Transformations, symmetries and Noether theorems for differential-difference equations

TL;DR

This paper develops a geometric framework for differential-difference equations, establishing symmetry properties, and extending Noether's theorems to this setting, with applications to physics.

Contribution

It introduces a geometric approach to differential-difference equations, proves symmetry commutation conditions, and extends Noether's theorems to these equations.

Findings

No failure of symmetry dependence on discrete variables for structure-preserving mappings

A class of equations allowing greater symmetry freedom

Extension of Noether's theorems to differential-difference equations

Abstract

The first part of this paper develops a geometric setting for differential-difference equations that resolves an open question about the extent to which continuous symmetries can depend on discrete independent variables. For general mappings, differentiation and differencing fail to commute. We prove that there is no such failure for structure-preserving mappings, and identify a class of equations that allow greater freedom than is typical. For variational symmetries, the above results lead to a simple proof of the differential-difference version of Noether's Theorem. We state and prove the differential-difference version of Noether's Second Theorem, together with a Noether-type theorem that spans the gap between the analogues of Noether's two theorems. These results are applied to various equations from physics.