On the Stability of the Homographic Polygon Configuration in the Many-Body Problem

In this paper the stability of a new class of exact symmetrical solutions in the Newtonian gravitational (n+1)-body problem is studied. This class of solution follows from a suitable geometric distribution of the (n+1)-bodies, and initial conditions, so that the solution is represented geometrically by an oscillating regular polygon with n sides rotating non-uniformly about its center. The body having a mass m_0 is at the center of the polygon, while nbodies having the same mass m are at the vertices of the polygon and move about the central body in identical elliptic orbits. It is proved that for n=2 and for regular polygons 3<= n<= 6 each corresponding solution is unstable for any value of the central mass m_0. For n=>7 the solution is linearly stable and the eccentricity of the particles’ orbits if both \mu = m_0 / m > 141.477 and the eccentricity of the particles' orbits e is sufficiently small