具有固定角的严格凸四边形中心构型的唯一性研究
On the uniqueness of the strictly convex quadrilateral central configuration with a fixed angle
AI总结：
针对天体力学牛顿四体问题中严格凸四边形中心构型唯一性的公开难题，借助相互距离坐标与Morse临界点理论，证明了给定一组对边夹角时平面严格凸四体中心构型至多一个的唯一性结论。
摘要：
牛顿四体问题中严格凸四边形中心构型的存在性与唯一性猜想，是天体力学经典n体问题研究中最受关注的公开问题之一。MacMillan和Bartky在20世纪30年代首次证明了其一般存在性，并给出了唯一性的一个特例。但自20世纪80年代Simó和Yoccoz开展相关研究、2008年Albouy和Fu在正式出版物中首次提及该问题以来，其一般情形仍未得到完美解决。我们利用相互距离坐标与Morse临界点理论，证明了当给定一组对边的夹角时，平面严格凸四体中心构型的唯一性（至多一个）。
英文摘要：
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.
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