Finding nonlocal Lie symmetries algorithmically

TL;DR

This paper introduces an algorithmic method to compute various types of Lie symmetries, including nonlocal ones, for rational second order ODEs, aiding their integration and parameter analysis.

Contribution

It presents a novel approach leveraging the formal equivalence between the total derivative and the associated vector field to compute symmetries, including nonlocal symmetries, algorithmically.

Findings

Effective computation of nonlocal Lie symmetries for rational 2ODEs.

Ability to analyze parameter regions for integrability.

Enhanced integration techniques for second order ODEs.

Abstract

Here we present a new approach to compute symmetries of rational second order ordinary differential equations (rational 2ODEs). This method can compute Lie symmetries (point symmetries, dynamical symmetries and non-local symmetries) algorithmically. The procedure is based on an idea arising from the formal equivalence between the total derivative operator and the vector field associated with the 2ODE over its solutions (Cartan vector field). Basically, from the formal representation of a Lie symmetry it is possible to extract information that allows to use this symmetry practically (in the 2ODE integration process) even in cases where the formal operation cannot be performed, i.e., in cases where the symmetry is nonlocal. Furthermore, when the 2ODE in question depends on parameters, the procedure allows an analysis that determines the regions of the parameter space in which the integrable cases are located.