How Scaling Symmetry Solves a Second-Order Differential Equation

TL;DR

This paper demonstrates how scaling symmetry simplifies second-order differential equations, like the Lane-Emden equation in stellar physics, into first-order equations using invariants, facilitating solutions and understanding of polytropic models.

Contribution

It applies Lie's theorem to reduce the Lane-Emden equation via scale invariance, providing a new perspective on solving and analyzing stellar structure equations.

Findings

Reduction of Lane-Emden equation to first-order form

Graphical illustration of scale transformations on solutions

Unified phase diagram encapsulating polytrope properties

Abstract

While not generally a conservation law, any symmetry of the equations of motion implies a useful reduction of any second-order equationto a first-order equation between invariants, whose solutions (first integrals) can then be integrated by quadrature (Lie's Theorem on the solvability of differential equations). We illustrate this theorem by applying scale invariance to the equations for the hydrostatic equilibrium of stars in local thermodynamic equilibrium: Scaling symmetry reduces the Lane-Emden equation to a first-order equation between scale invariants un; vn, whose phase diagram encapsulates all the properties of index-n polytropes. From this reduced equation, we obtain the regular (Emden) solutions and demonstrate graphically how they transform under scale transformations.