On Periodic Perturbations of Uniform Motion of Maxwell's Planetary Ring

We examine the case when equally sized small moons arrange themselves on the vertices of a regular n-gon for n 7. For n 4, there are at least 3 pure imaginary characteristic exponents, each of which has multiplicity = 1, a surprising result that makes it possible to apply the Lyapunov center theorem to verify the existence of some periodic perturbations. For sufficiently large n, when the regular n-gon is the unique central configuration, the number of families of periodic perturbations is at least equal to 2n – (n + 1)/4, where x is the greatest integer less than or equal to x.