The Inverse Problem for Nested Polygonal Relative Equilibria

TL;DR

This paper proves that for certain potentials, relative equilibria with two homothetic regular polygons require equal masses at each polygon, and such configurations always exist under specified conditions.

Contribution

It establishes conditions under which nested polygonal relative equilibria occur, including the necessity of equal masses and existence criteria for large radius ratios.

Findings

Equal masses are required for relative equilibria with two homothetic regular polygons.

Such configurations always exist when the ratio of radii is sufficiently large.

The results apply to potentials including Newtonian and Helmholtz vortex potentials.

Abstract

We prove that for some potentials (including the Newtonian one, and the potential of Helmholtz vortices in the plane) relative equilibria consisting of two homothetic regular polygons of arbitrary size can only occur if the masses at each polygon are equal. The same result is true for many regular polygons as long as the ratio between the radii of the polygons are sufficient large. Moreover, under these hypotheses, the relative equilibrium always exist.