Brouwer’s rotating vessel I: stabilization

The paradox of destabilization of a conservative or non-conservative system by small dissipation, or Ziegler’s paradox (1952), has stimulated an interest in the sensitivity of reversible and Hamiltonian systems with respect to dissipative perturbations. We discuss the motion of a particle in Brouwer’s rotating vessel, a typical gyroscopic system, that has an unstable equilibrium caused by internal damping for a wide range of rotation velocities. Using quasi-periodic averagingnormalization by Mathematica, we find that modulation of the rotation frequency in the cases of single-well and saddle equilibria stabilizes the system for a number of combination resonances, thus producing quenching of the unstable motion.