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. © 1998 Plenum Publishing Corporation.