數百年以來,
有關N體的週期解存在性問題已經被廣大的研究.
當n=2時,
Kepler問題已經被解決了.
當n大於等於3時刺激了一系列天體問題的研究,
Euler(1767)、
Lagrange(1772)都有重要的貢獻
在2000年,
Chenciner和Montgomery用變分法證明三體問題裡面了著名的figure-8 obrit[5]存在性的問題.
在這之後,
許多N體問題解的存在性陸陸續續的被解決.
由於使用的方法是變分法,
必須要使用到質量對稱的條件,
所以能夠解決的問題仍舊侷限於某些質量有對稱的情況.
但在2006年,
陳國璋教授解決了三體問題反轉解的存在性問題[1].
而這個結果,
對於不同質量的三體問題解的存在性是一個重大的突破.

我們除了對N體問題解的存在性有興趣外,
我們也希望可以找出這些解的一些定量的性質.
這一篇文章,
三體問題反轉解之間質點的相互距離給出一個上界的估計.
而這個估計會和質點的質量有關.

在這一篇文章裡面,
第一節主要在說明N體問題的一些基本的方程式以及符號.
第二節在說明三體問題反轉解的存在性問題,
這部分的討論大部分來自陳國璋教授的[1]這篇論文,
第三節是這篇文章主要的結果,
利用對稱,三角不等式以及疊代的辦法,
求出質點相互距離的上界估計.
第四節是把第三節所得到的結果,
實際上的數值狀況顯示出來.

The existence of periodic solutions of the N-body problem has been widely studied.
The case n=2,
called the Keplerian problem,
is well-understood.
When n is equal or bigger than 3,
this problem has stimulated a series of research about the N-body problem.
Euler(1767) and Lagrange(1772) made some fundamental contributions.
In 2000 [5],
Chenciner and Montgomery used calculus of variations to prove the existence of the figure-8 orbit of the three-body problem.
After this,
the existence of many N-body problems were obtained.
Because the method used is calculus of variations on symmetric path spaces,
it relies on some equal-mass conditions.
Therefore,
the problems solved are confined to certain cases with symmetry assumptions.
In 2006 [1],
professor K.-C Chen proved the existence of retrograde orbits of the three-body problem.
This result is a significant breakthrough regarding the existence of solutions of the three-body problem.

In addition to the existence for the N-body problem,
we are also interested in some quantitative properties.
In this article,
we obtain upper bound estimates for mutual distances of action-minimizing retrograde orbits of the three-body problem.

In section 1,
we briefly introduce the N-body problem, some basic equations and notations.
In section 2,
we explain the existence of retrograde orbits of the three-body problem.
This part of discussion mainly comes from [1].
Section 3 contains main results of this article.
We use symmetry,
the triangle inequality,
and the method of iterations to find out the upper bound estimates for the mutual distances of particle.
In section 4,
we use some numerical method to calculate these upper bound estimates.

1. Chen, k.-C. , Existence and minimizing properties of retrograde orbits to the three body problem with various choices of masses. Annals of Math., to appear.
2. Chen, K.-C., Binary decompositions for planar N-body problems and symmetric periodic solutions. Arch. Ration. Mech. Anal ,170 ,2003, 247--276.
3. Gordon, William B. , A minimizing property of Keplerian orbits. Amer. J. Math. ,99 ,1977, 5, 961--971
4. Palais, Richard S. , The principle of symmetric criticality. Comm. Math. Phys, .69 ,1979, 1, 19--30.
5. A. Chenciner, R. Montgomery, A remarkable periodic solution of the three body problem in the case of equal masses. Annals of Mathematics. ,152 ,1999, 881--901