The hierarchical approach to proving the existence of generalized planar nested central configurations on some versions of the general (pn+1)-body problem

A hierarchical approach to proving of existence in the general (pn+1)-body exact partial solutions is presented, the so called generalized planar nested central configurations in a form of consequently nested in each other convex n -gons with nonequal in general masses in the vertices and a nonspherical body in the centre. Flat nest-shaped central configurations in the form of convex quadrilaterals of mixed shapes nested one into another of the type square + rhombus + deltoid + trapezoid + central body within the frame-work of the general problem of (4n+1)-bodies of celestial mechanics were measured. The given general conditions of existence are valid for any nest-shaped planar central configurations within the framework of the (4n+1)-bodies problem. Symbolic calculations of the Maple mathematical package are used to solve the system of equations. The system of algebraic equations has a hierarchical structure similar to the obtained direct transformations to the system of algebraic equations within the process of solving systems of linear equations by the Gauss method. The cases of a central body in the form of a spherical (a ball) and a non-spherical (an ellipsoid of rotation or a triaxial ellipsoid) structures are considered. In each of the cases, the corresponding necessary and sufficient conditions for the existence of central configurations of various types are given.