Linearizability Criteria for Systems of Two Second-Order Differential Equations by Complex Methods

TL;DR

This paper extends Lie's linearizability criteria to systems of two second-order differential equations using complex methods, enabling linearization and solution derivation for nonlinear systems, including applications to mechanical and astrophysical models.

Contribution

It introduces complex methods for linearizing systems of two second-order differential equations, expanding geometric approaches to new classes of equations.

Findings

Successfully linearized systems of two second-order equations

Derived solutions for nonlinear systems using geometric methods

Applied techniques to mechanical and stellar structure models

Abstract

Lie's linearizability criteria for scalar second-order ordinary differential equations had been extended to systems of second-order ordinary differential equations by using geometric methods. These methods not only yield the linearizing transformations but also the solutions of the nonlinear equations. Here complex methods for a scalar ordinary differential equation are used for linearizing systems of two second-order ordinary and partial differential equations, which can use the power of the geometric method for writing the solutions. Illustrative examples of mechanical systems including the Lane-Emden type equations which have roots in the study of stellar structures are presented and discussed.