Symmetry Classification of First Integrals for Scalar Linearizable Second-Order ODEs

E Momoniat,K S Mahomed

doi:10.1155/2012/847086

Abstract

Symmetries of the fundamental first integrals for scalar second-order ordinary differential equations (ODEs) which are linear or linearizable by point transformations have already been obtained. Firstly we show how one can determine the relationship between the symmetries and the first integrals of linear or linearizable scalar ODEs of order two. Secondly, a complete classification of point symmetries of first integrals of such linear ODEs is studied. As a consequence, we provide a counting theorem for the point symmetries of first integrals of scalar linearizable second-order ODEs. We show that there exists the 0-, 1-, 2-, or 3-point symmetry cases. It is shown that the maximal algebra case is unique.

Highlights

First integrals or constants of the motion of ordinary differential equations ODEs are quite an active and interesting area of research at the present time
In this work we have provided the algebraic structure of first integrals of the free particle or any scalar linearizable, via point transformation, ODE
We derived the relationship between the symmetries and the first integrals of the free particle equation

Summary

Introduction

First integrals or constants of the motion of ordinary differential equations ODEs are quite an active and interesting area of research at the present time. Journal of Applied Mathematics contribution by Leach and Mahomed 4 , in which they have found that the Lie algebra of the fundamental first integrals and their quotient of scalar linear second-order ODEs are three-dimensional and have very interesting properties This applies to linearizable by invertible transformations second-order ODEs which are given as examples in their paper. None of these authors consider the classification of the symmetries of first integrals of scalar linear nthorder ODEs, n ≥ 1, nor even investigate what could be the maximal numbers of symmetries for the first integrals of these linear or linearizable equations They do give insights into the algebraic structure of the fundamental first integrals and in some cases their quotients. We give the classifying relation for the symmetries of the first integrals of the free particle equation

Symmetries of the Fundamental First Integrals

Classifying Relation for the Symmetries

Symmetry Structure of First Integrals

Concluding Remarks