Relativistic Ermakov-Milne-Pinney Systems and First Integrals

TL;DR

This paper develops a relativistic extension of the Ermakov-Milne-Pinney system, deriving new invariants and properties, and explores their implications in physics with explicit time dependence.

Contribution

It introduces the special relativistic analog of the Ermakov-Milne-Pinney equation and its invariant, extending the classical system to relativistic contexts.

Findings

Derived the relativistic Ermakov-Milne-Pinney equation and invariant.

Analyzed properties and reduced the problem to an effective Newtonian form.

Identified a relativistic nonlinear superposition law.

Abstract

The Ermakov-Milne-Pinney equation is ubiquitous in many areas of physics that have an explicit time-dependence, including quantum systems with time dependent Hamiltonian, cosmology, time-dependent harmonic oscillators, accelerator dynamics, etc. The Eliezer and Gray physical interpretation of the Ermakov-Lewis invariant is applied as a guiding principle for the derivation of the special relativistic analog of the Ermakov-Milne-Pinney equation and associated first integral. The special relativistic extension of the Ray-Reid system and invariant is obtained. General properties of the relativistic Ermakov-Milne-Pinney are analyzed. The conservative case of the relativistic Ermakov-Milne-Pinney equation is described in terms of a pseudo-potential, reducing the problem to an effective Newtonian form. The non-relativistic limit is considered as well. A relativistic nonlinear superposition law for relativistic Ermakov systems is identified. The generalized Ermakov-Milne-Pinney equation has additional nonlinearities, due to the relativistic effects.