On the Damped Pinney Equation from Noether Symmetry Principles

TL;DR

This paper explores the derivation of damped Pinney equations using Noether symmetry principles, extending the approach to nonlinear damped Ermakov systems and showing how damping can be removed via time-rescaling.

Contribution

It demonstrates that Noether's theorem can be applied to derive damped Pinney equations and extends the method to general nonlinear damped Ermakov systems.

Findings

Noether symmetry guides the derivation of damped Pinney equations.

Damping can be eliminated through a specific time-rescaling.

The approach applies to general nonlinear damped Ermakov systems.

Abstract

There are several versions of the damped form of the celebrated Pinney equation, which is the natural partner of the undamped time-dependent harmonic oscillator. In this work these dissipative versions of the Pinney equation are briefly reviewed. We show that Noether's theorem for the usual time-dependent harmonic oscillator as a guiding principle for derivation of the Pinney equation also works in the damped case, selecting a Noether symmetry based damped Pinney equation. The results are extended to general nonlinear damped Ermakov systems. A certain time-rescaling always allows to remove the damping from the final equations.