The relation between screw symmetric gravitational waves and electromagnetic vortices was pointed out recently. In this talk I discuss the similarities between the gravitational waves and other physical systems, like ion traps in $2$ and $3$ dimensions by using the Bargmann framework.
In particular, I will show that the Eisenhart-Duval lift of the $2$ dimensional Paul trap is a linearly polarized periodic gravitational wave. A modified version of the $3$ dimensional Penning trap is related in turn to circularly polarized periodic gravitational waves. The reason behind the relation between so remote systems lies in the fact that the motions in both systems can be understood in terms of anisotropic (and possibly time dependent) harmonic oscillators.
I will also present an analytic solution for bounded motion of a test particle under circularly polarized gravitational wave. By using this solution, one can also solve the associated Sturm-Liouville problem.