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 m0 is at the center of the polygon, while n bodies 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 m0 . For n => 7 the solution is linearly stable if both