Legendre Polynomial and Nonlinear Oscillating Point-Like Charged Particle

Abstract

In this article we explore the kinematics of a point-like charged particle placed within the interior plane of a charged ring. Analytically we formulate the electric field of the ring along a representative diagonal. Graph of the field as a function of the distance from the center of the ring assists foreseeing oscillating movement of the charged particle. We formulate the equation of motion; this is a nonlinear differential equation. Applying Computer Algebra System (CAS), specifically Mathematica [1] we solve the equation numerically. Utilizing the solution we quantify the kinematic quantities of interest including oscillations period. Although the equation of motion is nonlinear its period is regulated. For better understanding we take an advantage of Mathematica animation features animating the nonlinear oscillations.