Four-body co-circular central configurations

Abstract

We classify the set of central configurations lying on a common circle in the Newtonian four-body problem. Using mutual distances as coordinates, we show that the set of four-body co-circular central configurations with positive masses is a two-dimensional surface, a graph over two of the exterior side-lengths. Two symmetric families, the kite and isosceles trapezoid, are investigated extensively. We also prove that a co-circular central configuration requires a specific ordering of the masses and find explicit bounds on the mutual distances. In contrast to the general four-body case, we show that if any two masses of a four-body co-circular central configuration are equal, then the configuration has a line of symmetry.MSC Classifications: 70F10, 70F15, 37N05

New classes of spatial central configurations for the -body problem

“…We also believe that similar results can be obtained taking the same structure with two equal co-circular central configurations (see Cors and Roberts 2012), instead of two equilateral triangles.…”

Bifurcation of Relative Equilibria of the (1+3)-Body Problem

“…The characterization of the convex central configurations with an axis of symmetry and the concave central configurations of the 4-body problem when the masses satisfy that m 1 = m 2 ̸ = m 3 = m 4 is done inÁlvarez and Llibre [6]. Results on the co-circular 4-body problem can be found in [14].…”

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