Brouwer's problem on a heavy particle in a rotating vessel: wave propagation, ion traps, and rotor dynamics

TL;DR

This paper revisits Brouwer's 1918 stability problem of a heavy particle in a rotating vessel, revealing deep connections with rotor dynamics, fluid mechanics, and modern physics, and analyzing the stability boundary's complex singularities.

Contribution

It uncovers the intricate stability boundary structure of Brouwer's problem, including singularities like Swallowtail and Whitney umbrellas, linking classical mechanics with modern physics and stability theory.

Findings

The stability domain boundary has a Swallowtail singularity.

Presence of Whitney umbrellas on the stability boundary.

Existence of eigenvalues purely imaginary despite damping and forces.

Abstract

In 1918 Brouwer considered stability of a heavy particle in a rotating vessel. This was the first demonstration of a rotating saddle trap which is a mechanical analogue for quadrupole particle traps of Penning and Paul. We revisit this pioneering work in order to uncover its intriguing connections with classical rotor dynamics and fluid dynamics, stability theory of Hamiltonian and non-conservative systems as well as with the modern works on crystal optics and atomic physics. In particular, we find that the boundary of the stability domain of the undamped Brouwer's problem possesses the Swallowtail singularity corresponding to the quadruple zero eigenvalue. In the presence of dissipative and non-conservative positional forces there is a couple of Whitney umbrellas on the boundary of the asymptotic stability domain. The handles of the umbrellas form a set where all eigenvalues of the system are pure imaginary despite the presence of dissipative and non-conservative positional forces.