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

Abstract

Exact solutions to the problem of the Newtonian gravitational interaction of N material points moving around N2 concentric circular orbits are considered. Each circular orbit contains N3 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.

Computing the Bodies Motions in the Space and Long-term Changes in the Earth's Climate

“…Also, J.J. Smulsky has analytically exactly solved some previously unsolved N-body problems, including the following ones: what motions are exercised by bodies on a ring [7]? Can a system of bodies involving several of rings rotate as an entity [8]? Those and other exact solutions have allowed J.J. Smulsky to solve several previously unsolved problems: about an optimum mission to the Sun [9], about the motion of Mercury's perihelion [10], about the evolution of the Earth axis [11,12], and so forth.…”

“…The structures were obtained by integration of differential motion equations performed using the Galactica program: N 2 = 15; N 3 = 30; period P rd =1 year; the mass of the central body is equal to the Sun mass. Given in the table are the relative radii of the rings and the masses of one body in the rings[8].Here (Fig. 5), results of an exact solution of the problem of N-bodies located in N 2 layers around a central body are shown.…”

Advances in Mechanics and Outlook for Future Mankind Progress

“…The layers are counted from the center, and the bodies in each layer are counted from the x o axis. Such notation permits an easy determination of the distances between any two bodies and the angle between their central radii [20]. Such analysis of the problem has allowed us to write the following simple expression for the force of impact of bodies on one first body in the j-th layer: …”

“…For constructing the rotating systems, a program RtCrcSt2.for was developed [20]; this program can be accessed at www.ikz.ru/~smulski/Data/RtCrcStr/. Structures with different total numbers of layers N 2 and with different total numbers of bodies in the layers N 3 were obtained.…”

“…Methods to investigate into the dynamics and evolution of the structures of interest were developed in [18] - [20].…”