Moving Frames and Noether's Finite Difference Conservation Laws II

TL;DR

This paper develops a framework for deriving and integrating finite difference Euler-Lagrange equations and conservation laws invariant under specific Lie group actions, extending previous work with new methods for solving these equations.

Contribution

It introduces a systematic approach to find invariants, formulate invariant Euler-Lagrange equations, and perform the integration step for Lagrangians invariant under SL(2), area-preserving, and projective actions.

Findings

Derived explicit invariants for the group actions.

Formulated invariant Euler-Lagrange difference equations.

Provided methods for integrating these equations using Lie group theory.

Abstract

In this second part of the paper, we consider finite difference Lagrangians which are invariant under linear and projective actions of $SL(2)$, and the linear equi-affine action which preserves area in the plane. We first find the generating invariants, and then use the results of the first part of the paper to write the Euler--Lagrange difference equations and Noether's difference conservation laws for any invariant Lagrangian, in terms of the invariants and a difference moving frame. We then give the details of the final integration step, assuming the Euler Lagrange equations have been solved for the invariants. This last step relies on understanding the Adjoint action of the Lie group on its Lie algebra. We also use methods to integrate Lie group invariant difference equations developed in Part I. Effectively, for all three actions, we show that solutions to the Euler--Lagrange equations, in terms of the original dependent variables, share a common structure for the whole set of Lagrangians invariant under each given group action, once the invariants are known as functions on the lattice.