﻿ 两对相邻等质量的四体共圆中心构型 Four-Body Co-Circular Central Configurations with Two Pairs of Adjacent Equal Masses

Modern Physics
Vol.07 No.03(2017), Article ID:20524,7 pages
10.12677/MP.2017.73006

Four-Body Co-Circular Central Configurations with Two Pairs of Adjacent Equal Masses

Mingjun Ma*, Yiyang Deng

Department of Mathematics, Sichuan University, Chengdu Sichuan

Received: Apr. 21st, 2017; accepted: May 13th, 2017; published: May 16th, 2017

ABSTRACT

In 2012, Cors and Roberts [1] showed that the four-body co-circular central configuration is an isosceles trapezoid when two pairs of adjacent masses are equal. However, their proof is very complicated. In this paper, we give a simpler proof by using the method of oriented areas and mutual coordinates.

Keywords:Four-Body Co-Circular Problem, Central Configurations, Isosceles Trapezoid

Cors和Roberts在2012年的文章 [1] 中证明了：在四体共圆中心构型中，若两对相邻质点的质量相等，则该共圆中心构型一定是等腰梯形。但是证明方法很复杂，本文采用有向面积方法并结合相对距离坐标给出了一个简洁证明。

1. 引言

(1)

(2)

(3)

(4)

Cors和Roberts的证明仅从相对距离坐标入手，利用四个质点共圆的几何条件，通过复杂的计算和证明，建立四体共圆问题的中心构型方程，然后再经过一系列的讨论和化简后，利用质量的等量关系

Figure 1. Planar four-body co-circular central configurations

2. 一个重要的引理

(5)

，令表示除去第个顶点后剩余3个顶点所形成的三角形的面积。我们定义有向面积如下：

(6)

. (7)

(8)

1900年，Dziobek [4] 证明了

(9)

，则整个系统的势函数与转动惯量可分别表示为

(10)

(11)

(12)

(13)

(14)

，整理得

(15)

(16)

(17)

(18)

(19)

(20)

(21)

3. 定理1.2的证明

，即

.

(i);

(ii).

(22)

(23)

(24)

(25)

(26)

(27)

(28)

(29)

(30)

(31)

(32)

(33)

(34)

(35)

，与关系式(30)矛盾。

4. 结束语

Four-Body Co-Circular Central Configurations with Two Pairs of Adjacent Equal Masses[J]. 现代物理, 2017, 07(03): 37-43. http://dx.doi.org/10.12677/MP.2017.73006

1. 1. Cors, M. and Roberts, E. (2012) Four-Body Co-Circular Central Configurations. Nonlinearity, 25, 343-370. https://doi.org/10.1088/0951-7715/25/2/343

2. 2. Wintner, A. (1941) The Analytical Foundations of Celestial Mechanics. Princeton Math. Series 5. Princeton University Press, Princeton.

3. 3. Cayley, A. (1841) On a Theorem in the Geometry of Position. Cambridge Mathematical Journal, 2, 267-271.

4. 4. Dziobek, O. (1900) Über Einen Merkwürdigen Fall des Viclkörpronlems. Astronomische Nachrichten, 152, 32-46. https://doi.org/10.1002/asna.19001520302

5. 5. Perez-Chavela, E. and Santoprete, M. (2007) Convex Four-Body Central Configurations with Some Equal Masses. Archive for Rational Mechanics and Analysis, 185, 481-494. https://doi.org/10.1007/s00205-006-0047-z

6. 6. Deng, Y., Li, B. and Zhang, S. (2017) Four-Body Central Configurations with Adjacent Equal Masses. Journal of Geometry and Physics, 114, 329-335. https://doi.org/10.1016/j.geomphys.2016.12.009

7. NOTES

*通讯作者。