In this paper, we consider the elliptic relative equilibria of the restricted $4$-body problems, where the three primaries form an Euler collinear configuration and the four bodies span $\mathbf{ R}^2$. We obtain the symplectic reduction to the general restricted $N$-body problem. By analyzing the relationship between this restricted $4$-body problems and the elliptic Lagrangian solutions, we obtain the linear stability of the restricted $4$-body problem by the $\omega$-Maslov index. Via numerical computations, we also obtain conditions of the stability on the mass parameters for the symmetric cases.
Categories: math.DS math-ph math.MP