The Ermakov-Pinney Equation: its varied origins and the effects of the introduction of symmetry-breaking functions

TL;DR

This paper explores the Ermakov-Pinney Equation's origins, examines how introducing time-dependent functions affects its symmetry properties, and identifies conditions under which certain symmetries are preserved or broken.

Contribution

It analyzes the impact of time-dependent functions on the symmetry structure of the Ermakov-Pinney Equation, revealing conditions for symmetry preservation and destruction.

Findings

Time-dependent  functions do not affect the algebra of Lie point symmetries.

Introducing a specific relationship between functions preserves a single symmetry.

The form of the autonomous equation is derived for cases with preserved symmetry.

Abstract

The Ermakov-Pinney Equation, $$\ddot{x}+\omega^2 x=\frac{h^2}{x^3},$$ has a varied provenance which we briefly delineate. We introduce time-dependent functions in place of the $\omega^2$ and $h^2$. The former has no effect upon the algebra of the Lie point symmetries of the equation. The latter destroys the $sl(2,\Re)$ symmetry and a single symmetry persists only when there is a specific relationship between the two time-dependent functions introduced. We calculate the form of the corresponding autonomous equation for these cases.