on any higher order jet bundle automatically reduces to equivalence on J(r). As a consequence, we deduce the existence of "derivative covariants", which are certain functions of the partial derivatives of a suitably nondegenerate r-th order Lagrangian whose transformation rules are the same as those of the n-th order derivatives for any n > r. This implies that any such Lagrangian determines an invariantly defined system of n-th order differential equations for any n > r, generalizing the Euler-Lagrange equations.