Linearization from complex Lie point transformations

TL;DR

This paper explores how complex Lie point transformations can be used to linearize certain second order ODE systems with maximum Lie algebra dimension, using geometric structures to extend linearizability criteria.

Contribution

It introduces a geometric approach to identify and linearize second order ODEs via complex Lie transformations, extending criteria from 2D to 3D.

Findings

Identifies classes of second order ODEs linearizable by complex Lie transformations.

Provides a geometric construction for linearizability criteria in three dimensions.

Extends linearization techniques from two to three dimensions using complex structures.

Abstract

Complex Lie point transformations are used to linearize a class of systems of second order ordinary differential equations (ODEs) which have Lie algebras of maximum dimension $d$, with $d\leq 4$. We identify such a class by employing complex structure on the manifold that defines the geometry of differential equations. Furthermore we provide a geometrical construction of the procedure adopted that provides an analogue in $\R^{3}$ of the linearizability criteria in $\R^2$.