Homothetic radial solutions of the Newtonian general spatial (N + 1)-body problem

The equations of motion are derived, and the existence of symmetrical radial solutions of the general spatial (N + 1)-body problem, in which N bodies of identical mass m are found at the initial instant of time at the vertices of polyhedrons known as Platonic bodies (N = 4, 6, 8, 12, 20) and the (N + 1)-th body of mass M is found at the geometrical centre of the polyhedron, is proved rigorously, analytically and numerically. It is assumed that all the bodies are attracted according to Newton's law and that the initial velocities of the bodies with mass m are directed along a radius vector. The solutions found are homothetically expanding (contracting) central configurations. A geometrical image of the solutions is presented for a hexahedron, and their evolution is described for all the Platonic bodies. A distinguishing characteristic of the solutions is the value of the rate of expansion (contraction) of a configuration, which depends on the attraction law, the number of vertices in the polyhedron and its configuration.