ROBERT HOOKE'S THREE-BODY PROBLEM

Abstract

During the winter 1679, R. Hooke challenged I. Newton to predict the dynamics of an object submitted to a constant radial force. This correspondence made a strong impact on I. Newton, who wrote four years later "De Motu", the real ancestor of "The Principia", published in 1687. R. Hooke's problem can be physically linked to the dynamics of a sphere sliding on an inverted cone due to gravitational effects. If the symmetry axis of the cone is parallel to the gravitational field, the ball executes stable precessions. Breaking this symmetry induces the appearance of chaotic motions. After having derived the equations related to the position of the sphere, we analyze its dynamics, and we perform an approximated Floquet analysis that is compared to our numerical results.