EXPLICIT SOLUTION OF THE FOCUS LOCUS PROBLEM FOR THE HARMONIC OSCILLATOR ORBITS IN THE PLANE

Dynamical orbits of the harmonic oscillator potential in the plane are ellipses which depend on a real parameter. Some time ago in this journal it has been proven by purely geometrical methods that the locus of the focuses of these ellipses are Cassinian ovals.Here we present several explicit analytic parameterizations of these remarkable curves. Nominally, their forms depend on the magnitude of the initial distance from the center of attraction and the magnitude of the initial velocity. We have found a few parameterizations in which the roles of the size and shapes can be clearly distinguished.

Automated summary

This paper presents the dynamical orbits of the harmonic oscillator potential in the plane as ellipses, and proves that the locus of the focuses of these ellipses are Cassinian ovals. Several explicit analytic parameterizations are also provided, which can clearly distinguish the roles of size and shape.