First Integrals and Hamiltonians of Some Classes of ODEs of Maximal Symmetry

J. C. Ndogmo

doi:10.1155/2017/7302081

Abstract

Complete sets of linearly independent first integrals are found for the most general form of linear equations of maximal symmetry algebra of order ranging from two to eight. The corresponding Hamiltonian systems are constructed and it is shown that their general solutions can also be found by a simple superposition formula from the solutions of a scalar second-order source equation.

Highlights

An important class of linear ordinary differential equations (LODEs) consists of those equations having a symmetry algebra of maximal dimension. This is partly due to the simple characterization of such a class of equations [1, 2], according to which they are precisely the iterative equations, and equivalently they can be reduced by a point transformation to the canonical form y(n) = 0
Despite the fact that linear equations are the simplest types of differential equations, they are far from being completely understood and yet they frequently appear in the study of all other types of equations and in particular as canonical or reduced form of nonlinear or partial differential equations with more complex structures
Nœther’s theorem [7, 8] is a powerful tool that allows us to associate each variational symmetry of the equation with a conservation law, and, in the case of ordinary differential equations (ODEs), these conservation laws correspond to first integrals which in this instance are true constants of motion

Summary

Introduction

An important class of linear ordinary differential equations (LODEs) consists of those equations having a symmetry algebra of maximal dimension. Nœther’s theorem [7, 8] is a powerful tool that allows us to associate each variational symmetry of the equation with a conservation law, and, in the case of ordinary differential equations (ODEs), these conservation laws correspond to first integrals which in this instance are true constants of motion. The role of these first integrals is often crucial in the study of solutions and properties of differential equations, including questions related to stability [9, 10] or integrability [11,12,13,14]. We give the Hamiltonian formulation for the corresponding class of scalar equations and show amongst others that their general solution can be obtained by a simple superposition formula from those of a scalar second-order source equation

Basic Properties of Linear Iterative Equations

Variational Symmetries and First Integrals

Hamiltonian Formulation

Concluding Remarks