Applications of Classical Scaling Symmetry

TL;DR

This paper explores the role of classical scaling symmetry in differential equations, extending Noether's theorem to non-variational symmetries, and applies these ideas to derive properties of polytropes and solutions of the Lane-Emden equation.

Contribution

It extends Noether's theorem to include non-variational symmetries and provides a variational formulation for spherical hydrostatics, leading to new insights into polytropes and their properties.

Findings

Derived conservation laws from non-variational symmetries.

Presented a variational formulation of hydrostatics.

Obtained solutions and approximations for Lane-Emden equations.

Abstract

Any symmetry reduces a second-order differential equation to a first-order equation: variational symmetries of the action (exemplified by central field dynamics) lead to conservation laws, but symmetries of only the equations of motion (exemplified by scale-invariant hydrostatics), yield first-order {\em non-conservation laws} between invariants. We obtain these conservation laws by extending Noether's Theorem to non-variational symmetries, and present a variational formulation of spherical adiabatic hydrostatics. For scale-invariant hydrostatics, we directly recover all the published properties of polytropes and define a {\em core radius}, a new measure of mass concentration in polytropes of index n. The Emden solutions (regular solutions of the Lane-Emden equation) are finally obtained, along with useful approximations. An appendix discusses the special n=3 polytrope, emphasizing how the same mechanical structure allows different {\em thermostatic} structures in relativistic degenerate white dwarfs and and zero age main sequence stars.