Periodic Solutions with Alternating Singularities in the Collinear Four-body Problem

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This paper shows the existence of a periodic orbit with singularity in the symmetric collinear four body problem. In each period of the 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 proven by using the implicit function theorem and a continuity argument on differential equations of the regularized Hamiltonian.