Symmetry Reduction of Lane-Emden Equation for Polytropes

TL;DR

This paper presents a symmetry reduction approach for the Lane-Emden equation with arbitrary polytropic index, revealing limitations of standard methods and deriving some special solutions.

Contribution

It introduces an ansatz for symmetry reduction of the Lane-Emden equation with one symmetry generator, highlighting challenges in finding non-trivial symmetries.

Findings

Standard reduction methods do not admit non-trivial Lie symmetries for the reduced equation.

Special solutions for the reduced differential equation are obtained.

The approach provides insights into symmetry properties of polytropic models.

Abstract

We describe an ansatz for symmetry reduction of the Lane-Emden equation for an arbitrary polytropic index n, admitting only one symmetry generator. For the reduced first order differential equation it is found that standard reduction procedure do not admit any non-trivial Lie point symmetry. However some special solutions for the differential equation are obtained.