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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]

This paper gives an analytic proof of the existence of Schubart-like orbit, a periodic orbit with singularities in the symmetric collinear four-body problem. In each period of the Schubart-like orbit, there is a binary collision (BC) between the inner two bodies and a simultaneous binary collision (SBC) of the two clusters on both sides of the origin. The system is regularized and the existence is proved by using a 'turning point' technique and a continuity argument on differential equations of the regularized Hamiltonian. [ABSTRACT FROM AUTHOR]

For each given n ≥ 2 , we study the variational existence of a new spatial periodic orbit in the 2n-body problem. Besides excluding possible collision singularities, the main challenges left are to show that the orbit is not a relative equilibrium and it is spatial. By introducing a new estimate in the collinear central configuration, we can prove that it is not a relative equilibrium in the 2n-body problem. Furthermore, with the help of numerical calculation, the orbit is spatial in the equal-mass four-body problem. [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]