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Abstract: In this paper, we study the existence and linear stability of the Broucke periodic orbit in the planar three-body problem. In each period of this orbit, there are two binary collisions (or BC for short) between the outer bodies, while the inner body reaches its minimum or maximum at the time of each BC. A surprising simple existence proof of this orbit is given. The initial condition of this orbit is shown to be a supremum of some well-chosen set. The linear stability is then analyzed by Robertsʼ symmetry reduction method. It is shown that the Broucke periodic orbit with equal masses is linearly stable. [Copyright &y& Elsevier]

Abstract: In this paper, we study the existence and linear stability of the rhomboidal periodic orbit in the planar equal mass four-body problem. The Hamiltonian of the differential system is regularized by a Levi–Civita type transformation and an appropriate scaling of time. The initial condition of this orbit is shown to be the infimum of some well-chosen set. This existence proof is direct and surprisingly simple. Further, a careful study shows that this orbit has a symmetry group isomorphic to the dihedral group . Then Robertsʼ symmetry reduction method is applied to show the linear stability. It turns out that the rhomboidal periodic orbit in the planar equal mass four-body problem is linearly stable. [Copyright &y& Elsevier]

This paper gives an analytic existence proof of the Schubart periodic orbit with arbitrary masses, a periodic orbit with singularities in the collinear three-body problem. A 'turning point' technique is introduced to exclude the possibility of extra collisions and the existence of this orbit follows by a continuity argument on differential equations generated by the regularized Hamiltonian. [ABSTRACT FROM AUTHOR]

We analytically prove the existence of a symmetric periodic simultaneous binary collision orbit in a regularized planar pairwise symmetric equal mass four-body problem. This is an extension of our previous proof of the analytic existence of a symmetric periodic simultaneous binary collision orbit in a regularized planar fully symmetric equal mass four-body problem. 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. Numerical estimates of the the characteristic multipliers show that these periodic orbits are linearly stability when 0.54 ≤ m ≤ 1, and are linearly unstable when 0 < m ≤ 0.53. [ABSTRACT FROM AUTHOR]