Solution of the inverse problem of the calculus of variations

Abstract

Given a family of curves constituting the general solution of a system of ordinary differential equations, the natural question occurs whether the family is identical with the totality of all extremals of an appropriate variational problem. Assuming the regularity of the latter problem, effective approaches are available but they fail in the non-regular case. However, a rather unusual variant of the calculus of variations based on infinitely prolonged differential equations and systematic use of Poincaré-Cartan forms makes it possible to include even all constrained variational problems. The new method avoids the use of Lagrange multiplitiers. For this reason, it is of independent interest especially in regard to the 23rd Hilbert's problem.