An integrable family of torqued, damped, rigid rotors

TL;DR

This paper introduces a new family of integrable damped rigid body rotation systems, characterized by cubic equations and nested quadric surfaces, with fixed points that can attract trajectories along principal axes.

Contribution

It constructs a generalized, integrable damping model for rigid body rotation that extends classical Euler equations, revealing new dynamical behaviors and fixed point structures.

Findings

Trajectories confined to nested quadric surfaces in angular momentum space

Existence of fixed points attracting trajectories along principal axes

Limiting cases recover energy or angular momentum conservation

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.