A Semi-Algorithmic Search for Lie Symmetries

TL;DR

This paper explores the relationship between S-functions and Lie symmetries in rational second order ODEs, introducing a semi-algorithmic method to find symmetries even when none are apparent as Lie point symmetries.

Contribution

It establishes a link between S-functions and Lie symmetries, enabling a semi-algorithmic approach to identify symmetries in complex differential equations.

Findings

Method can find Lie symmetries without Lie point symmetries.

Provides a semi-algorithmic procedure for symmetry detection.

Enhances understanding of symmetries in rational 2ODEs.

Abstract

In [Solving second order ordinary differential equations by extending the Prelle-Singer method, J. Phys. A: Math.Gen., 34, 3015-3024 (2001)] we defined a function (we called S) associated to a rational second order ordinary differential equation (rational 2ODE) that is linked to the search of an integrating factor. In this work we investigate the relation between these $S$-functions and the Lie symmetries of a rational 2ODE. Based on this relation we can construct a semi-algorithmic method to find the Lie symmetries of a 2ODE even in the case where it presents no Lie point symmetries.