  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.
