On a Generalisation of the Poincare-Cartan Form to Classical Field Theory

We present here a possible generalisation of the Poincar\'e-Cartan form in classical field theory in the most general case: arbitrary dimension, arbitrary order of the theory and in the absence of a fibre bundle structure. We use for the kinematical description of the system the $(r,n)$-Grassmann manifold associated to a given manifold $X$, i.e. the manifold of $r$-contact elements of $n$-dimensional submanifolds of $X$. The idea is to define globally a $n+1$ form on this Grassmann manifold, more precisely its class with respect to a certain subspace and to write it locally as the exterior derivative of a $n$ form which is the Poincar\'e-Cartan form. As an important application we obtain a new proof for the most general expression of a variationally trivial Lagrangian.