An integrable family of torqued, damped, rigid rotors

Abstract

Expositions of the Euler equations for the rotation of a rigid body often invoke the idea of a specially damped system whose energy dissipates while its angular momentum magnitude is conserved in the body frame. An attempt to explicitly construct such a damping function leads to a more general, but still integrable, system of cubic equations whose trajectories are confined to nested sets of quadric surfaces in angular momentum space. For some choices of parameters, the lines of fixed points along both the largest and smallest moment of inertia axes can be simultaneously attracting. Limiting cases are those that conserve either the energy or the magnitude of the angular momentum. Parallels with rod mechanics, micromagnetics, and particles with effective mass are briefly discussed.