Harmonic motion and Cassini ovals

TL;DR

This paper explores the geometric properties of harmonic oscillator orbits, revealing that the foci of all possible elliptical trajectories form a Cassini oval, with the specific type depending on initial velocity.

Contribution

It demonstrates that the locus of foci of harmonic oscillator orbits forms Cassini ovals, connecting harmonic motion with classical geometric curves.

Findings

Foci of all elliptical orbits form a Cassini oval.

Different initial velocities produce different types of Cassini ovals.

The envelope of orbits is an ellipse with foci on the Cassini oval.

Abstract

We consider a two-dimensional free harmonic oscillator where the initial position is fixed and the initial velocity can change direction. All possible orbits are ellipses and their enveloping curve is an ellipse too. We show that the locus of the foci of all elliptical orbits is a Cassini oval. Depending on the magnitude of the initial velocity we observe all three kinds of Cassini ovals, one of which is the lemniscate of Bernoulli. These Cassini ovals have the same foci as the enveloping ellipse.