The point charge oscillator: Qualitative and analytical investigations

Abstract

We determine the global phase portrait of a mathematical model describing the point charge oscillator. It shows that the family of closed orbits describing the point charge oscillations has two envelopes: an equilibrium point and a homoclinic orbit to an equilibrium point at infinity. We derive an expression for the growth rate of the primitive period Ta of the oscillation with the amplitude a as a tends to infinity. Finally, we determine an exact relation between period and amplitude by means of the Jacobi elliptic function cn.