這篇論文的目的是去研究區域形變跟整體形變方法在多體多中心問題上面的應用。

最近幾十年來，變分法已經被廣泛應用在建構多體問題和多中心問題的週期解以及相對週期解上面。在排除解的碰撞的部分，總是用到區域形變或整體形變方法之中其中一個方法，但不是兩者。所以我們會感興趣的去問：是不是存在著一種較困難的問題，是必須同時使用兩種方法才能解開？

多體多中心問題就是這樣的例子。在這篇論文的第五章，我們就有詳細去描述如何用這兩個方法得出解的存在性。

多體多中心問題是多體問題跟多中心問題的組合，而多中心問題是多體問題簡化的形式，因此我們也會有興趣去研究界在它們之間是否還存在哪些有趣的問題。

基於這些原因，所以我們就聚焦在n體m中心系統，這個系統包含 n個自由質點跟 m個固定質點，然後我們去研究n個自由質點在整個系統的引力下，他們的軌跡跟行為。在這篇文章裡，我們證明了其中一些型態的解的存在性。

The objective of this thesis is to study the applications of local deformation methods and global estimates in the few-body-few-center problem.
In recent years, variational methods have been applied to construct many periodic and relative periodic solutions for both the n-body and the n-center problem. Collision avoidance is often carried out by either local deformation or global estimate for the action functional, but not both. It would be interesting to find examples which require both methods, we will show that the fewbody-few-center systems are such examples. In Chapter 5, we present examples for which both techniques are needed. These examples are combinations of n-body and n-center problems, that is, the few-body-few-center problem.
Another motivation for our study of the few-body-few-center systems is due to the fact that n-body systems are generally much more complex than n-center systems, as the later is often served as simplification of the former, it would be interesting to understand the intermediate case.
Due to reasons above, here we focus on the n1-body-n2-center system, meaning the system has n1 free bodies and n2 fixed ones. We want to study the motions of n1 free bodies under the gravitational influence of the whole system. Comparing with n-center systems, some nice features of action minimizers are no longer valid in n1-body-n2-center systems, such as collision reflection property. We will show existence of some collision-free minimizers for such systems.

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