A Novel Potential Featuring Off-Center Circular Orbits

In Book 1, Proposition 7, Problem 2 of his 1687 PhilosophiæNaturalis 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 on that 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.