标题： 三体舞蹈在双纽线上 显示英文标题

标题： Choreographic Three Bodies on the Lemniscate

摘要： 我们证明了在双纽线上周期为T的舞蹈三体{x(t), x(t+T/3), x(t-T/3)}，其参数由雅可比椭圆函数sn和cn表示，模数k^2 = (2+sqrt{3})/4，保持质心和角动量不变，其中x-hat和y-hat是定义运动平面的正交单位向量。它们还保持转动惯量、动能、曲率平方和、距离乘积以及物体之间距离的平方和。 我们发现它们在势能 sum_{i<j}(1/2 ln r_{ij} -sqrt{3}/24 r_{ij}^2) 或 sum_{i<j}1/2 ln r_{ij}-sum_{i}sqrt{3}/8 r_{i}^2 下满足运动方程，其中 r_{ij}是物体 i 和 j 之间的距离，r_{i}是从原点的距离。 势能的第一项是二维的牛顿引力，而第二项分别是相互排斥力或来自原点的排斥力。 然后给出了用于确定舞蹈三体位置的几何构造方法。 显示英文摘要

摘要： We show that choreographic three bodies {x(t), x(t+T/3), x(t-T/3)} of period T on the lemniscate, x(t) = (x-hat+y-hat cn(t))sn(t)/(1+cn^2(t)) parameterized by the Jacobi's elliptic functions sn and cn with modulus k^2 = (2+sqrt{3})/4, conserve the center of mass and the angular momentum, where x-hat and y-hat are the orthogonal unit vectors defining the plane of the motion. They also conserve the moment of inertia, the kinetic energy, the sum of square of the curvature, the product of distance and the sum of square of distance between bodies. We find that they satisfy the equation of motion under the potential energy sum_{i<j}(1/2 ln r_{ij} -sqrt{3}/24 r_{ij}^2) or sum_{i<j}1/2 ln r_{ij} -sum_{i}sqrt{3}/8 r_{i}^2, where r_{ij} the distance between the body i and j, and r_{i} the distance from the origin. The first term of the potential energies is the Newton's gravity in two dimensions but the second term is the mutual repulsive force or a repulsive force from the origin, respectively. Then, geometric construction methods for the positions of the choreographic three bodies are given.