Investigation of the Stability Problem for the Critical Cases of the Newtonian Many-Body Problem

Abstract

Using the computer algebra methods, different authors have proved the existence of new classes of the homografic solutions, in the Lagrange-Wintner sense [1], in the Newtonian many-body problem [2-6]. E.A. Grebenicov has also shown that any homographic solution of the n-body problem generates a new dynamical model, namely, “the restricted Newtonian (n + 1)-body problem”. These problems are similar to the famous “restricted three-body problem” which was proposed for the first time by K. Jacobi [7]. Then the theorems of the existence of stationary solutions (the equilibrium positions) for some fixed values of the parameter n were proved [8-10], and the problem of studying the stability of these solutions in the Lyapunov sense was formulated. The study of this problem can be done only on the basis of the KAM-theory [11-13] and only for the dynamical systems with two degrees of freedom it can be realized. We have shown that all the planar restricted n-body problems belong to this class for any n. The situation is essentially complicated for the critical (resonance) cases, when the eigenvalues of the linearized system of differential equations in the neighborhood of the stationary solution are rationally commensurable. In these critical cases the stability problem for hamiltonian dynamics may be studied only on the basis of Markeev and Sokolsky theorems [14,15]. These theorems contain mathematical estimations of the influence of so-called “non-annihilable resonance terms” in the Poincare-Birkho. normalizing transformations, which must be taken into account in the theorems on the stability of stationary solutions of hamiltonian equations in critical cases [16].