Central Configurations For The Planar Four-body Problem

N-body problem is very important in the research of celestial mechanics, while there are still many mysterious areas for the mathematician to study hard. N-body problem is a very complex and difficult issue, and there are still many questions unanswered. Until now, only the two-body problem has been fully resolved. The central configuration is an important tool for the N-body problem,although with more than a hundred years studying, the results are still incomplete. Only the central configurations of the two-body and three-body problem have fully results. As is well known that, the classification of the four-body problem is limited, but still can not fully determine the specific form. The plane central configurations of four-body problem and central configurations of spatial five-body problem are closely linked, and this is the reason why scholars study planar four-body problem deeply. The main study of this paper is to study the shape of the central configurations for the planar four-body problem with some equal masses.According to Newton's gravitational law, the equations of motion are given by for m1 is the masses of point particles, and ri∈Rk, k= 2,3 is the position vector of the punctual mass mi, and |ri-rj|is Euclidean distance. G is constant of Gravitational.Definition 1 A solution of the N-body problem is called Homographic if in a barycen-tric reference frame the configurations formed by the bodies remain similar to each other at all times.Definition 2 Let A configuration of N bodies r= (r1, r2,…,rn)∈X\Δis called central configuration if and only if there exists a A, such that for A uniquely determined by and I satisfy the equation I= (?)Lemma 1 If a configuration r= r(t)∈X\Δwith m=(m1,m2,…, mn) is a solution of central configuration, and E∈SO(3)={G∈GL(3)|GTG= I}, x> 0, then xEr(t) is also a solution of central configuration with m=(m1, m2,…, mn).It means that the central configurations in the same equivalence class are invariant un-der rotations. So we only need to study one class in the classes of central configurations defined by the above equivalence relation. This greatly simplifies the study of the central configurations.Although most solutions of the central configurations are based on the center of inertial system, but in the studying of equivalence class of central configurations,to be simplicity, we need a general equation of central configuration.So we need the lemma as the following.Lemma 2 If a configuration r with m= (m1, m2,…, mn) is not inertial system, then we have the equivalent equation and rG=(?) is the center of mass with m=(m1, m2,…, mn).Over the years, for hard efforts of many mathematicians, the research on the central configurations in many areas has made remarkable achievements, for example, the number of central configurations, the shape of central configurations,topological properties and so on. This paper is the study of central configurations of planar four-body problem, and given the theorems of this problem. The following theorem is give by using Dziobek equations.Theorem 1 let r= (r1,r2, r3,r4)∈(R2)4, be a non-collinear central configuration in the planar with m1=m2, if the configuration r is a convex configuration, then the configuration q must possess a symmetry, and either forms a rhombus with m1= m2, m3= m4,all angle belonging to (π/3,2π/3),or forms a kite with, m1=m2,∠1=∠2∈(π/3, 2π/3); if the configuration r is a concave configurations,then the configuration r must possess a symmetry,and the other two point must have one which located inside the equilateral triangle which has the side of a pair symmetrical particle.