Towards a New ODE Solver Based on Cartan's Equivalence Method

TL;DR

This paper introduces a novel ODE solver based on Cartan's equivalence method, aiming to enhance the handling of second order differential equations by leveraging symmetry groupoids and coordinate transformations.

Contribution

It proposes an algorithm that improves existing ODE solvers for second order equations and establishes a theoretical link between coordinate changes and symmetry groupoids.

Findings

New algorithm for second order ODEs based on Cartan's method

Theoretical relationship between coordinate transformations and symmetry groupoids

Potential improvements in solving complex differential equations

Abstract

The aim of the present paper is to propose an algorithm for a new ODE--solver which should improve the abilities of current solvers to handle second order differential equations. The paper provides also a theoretical result revealing the relationship between the change of coordinates, that maps the generic equation to a given target equation, and the symmetry $\D$-groupoid of this target.