1 - Lagrangian and Hamiltonian Formalism for Discrete Equations: Symmetries and First Integrals

Summary

Abstract

In this chapter the relation between symmetries and first integrals of discrete Euler–Lagrange and discrete Hamiltonian equations is considered. These results are built on those for continuous Euler–Lagrange and canonical Hamiltonian equations. First, the well-known Noether theorem which provides conservation laws for continuous Euler–Lagrange equations is reviewed. Then, its discrete analog is presented. Further, it is mentioned that continuous and discrete Hamiltonian equations can be obtained by the variational principle from action functionals. This is used to develop Noether-type theorems for canonical Hamiltonian equations and their discrete counterparts (discrete Hamiltonian equations). The approach based on symmetries of the discrete action functionals provides a simple and clear way to construct first integrals of discrete Euler–Lagrange and discrete Hamiltonian equations by means of differentiation of discrete Lagrangian (or Hamiltonian) and algebraic manipulations. It can be used to conserve structural properties of underlying differential equations under a discretization procedure that is useful for numerical implementation. The results are illustrated by a number of examples.

Introduction

It has been known since E. Noether's fundamental work that conservation laws of differential equations are connected with their symmetry properties [28]. For convenience we present here some well-known results (see also, for example, [1, 3, 18]) for the Lagrangian approach to conservation laws (first integrals). We restrict ourselves to the case with one independent variable.