Inverse problem in the calculus of variations -- functional and antiexact forms

TL;DR

This paper explores the relationship between functional forms and antiexact differential forms in the calculus of variations, revealing how antiexact forms obstruct variationality and proposing a new variational principle for non-variational differential equations.

Contribution

It introduces a novel connection between variational bicomplex theory and antiexact forms, providing a new approach to formulate variational principles for non-variational equations.

Findings

Identifies antiexact forms as obstructions to variationality.

Formulates variational principles for non-variational equations.

Illustrates the theory with heat, Navier-Stokes, and KdV equations.

Abstract

We connect the well-known theory of functional forms of variational bicomplex with the theory of antiexact differential forms. We identify antiexact functional forms as an obstruction to the variationality of differential equations. The most prominent result of this observation is the formulation of the variational problem for some differential equations that are not variational and neither have a variational multiplier. Heat, Navier-Stokes, and KdV equations illustrate this new variational principle.