Periodic Solutions And Central Configurations Of The N-body Problem

N-body problem is a classic problem in celestial dynamics and astronomy, which has close relations with mathematics and mechanics. Intuitively, n-body problem refers to the research of the motion of N mass in the space that having been given the initial positions and the velocity, when considering the gravity among them only.Base on the results of the N-body problem in recent years, this article summarized the latest progress in the existence and stability together with the central configurations of the N-body problem in recent years.Through the mathematical model of n-body problems, we know that the motions of the plants are described by the Hamiltonian Dynamical Systems, and the periodic motion is the common form of motions, then we can come to the conclusion that our object is to research the periodic solutions of the Hamiltonian dynamical systems. In chapter one, in order to summarize the studies in the existence and the stability of the Hamiltonian system, we introduced the nearly integrable Hamiltonian system with perturbations. In the existence, we always focus on the existence of the Lagrange functional critical point instead, by variation principle. In the last of this chapter, we summarized the studies of the existence of the periodic solutions of the Hamiltonian systems in different forms. Through the summaries, we get the importance of the mountain pass theorem and saddle point theorem in application; in the stability, we summarize some perfect results on the existence of invariant torus of different type of the Hamiltonian. And at last we show some studies of Jaume Llibre and Cristina Stoicaon on Comet- and Hill-type periodic orbits in restricted (N+1)-body problems. In the last chapter, we introduce the equivalent definitions on central configurations raised by Long Yiming, Sun Shanzhong and Zhang Shiqing. With the definition we introduce some application in proving the existence of some central configuration. In the end we give some important proof in recent years.