Two forms of the action variable for the relativistic harmonic oscillator

TL;DR

This paper constructs two equivalent action variables for a relativistic harmonic oscillator, enabling frequency determination without solving complex equations, and compares this approach with traditional hypergeometric series solutions.

Contribution

It introduces two novel forms of the action variable for the relativistic harmonic oscillator, simplifying frequency calculation without hypergeometric functions.

Findings

Two equivalent action variables are constructed for the relativistic harmonic oscillator.

The energy-dependent frequency is obtained directly from the action variables.

The approach is compared with traditional hypergeometric series solutions, showing its effectiveness.

Abstract

The frequency of a classical periodic system can be obtained using action variables without solving the dynamical equations. We demonstrate the construction of two equivalent forms of the action variable for a one dimensional relativistic harmonic oscillator and obtain its energy dependent frequency. This analysis of oscillation is compared with the traditional solution of the problem which requires the use of hypergeometric series.