Geometric Linearization of Ordinary Differential Equations

TL;DR

This paper develops explicit criteria for determining when certain second order differential equations can be linearized, using geometric methods related to symmetries and isometries of geodesic equations, and proves a related conjecture.

Contribution

It provides explicit criteria for linearizability of second order ODEs based on geometric symmetry analysis and proves a conjecture linking these criteria to maximally symmetric spaces.

Findings

Criteria established for second order quadratically and cubically semi-linear equations.

Connection between symmetries and isometries of geodesic equations confirmed.

Conjecture relating linearizability criteria to maximally symmetric spaces proved.

Abstract

The linearizability of differential equations was first considered by Lie for scalar second order semi-linear ordinary differential equations. Since then there has been considerable work done on the algebraic classification of linearizable equations and even on systems of equations. However, little has been done in the way of providing explicit criteria to determine their linearizability. Using the connection between isometries and symmetries of the system of geodesic equations criteria were established for second order quadratically and cubically semi-linear equations and for systems of equations. The connection was proved for maximally symmetric spaces and a conjecture was put forward for other cases. Here the criteria are briefly reviewed and the conjecture is proved.