Magnets at the corners of polygons

We consider the statics and dynamics of $N$ identical magnetic dipoles, with inertia, pinned at the corners of a regular polygon with $N$ sides. We show that at equilibrium all dipoles are aligned tangent to the circumscribed circle. We find the normal modes and frequencies for oscillations about equilibrium, and discuss the dispersion relation. In the highest frequency normal mode the dipoles oscillate in phase with equal amplitudes, so no magnetic dipole radiation is emitted. In the limit $N→∞,$ the equilibrium energy and the frequency of the symmetric mode are both related to ζ(3), where ζ is the Riemann zeta function.