# Relativistic Ermakov–Milne–Pinney Systems and First Integrals

## Abstract

**:**

## 1. Introduction

## 2. The Eliezer and Gray Physical Interpretation

## 3. A Relativistic Ermakov–Milne–Pinney System

## 4. Derivation from a Dynamical Rescaling of Time

**A**is the vector potential and $\varphi $ is the scalar potential of the electromagnetic field) in the two-spatial dimensions case, with $\phi \sim ({x}^{2}+{y}^{2})$ together with $\mathbf{A}=0$. This is the Lagrangian (13) assuming unit rest mass without loss of generality. In addition, the relativistic harmonic oscillator and REMP equations reproduce the Newtonian systems in the limit $u/c\to 0$, where u is a measure of the maximal velocity of the problem. For instance, if ${A}_{0}$ is the amplitude of the motion under a linear force $F=-{\kappa}^{2}x$, the relativistic effects become negligible provided $\kappa {A}_{0}/c\to 0$. The case of a very strong external field acting on a charged particle [29] is a suitable system to probe the relativistic effects. In this context, since the non-relativistic Ermakov system has found applications in accelerator physics [11], the relativistic version has potential applications for charged particle motions under high-intensity external fields.

## 5. The $J=0$ Case

## 6. Relativistic Conservative Ermakov–Milne–Pinney Equation

## 7. Nonlinear Superposition Law

## 8. Conclusions

## Funding

## Conflicts of Interest

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**Figure 1.**Phase-space contour plots of the energy first integral (32) for the 1D conservative relativistic harmonic oscillator described by Equation (31) with constant $\kappa $, for $\overline{x}=\kappa x/c,\phantom{\rule{0.166667em}{0ex}}\overline{v}=\dot{x}/c$ and different values of ${H}_{1D}$, as indicated.

**Figure 2.**Pseudo-potential ${V}_{1D}\left(\overline{x}\right)$ from Equation (34) and different values of ${H}_{1D}$, as indicated.

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Haas, F. Relativistic Ermakov–Milne–Pinney Systems and First Integrals. *Physics* **2021**, *3*, 59-70.
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Haas F. Relativistic Ermakov–Milne–Pinney Systems and First Integrals. *Physics*. 2021; 3(1):59-70.
https://doi.org/10.3390/physics3010006

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Haas, Fernando. 2021. "Relativistic Ermakov–Milne–Pinney Systems and First Integrals" *Physics* 3, no. 1: 59-70.
https://doi.org/10.3390/physics3010006