Helmholtz's inverse problem of the discrete calculus of variations

TL;DR

This paper derives a discrete Helmholtz condition that characterizes second order finite difference equations with a Lagrangian formulation, providing a comprehensive classification of all such formulations.

Contribution

It introduces a discrete Helmholtz condition and characterizes all second order finite difference equations that admit a Lagrangian formulation.

Findings

Derived the discrete Helmholtz condition for finite difference equations

Characterized all second order difference equations with a Lagrangian formulation

Provided the class of all possible Lagrangian formulations in the discrete setting

Abstract

We derive the discrete version of the classical Helmholtz condition. Precisely, we state a theorem characterizing second order finite differences equations admitting a Lagrangian formulation. Moreover, in the affirmative case, we provide the class of all possible Lagrangian formulations.