Off-Center Circular Orbits

Off-center circular orbits and non-Euclidean geometry

Following Newton [I. Newton, Philosophiae Naturalis Principia Mathematica, Book 1, Prop. 7, Pr. 2], we pose and solve the following problem: find a central potential whose zero-energy orbits are represented by off-center circles[O23]. The solution turns out to have deep connections to free motion on a spherical or hyperbolic surface, for the potential center inside or outside the orbit respectively. The connection is a non-Euclidean generalization of the so called Bohlin-Arnold-Vassiliev duality [V. Arnold, Huygens and Barrow, Newton and Hooke, Birkhäuser, Basel, 1990.].