Co-circular and Co-spherical Kite Central Configurations

Abstract

In this article we give a simple proof of the existence of kite central configurations in the planar four-body problem which lie on a common circle. We also give a simple proof of the existence of kite central configurations in the spatial five-body problem which lie on a common sphere.

Regions of Central Configurations in a Symmetric 4 + 1-Body Problem

“…In this paper, we ask the inverse of the question, that is, given a four-or five-body configuration, if possible, find positive masses for which it is a central configuration. Similar question has been answered by Ouyang and Xie [18] for a collinear four-body problem and by Mello and Fernandes [19] for a rhomboidal four-and five-body problems.…”

Convex Central Configurations of the 4-Body Problem with Two Pairs of Equal Adjacent Masses

“…This kind of central configurations are called kite central configurations. Several papers were written studying kite central configurations and their properties, see [5,14,17,18] and references therein. In [21] Perez-Chavela and Santoprete proved that the unique convex planar central configuration with two opposite equal masses is the kite central configuration or the rhombus central configuration when the other two masses are also equal.…”

Four-body co-circular central configurations

“…First note that under these assumptions, m 2 = m 4 , a result we expect from symmetry. Expressions (16), (18) and (19) Proof: The only item remaining to show is the ordering of the masses, which is clear from Figure 3, but can be shown rigorously with straight-forward analysis. We first show m 3 ≤ m 2 by verifying that α ≤ 1.…”

“…In [10], Hampton shows that only in the case of four equal masses positioned at the vertices of square, does the center of mass coincide with the center of the circumcircle, answering a question posed by Alain Chenciner. Some recent work in [18] proves the existence of a family of co-circular kite c.c. 's using Cartesian coordinates.…”