A Novel Potential Featuring Off-Center Circular Orbits

TL;DR

This paper introduces a new potential that produces off-center circular orbits at zero energy, explores their duality with finite-energy orbits via stereographic projection, and identifies an integral of motion analogous to the Runge-Lenz vector.

Contribution

It presents a novel potential for off-center circular orbits, maps zero-energy orbits to finite-energy ones on a sphere, and finds a new integral of motion similar to the Runge-Lenz vector.

Findings

Identified a potential producing off-center circular orbits at zero energy.

Mapped zero-energy orbits to finite-energy orbits via inverse stereographic projection.

Discovered an integral of motion analogous to the Runge-Lenz vector.

Abstract

In Book 1, Proposition 7, Problem 2 of his 1687 Philosophiae Naturalis Principia Mathematica, Isaac Newton poses and answers the following question: Let the orbit of a particle moving in a central force field be an off-center circle. How does the magnitude of the force depend on the position of the particle onthat circle? In this article, we identify a potential that can produce such a force, only at zero energy. We further map the zero-energy orbits in this potential to finite-energy free motion orbits on a sphere; such a duality is a particular instance of a general result by Goursat, from 1887. The map itself is an inverse stereographic projection, and this fact explains the circularity of the zero-energy orbits in the system of interest. Finally, we identify an additional integral of motion - an analogue of the Runge-Lenz vector in the Coulomb problem - that is responsible for the closeness of the zero-energy orbits in our problem.