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

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

1998-01-15

arxiv.org/abs/math/9801073v1

Mathematics

Differential Geometry

42 pages, LATEX2e

Scientific paper

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.

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