Continuous piecewise linearization method for approximate periodic solution of the relativistic oscillator

Abstract

This paper presents an approximate periodic solution to the vibration of the relativistic oscillator using a novel analytical method called continuous piecewise linearization method. First, an equivalent conservative equation for the vibration of the relativistic oscillator was derived in a simple straightforward manner that elucidates the physical meaning of the conservative equation. The continuous piecewise linearization method was then applied to derive periodic solutions for the displacement and velocity of the relativistic oscillator based on the conservative equation. The results of the present method were compared with results of published methods and exact numerical solution and the maximum error of the present method was less than 0.002%. The model derivations and the solutions presented in this paper are considerably simple and very accurate and can be used to introduce the relativistic oscillator in relevant undergraduate courses on dynamics. Essentially, knowledge of freshman calculus is sufficient to comprehend and implement the continuous piecewise linearization method for the relativistic oscillator.