On the inverse problem of the discrete calculus of variations

TL;DR

This paper introduces an algorithm to determine the discrete Lagrangian for autonomous recurrence relations of even order, solving the inverse calculus of variations problem using annihilation operators and linear PDEs.

Contribution

It provides a novel method based on annihilation operators to solve the inverse problem for discrete variational calculus of arbitrary even order.

Findings

Algorithm successfully finds discrete Lagrangians for given recurrence relations.

Method reduces the inverse problem to solving linear PDE systems.

Applicable to autonomous recurrence relations of any even order.

Abstract

In this paper we present an algorithm to find the discrete Lagrangian for an autonomous recurrence relation of arbitrary even order $2k$ with $k>1$. The method is based on the existence of a set of differential operators called annihilation operators which can be used to convert a functional equation into a system of linear partial differential equations. This completely solves the inverse problem of the calculus of variations in this setting.