A classification of second order equations via nonlocal transformations.

Abstract

The study of second order ordinary differential equations is vital given their proliferation in

mechanics. The group theoretic approach devised by Lie is one of the most successful techniques

available for solving these equations. However, many second order equations cannot be reduced

to quadratures due to the lack of a sufficient number of point symmetries. We observe that

increasing the order will result in a third order differential equation which, when reduced via an

alternate symmetry, may result in a solvable second order equation. Thus the original second

order equation can be solved.

In this dissertation we will attempt to classify second order differential equations that can

be solved in this manner. We also provide the nonlocal transformations between the original

second order equations and the new solvable second order equations.

Our starting point is third order differential equations. Here we concentrate on those invariant

under two- and three-dimensional Lie algebras.