Exact solution to the problem of N bodies forming a multi-layer rotating structure

Exact solutions to the problem of the Newtonian gravitational interaction of N material points moving around N(2) concentric circular orbits are considered. Each circular orbit contains N(3) axisymmetrically located bodies having identical masses. The structure as a whole rotates around its symmetry axis. Such structures are identical to the homographic-dynamics configurations, or planar central configurations, known from literature. Conceptually, those structures can be considered as structures formed by mutually embedded polygons with point bodies placed at polygon vortices. For structures involving less than 20 bodies, solutions were obtained using Hamiltonian-mechanics methods. In the study, the forces acting on each body in the rotating structure from the side of all other bodies were found. The differential motion equations of the bodies were reduced to a system of linear algebraic equations for the body masses. Solutions in various forms were obtained. For specifying the initial parameters and for calculating all other characteristics of the structures, a computer program RtCrcSt2.for has been developed. Structures comprising up to one million bodies have been calculated. Graphical images of obtained structures are presented, and their properties are described. Stability problems for examined structures are considered, and possible application of obtained results to celestial- and space-mechanics problems is discussed.