On a new filtration of the variational bicomplex

TL;DR

This paper introduces a jet order-based filtration on the variational bicomplex, enabling a module structure on graded components and simplifying the expression of vanishing conditions for functional forms.

Contribution

It defines a new filtration on the variational bicomplex that preserves structure at the graded level, facilitating module-based analysis of functional forms.

Findings

Filtration preserved by the interior Euler operator.

Graded components are isomorphic to modules.

Simplified module basis for vanishing conditions.

Abstract

We define a filtration on the variational bicomplex according to jet order. The filtration is preserved by the interior Euler operator, which is not a module homomorphism with respect to the ring of smooth functions on the jet space. However, the induced maps on the graded components of this filtration are. Furthermore, the space of functional forms in the image of the interior Euler operator inherits a filtration. Though the filtered subspaces are not submodules either, the graded components are isomorphic to linear spaces which do have module structures. This works for any fixed degree of the functional forms. In this way, the condition that a functional form vanishes can be stated concisely with a module basis. We work out explicitly two examples: one for functional forms of degree two in relation to the Helmholtz conditions and the other of arbitrary degree but with jet order one.