對稱的五體中心構型

Symmetrical central configurations of the five-body problem

蕭俊賢(Hsiao, Jun Shian)

指導教授 ： 陳國璋

國立清華大學/理學院/數學系/博士(2012年)

https://doi.org/10.6843/NTHU.2012.00661

摘要

中心構形是多體問體中研究中的一個重要主題 。在第一章我們將簡述其歷史及重要性 。在第二章我們介紹四體中心構型及其性質並給出一些最近的研究成果 。接著便進入我所研究的博士論文主題 ， 我考慮具有對稱的五體中心構型，並研究其可能的形狀 。在這個部分我給出了凸五體中心構型其邊長上的一些必要條件，以及關於嚴格凸五體中心構型的一種分類 。在第三章我給了一些極為有意義的凸五體中心構型例子。這些例子可以說明一篇著名論文[Williams 1938]中的一些結果是不對的。第二部分我考慮一個關於中心構型的猜想：在平面上有n個構成凸集的質點，那將不可能把第n+1個質點放在邊界上，使得它形成一個中心構型。首先，我們證明在某些對稱的情形之下，此猜想的確是對的 。最後，我們給了一個關於這個猜想的反例，說明有這樣的中心構型存在 。另外 , 關於對稱的凹中心構型，我考慮有4個質點形成凸4體構型而第5個質點位於其內部的狀況，並將它分為兩大類，討論了關於此種中心構型的存在性問體 。在最後一章，我列舉了一些關於中心構型的重要猜想，同時這也是我未來感興趣的方向 。

關鍵字

中心構型

並列摘要

Central configurations is an important subject in the n-body problem. The history of their study is summarized and we explain importance of central configurations in chapter 1. In chapter 2 we give an introduction on 4-body problem and important recent progress. Then we study the main topic:
Symmetrical five-body central configurations in the following chapters and divide our study into three parts based on the shape of central configurations. First, we discuss central configurations which are strictly convex. The main tool here is similar to Williams' approach, but we will point out some errors in Williams' paper and give some counter-examples. At the same time we give some numerical results about Chenciner's problem regarding center of masses of co-circular central configurations. Second, we study strictly concave central configurations where four of the particles are located at the vertices of a trapezoid or kite with the remaining mass in the interior of the quadrilateral. The main tool in this section is `Laura--Andoyer' equations and it is proved that except for classical central configurations, no other central configurations exist if the remaining mass lies on the intersection of the diagonals of the trapezoid or kite. At last we consider convex but not strictly convex central configurations. We give some examples and one of them is a degenerate central configuration which disproves a hypothesis proposed by Z. Xia. In the last chapter, we provide a list of some important problems about central configurations.

並列關鍵字

central configurations

參考文獻

Albouy, A.: The symmetrical central configurations of four equal masses. Contemp. Math. 198, 131--135 (1996)

Albouy, A.: Mutual distances in celestial mechanics, Nelin. Dinam. 361--386 (2006)

Albouy, A., Chenciner, A.: Le probleme des n corps et les distances mutuells. Invent. Math. 131, 151--184 (1998)

Albouy, A., Fu, Y., Sun, S.: Symmetry of planar four-body convex central configurations. Proc. R. Soc. Lond. (2008)

Albouy, A., Cabral, H. E., Santos, A.A. : Some problems on the classical n-body problem. Celest. Mech. Dyn. Astron. 113, 369-375 (2012)