Abstract. We prove the analytic existence of a symmetric periodic simultaneous binary collision orbit in a regularized planar pairwise symmetric equal mass four-body problem. We provide some analytic and numerical evidence for this periodic orbit to be linearly stable. We then use a continuation method to numerically find symmetric periodic simultaneous binary collision orbits in a regularized planar pairwise symmetric 1, m, 1, m four-body problem for m between 0 and 1. We numerically investigate the linear stability of these periodic orbits through long-term integration of the regularized equations, showing that linear stability occurs when 0.538 ≤ m ≤ 1, and instability occurs when 0 < m ≤ 0.537 with spectral stability for m ≈ 0.537.

We apply the analytic-numerical method of Roberts to determine the linear stability of time-reversible periodic simultaneous binary collision orbits in the symmetric collinear four-body problem with masses 1, m, m, 1, and also in a symmetric planar four-body problem with equal masses. In both problems, the assumed symmetries reduce the determination of linear stability to the numerical computation of a single real number. For the collinear problem, this verifies the earlier numerical results of Sweatman for linear stability with respect to collinear and symmetric perturbations.

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Analytical methods are used to prove the existence of a periodic, symmetric solution with singularities in the planar 4-body problem. A numerical calculation and simulation are used to generate the orbit. The analytical method easily extends to any even number of bodies. Multiple simultaneous binary collisions are a key feature of the orbits generated.

We apply the symmetry reduction method of Roberts to numerically analyze the linear stability of a one-parameter family of symmetric periodic orbits with regularizable simultaneous binary collisions in the planar pairwise symmetric four-body problem with a mass m ∈ (0, 1] as the parameter. This reduces the linear stability analysis to the computation of two eigenvalues of a 3 × 3 matrix for each m ∈ (0, 1] obtained from numerical integration of the linearized regularized equations along only the first one-eighth of each regularized periodic orbit. The results are that the family of symmetric periodic orbits with regularizable simultaneous binary collisions changes its linear stability type several times as m varies over (0, 1], with linear instability for m close or equal to 0.01, and linear stability for m close or equal to 1.