Linear stability of the elliptic relative equilibria for the restricted N -body problem: Two special cases

In this paper, we consider the elliptic relative equilibria of the restricted N-body problems, where the N − 1 primaries form an Euler-Moulton collinear central configuration or a 1 + n-gon central configuration. We obtain the symplectic reduction for the general restricted N-body problem. For the first case, by analyzing the relationship between these restricted N-body problems and the elliptic Lagrangian solutions, we distinguish the linear stability of the restricted N-body problem by the ω-Maslov index. Through numerical computations, we also determine the stability conditions in terms of the mass parameters for N = 4 and the symmetry of the central configuration. For the second case, there exist three positions S1, S2 and S3 for the massless body (up to rotations of angle 2πn). For m0m sufficiently large, we show that the elliptic relative equilibria is linearly unstable if the eccentricity 0 ≤ e < e0 and the massless body lies at S1 or S2; while the elliptic relative equilibria is linearly stable if the massless body lies at S3.