On the uniqueness of the strictly convex quadrilateral central configuration with a fixed angle

TL;DR

This paper proves the (at most) uniqueness of a strictly convex quadrilateral central configuration in the 4-body problem when one pair of opposite sides' angle is fixed, advancing understanding of this classical celestial mechanics problem.

Contribution

It establishes the (at most) uniqueness of such configurations under a fixed angle condition using Morse theory and mutual distance coordinates.

Findings

Proves at most one such configuration exists for given angle.

Uses Morse's critical point theory in celestial mechanics.

Provides a new approach to a long-standing open problem.

Abstract

The conjecture of the existence and the uniqueness of the strictly convex quadrilateral central configuration for the Newtonian 4-body problem is one of the most-talked open problems in the study of the classical n-body problems in celestial mechanics. MacMillan and Bartky first gave its general existence in the 1930s and a particular case for its uniqueness. Still, the general case has yet to be solved perfectly since it was considered by Sim'{o} and Yoccoz in the 1980s and was first mentioned by Albouy and Fu in 2008 in the formal publication. Using coordinates of mutual distances and Morse's critical point theory, we give the (at most) uniqueness of the planar strictly convex 4-body central configuration when the angle of one pair of the opposite sides is given.