A Novel Potential Featuring Off-Center Circular Orbits

• Published in: Symmetry, Integrability and Geometry: Methods and Applications
• Date: 2023
• Main Author: Olshanii, Maxim
• Format: Article
• Language: English
• Published: Інститут математики НАН України 2023
• Online Access: https://nasplib.isofts.kiev.ua/handle/123456789/211883
• Journal Title: Digital Library of Periodicals of National Academy of Sciences of Ukraine
• Cite this: A Novel Potential Featuring Off-Center Circular Orbits, Maxim Olshanii, SIGMA 19 (2023), 001, 8 pages

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.