Lagrangian formalism and the intrinsic geometry of PDEs

TL;DR

This paper introduces the concept of internal Lagrangians for differential equations, explores their spectral sequences, and links them to presymplectic structures, providing new insights into the geometry of PDEs and gauge theories.

Contribution

It presents a novel notion of internal Lagrangian, analyzes its spectral sequence, and connects it to presymplectic structures and gauge theories.

Findings

Defined internal Lagrangian for differential equations

Established spectral sequence related to internal Lagrangians

Connected internal Lagrangians with presymplectic structures

Abstract

A notion of internal Lagrangian for a system of differential equations is introduced. A spectral sequence related to internal Lagrangians is obtained. A connection between internal Lagrangians and presymplectic structures is investigated. An interpretation of the term $E^{3,\, n-2}_2$ of Vinogradov's $\mathcal{C}$-spectral sequence is given for irreducible gauge theories.