Abstract
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.