Two balls and a string: from ordered motion to chaos

Abstract

Two spherical balls are connected by a taught string passing through a small hole in a perfectly planar table: the first ball, subject to a central force, moves without friction on a two-dimensional plane, while the second ball moves only along the vertical axis directly below the hole. The pedagogical aspects of this novel two-body problem are given particular attention: Newton’s laws, central force motion, conservation laws, angular momentum, constraints, etc. The dynamics of the system is considered under various initial conditions wherein the ball on the table moves qualitatively in rotating ellipses or hypotrochoids. The conditions for closed or periodic orbits are examined. The more complex case of the inclined plane is then considered, revealing a rich variety of periodic, aperiodic and chaotic solutions as a function of the ball mass ratio and the plane inclination angle. The associated Poincaré phase-space maps are discussed.