具有Newton型弱力势的多体问题的研究

The n-body problem of celestial mechanics has a long history, which starts with Newton’s Principia in 1687. Since then, many famous mathematicians, astronomers and physicists brought valuable contributions towards understanding it. Thanks to this problem, they developed classical mathematical tools, which are now applied to many other areas of mathematics. Among them are the perturbation methods used by Laplace and Lagrange, which helped Adams and Leverrier discover the planet Neptune, as well as some variational methods, initially brought to celestial mechanics by Poincaré. In the past 20 years, the huge successful applications of variational methods with upper and lower bound estimates for action integrals, as well as the perturbation methods to generalized solutions, allowed us to obtain new orbits for the related n-body problems with weak forces. Nevertheless, many basic questions remain open, such as the classification and variational characterization of all solutions, the famous Saari and Wintner-Smale conjectures, not only in the Euclidean case, but also in spaces of constant curvature.. The goal of our project is to introduce new methods from nonlinear functional analysis, representation theory, dynamical systems and topology, algebraic geometry and differential geometry to study the existence and properties of many types of solutions for n-body problems (especially for n=3), and some applications.

天体力学中的n-体问题具有很长的历史，它始于1687年牛顿的巨著“原理”。自那以后，许多著名的数学家，天文学家和物理学家对于理解它做出了有价值的贡献。受益于该问题，他们发展了现在用在很多数学领域的经典数学工具。其中 Laplace 和 Lagrange使用过的扰动法帮助 Adams 和Leverrie发现了海王星，Poincaré将变分方法引入天体力学。近20年由于变分方法及对广义解的作用积分的上下界估计与扰动方法的巨大成功使得我们可以获得具有弱力势的多体问题的新轨道。但仍有许多基本问题没有解决，如多体问题的所有解的分类和变分刻划，著名的Sarri 猜想,Wintner-Smale猜想，不仅在欧式空间，还有常曲率空间情形。. 我们试图从非线性泛函分析、群表示论、动力系统与拓扑学、代数几何和微分几何中引进新的方法来研究多体问题的多种解的存在性和性质（尤其是n=3)以及一些应用。

张世清2018年在科学出版社出版的著作：“泛函分析及其应用” 受到同行专家和读者好评。 张世清与于翔合作的论文研究了固定端的1维多体问题的变分最小，对著名的Marchal定理做了补充, 论文发表在国际著名学术刊物Celestial Mechanics and Dynamical Astronomy，被他人在世界最顶尖数学刊物 Ann. of Math. (2)192 (2020), no. 2,499–550 引用。张世清近几年指导邓义杨和李秉宇研究4 体问题的中心构型的形状，论文被他人在 Nonlinearity 及 Arch. Ration. Mech. Anal.等引用。张世清与于翔合作的论文"Saari's conjecture for elliptical motions and minimizing solutions of the N-body problem, SIAM J. Math. Anal. 48 (2016), no. 1, 709–724.” 证明了D. Saari猜测的二个特殊情况：任意正质量的椭圆情形和限制N+1体情形，并得到一个重要应用，我们的论文受到“美国数学评论”的高度评价。张世清指导王智强并合作研究了3维空间中具有牛顿势的2n体问题新的周期解的存在性, 论文发表在国际著名刊物 Arch. Rational Mech. Anal. 219 (2016), 1185-1206, 我们的工作推广了意大利数学家Fusco等人2011年发表在国际顶尖数学刊物 Invention of Math.上的部分工作，我们的论文被他人在Arch. Rational Mech. Anal. 及 JDE 上引用。