On integrability of discrete variational systems. Octahedron relations

TL;DR

This paper investigates the integrability of discrete variational systems by analyzing corner equations and their relation to octahedron relations, providing explicit formulas and demonstrating their role in system consistency.

Contribution

It introduces the concept that the consistency of corner equations can be derived from two independent octahedron relations, with explicit formulas provided.

Findings

Corner equations' consistency follows from octahedron relations.

Explicit formulas for octahedron relations are derived.

The work links octahedron relations to the integrability of discrete systems.

Abstract

We elucidate consistency of the so-called corner equations which are elementary building blocks of Euler-Lagrange equations for two-dimensional pluri-Lagrangian problems. We show that their consistency can be derived from the existence of two independent octahedron relations. We give explicit formulas for octahedron relations in terms of corner equations.