A review on Poincaré research in celestial mechanics and dynamics (stable and unstable periodic motions)

Abstract

H. Poincaré considers in his Méthodes Nouvelles the asymptotic solutions obtained as particular solutions of Bohlin planetary perturbation method. He shows that several periodic solutions may appear when special invariant relationships are stated as initial conditions of this problem. The main results can be summarized as follows: I) Real (true) periodic solutions arise when the stated invariant relationships are equivalent to the conditions stated for obtaining periodic solutions of the first sort. These solutions are, then convergent (stable). II) If the initial conditions are different from the previous ones, periodic solutions can also be obtained. But, in this case, they are unstable. The essential difference between both cases rest upon the way in which the mean motions of masses are combined. Periodic solutions may also be obtained by applying the fixed point theorem. In fact, it may be shown that Riccati's equations in the complex domain can be used as perturbation equations. The dependent variables are the quantities p, q, r ( the instantaneous rotations of orbital system of coordinates, referred with respect to a fixed frame). These equations remain invariable under a bilinear transformations (Möbius). Since Möbius transformations may have a fixed stable point, the transformation theory shows that complete stability takes place in case of linear approximation. Then complete stability remains valid for the non-linear approximation, and the existence of a fixed point in the solutions is assured. Then, periodic orbits exist under such circumstances.