On central configurations of the κn-body problem

Abstract

We consider planar central configurations of the Newtonian κn-body problem consisting in κ groups of regular n-gons of equal masses, called (κ, n)-crown. We derive the equations of central configurations for a general (κ, n)-crown. When κ = 2 we prove the existence of a twisted (2, n)-crown for any value of the mass ratio. Moreover, for n = 3, 4 and any value of the mass ratio, we give the exact number of twisted (2, n)-crowns, and describe their location. Finally, we conjecture that for any value of the mass ratio there exist exactly three (2, n)-crowns for n ≥ 5. This is a preprint of: "On central configurations of the kn-body problem", Esther Barrabés, Josep Maria Cors, determined by three sequences, related to the masses, the radii of the κ circles where the different regular n-gons are inscribed, and the angles of rotation between the different κ n-gons. We also derive the equations for a (κ, n)-crown when all the angles of rotation are multiples of π/n, and we show two examples with κ = 3.Second, in Section 3, we consider the case of κ = 2 twisted rings, where the two gons are rotated an angle π/n. In Theorem 1 we prove that for any value of the mass ratio and n ≥ 3, there exists at least one (2, n)-crown. When n = 3 and n = 4, we give the exact number. More concretely, for n = 3, in Theorem 2, we show that the number varies between one and three; for n = 4, in Theorem 3, we show that for any value of the mass ratio there exists exactly three different crowns. Moreover, in both cases, n = 3, 4, we describe the set of admissible radii where the two twisted n-gons can be located. Finally, we conjecture that for any value of the mass ratio there exist exactly three (2, n)-crown for n ≥ 5. Some results when κ > 2 will be presented in a forthcoming paper.The paper include an Appendix where the detailed proof of some technical results are given.