Brouwerʼs problem on a heavy particle in a rotating vessel: Wave propagation, ion traps, and rotor dynamics

► We show how Brouwerʼs equations appear in optics, hydrodynamics and rotor dynamics. ► Swallowtail singularity exists on stability boundary of undamped Brouwer problem. ► Benjamin-Feir instability is interpreted as a destabilization by indefinite damping. ► Damping creates two Whitney umbrella singularities on the stability boundary. ► At the singular set eigenvalues are pure imaginary despite the presence of damping.