Complex Generated by Variational Derivatives. Lagrangian Formalism of Infinite Order and a Generalized Stokes' Formula

TL;DR

This paper develops a geometric framework for Lagrangians of multidimensional paths, introducing an exterior differential analog and a generalized Stokes' formula, applicable to manifolds and supermanifolds.

Contribution

It introduces a novel geometric approach to infinite-order Lagrangians, establishing a cochain complex structure and a generalized Stokes' formula for multidimensional variational calculus.

Findings

Exterior differential analog acts on Lagrangians

Space of Lagrangians forms a cochain complex

Generalized Stokes' formula proven geometrically

Abstract

We prove that an analog of the exterior differential acts on the space of arbitrary Lagrangians of multidimensional paths on any manifold or supermanifold, thus making this space into a cochain complex. An analog of the Stokes' formula holds. The construction and the proofs are purely geometrical, in terms of the variation of corresponding actions.