Relativistic potential energy of a bound particle (Non-dissipative relativistic harmonic oscillator)

TL;DR

This paper extends the concept of relativistic energy to a bound particle in a harmonic oscillator, deriving an analytical expression for the relativistic potential energy and confirming energy conservation across regimes.

Contribution

It provides the first analytical extension of potential energy for a relativistic bound harmonic oscillator, incorporating relativistic mass effects.

Findings

Relativistic potential energy is analytically derived for a harmonic oscillator.

Energy conservation holds in both non-relativistic and relativistic regimes.

The Lorentz factor transforms the oscillator's differential equation from linear to nonlinear.

Abstract

The well known relation of Einstein relativistic energy for a free particle is extended to cover the total relativistic energy of a bound particle by calculating the relativistic potential energy. A non dissipative harmonic oscillator (NDHO) is a fundamental bound system. Therefore, the potential energy of an NDHO is analytically extended from the non relativistic to the relativistic regime for the first time. This study is essentially concerned with the relativistic mass, where the Lorentz factor transforms the state of the second-order differential equation of an NDHO from linear into nonlinear. The results are finally confirmed by demonstrating energy conservation since the sum of kinetic and potential energies remains constant throughout the non relativistic and relativistic regimes.