Central configurations in the general four body coplanar problem with different masses

A generic method for finding central configurations in the general four body coplanar problem with four different masses is derived. New families of non-symmetrical central configurations with non-equal masses are found and all known symmetrical four body non-collinear central configurations are shown to be derivable subsets using this generic model. In this model, three masses are distributed on the vertices of a triangle. Then the fourth mass is allowed to be at any other point in the plane forming either convex or concave central configurations. In this general setting, a necessary condition is derived for the existence of central configurations and for requiring the masses to be positive. Using both analytical and numerical techniques, regions of possible central configurations can be derived. The special cases of four body central configurations investigated include isosceles trapezoids, right trapezoids, right quadrilaterals, convex kites and concave kites. In most of these special cases, the number of parameters of the problem can be reduced from four to two and hence the regions of existence can be shown graphically. A number of cases both convex and concave with no symmetry restrictions or equality of masses were also investigated to demonstrate the generality of the method for finding central configurations in the general four body coplanar problem with four different masses.