The nested planar central configurations of a trapezoid form in classical and generalized versions of the general (4n+1)-body problem

The study of central configurations, whose concepts and definitions were already formulated by the classics of celestial mechanics - Euler, Lagrange, Laplace and Liouville in the XVIII-XIX centuries, is of interest not only for celestial mechanics, but also for many sections of mathematical analysis, differential equations, analytical mechanics, stellar dynamics and space flight dynamics. In recent decades, there have been opportunities to use the concept of central configurations also in theoretical physics, chemistry, crystallography, etc. We consider planar central configurations, called nested, consisting of polygons sequentially nested one into another, at the vertices of which there are bodies (material points). The existence of nested planar central configurations of trapezoidal type with a sphere in the centre is proved. Early, it was found that abovementioned isolated central configurations exist in the heliocentric rotated coordinate systems. It was supposed only the Newton’s law of attraction is acting between bodies. The Maple software is used to derive the solution of this problem.