On the Existence of Symmetric Bicircular Central Configurations of the 3n-Body Problem

Abstract In this paper, we consider central configurations of the planar 3 n -body problem consisting of n masses at the vertices of a regular n -gon inscribed in a circle of radius r and 2 n masses at the vertices of a second (not necessarily regular) concentric 2 n -gon inscribed in a circle of radius ar which are symmetric in the sense that the set of positions of the 3 n masses and the set of the corresponding masses are invariant under the action of a finite subgroup of O (2). There are two different types of such configurations. In the first type, called regular bicircular central configurations of the 3n-body problem, the second 2 n -gon is regular, n of the vertices of the second n -gon are aligned with the vertices of the first regular n -gon and the masses at the vertices of this 2 n -gon alternate values. In the second type, called semiregular bicircular central configurations of the 3n-body problem, the second 2 n -gon is semiregular and the masses at its vertices are all of them equal. A semiregular 2 n -gon has n pair of vertices symmetric by a reflection of an angle β with respect to the axis of symmetry of the first regular n -gon. Our aim is to analyze the set of values of the parameter a for the regular 2 n -gon and of the parameters (a, ) (a, β) for the semiregular 2 n -gon providing symmetric bicircular central configurations. In particular, for all n 2 n ≥ 2 we prove analytically the existence of symmetric bicircular central configurations with a (respectively (a, ) (a, β) ) satisfying some particular conditions. Using either computer-assisted results or numerical results, we also describe the complete set of values of a (respectively (a, ) (a, β) ) providing symmetric bicircular central configurations for n=2, 3, 4, 5 n = 2, 3, 4, 5 and we give numerical evidences that the pattern for n>5 n > 5 is the same as the one for n=5 n = 5.