Symmetry of planar four-body convex central configurations

Abstract

We study the relationship between the masses and the geometric properties of central configurations. We prove that, in the planar four-body problem, a convex central configuration is symmetric with respect to one diagonal if and only if the masses of the two particles on the other diagonal are equal. If these two masses are unequal, then the less massive one is closer to the former diagonal. Finally, we extend these results to the case of non-planar central configurations of five particles.

Four-body co-circular central configurations

“…The fact that m 2 = m 4 implies the configuration is a kite is already a consequence of the main theorem in [4], where it is shown that this fact is true for any convex c.c., not just co-circular ones. ∂r 23 > 0 on the interior of D. By the implicit function theorem, it follows that ∂r 14 ∂r 23 < 0 on the interior of D. Since r 14 = 1 on the lower boundaries of D (the degenerate equilateral triangle or kite families), it follows that the minimum value for r 14 must occur on the upper boundary of D given by the curve r 23 = τ (r 34 ).…”

Co-circular and Co-spherical Kite Central Configurations

“…Without being exhaustive, we can mention the following: in [1] Albouy et al studied the symmetries of the central configurations of the four-body problem; in [2] Bernat et al studied planar central configurations of the four-body problem with three equal masses (these studies naturally lead to kite central configurations); in [6] Leandro proves the finiteness for a family of d-dimensional symmetrical configurations of d + 2 point masses (when d = 2 the studied case leads to a planar kite central configuration); in [7] Mello et al studied all the possible cases of planar kite central configurations.…”

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

“…(See Xia [42] for a simpler proof.) Albouy, Fu, and Sun [3] (see also [27,35]) stated the conjecture that there is a unique convex planar central configuration of the 4-body problem for each ordering of the masses in the boundary of its convex hull.…”

“…which is the center of mass equation (3) in the new variables. Finally, setting ε = 0, r i = 1 and eliminating y 0 using the center of mass equation, g i reduces to…”