Presymplectic current and the inverse problem of the calculus of variations

TL;DR

This paper explores how presymplectic forms in the variational bicomplex can determine whether a system of PDEs admits a variational formulation, extending previous results from ODEs to PDEs.

Contribution

It generalizes the inverse problem of the calculus of variations to PDEs using presymplectic forms, without restrictions on the order or variables involved.

Findings

Presymplectic forms enable construction of variational formulations for PDE subsystems.

The approach extends previous ODE results to PDEs.

Discussion on the uniqueness of the variational formulation.

Abstract

The inverse problem of the calculus of variations asks whether a given system of partial differential equations (PDEs) admits a variational formulation. We show that the existence of a presymplectic form in the variational bicomplex, when horizontally closed on solutions, allows us to construct a variational formulation for a subsystem of the given PDE. No constraints on the differential order or number of dependent or independent variables are assumed. The proof follows a recent observation of Bridges, Hydon and Lawson and generalizes an older result of Henneaux from ordinary differential equations (ODEs) to PDEs. Uniqueness of the variational formulation is also discussed.