On the local structure of the Euler-Lagrange mapping of the calculus of   variations
  The purpose of this paper is to announce some new results on the structure of the higher order Euler-Lagrange mapping of the multiple-integral variational calculus on fibered manifolds,namely a description of its kernel and its image,and an explicit characterization of the conditions under which a system of partial differential equations (of arbitrary order)is a system of the Euler--Lagrange equations. 
Demeter Krupka