Rotational random walk of the harmonic three body system

Abstract

When Robert Brown first observed colloidal pollen grains in water he inaccurately concluded that their motion arose "neither from currents in the fluid, nor from its gradual evaporation, but belonged to the particle itself". In this work we study the dynamics of a classical molecule consisting of three masses and three harmonic springs in free space that does display a rotational random walk "belonging to the particle itself". The geometric nonlinearities arising from the non-zero rest lengths of the springs connecting the masses break the integrability of the harmonic system and lead to chaotic dynamics in many regimes of phase space. The non-trivial connection of the system's shape space allows it, much like falling cats, to rotate with zero angular momentum and manifest its chaotic dynamics as an orientational random walk. In the transition to chaos the system displays random orientation reversals and provides a simple realization of L\'{e}vy walks.