Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 19 (2023), 001, 8 pages      arXiv:2207.09606      https://doi.org/10.3842/SIGMA.2023.001

A Novel Potential Featuring Off-Center Circular Orbits

Maxim Olshanii
Department of Physics, University of Massachusetts Boston, Boston Massachusetts 02125, USA

Received October 04, 2022, in final form January 03, 2023; Published online January 07, 2023

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.

Key words: off-center circular orbits; integrals of motion.

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