Simultaneous binary collisions in the equal-mass collinear four-body problem

In the four-body problem, it is not clear what initial conditions can lead to simultaneous binary collision (SBC), even in the collinear case. In this paper, we study SBC in the equal-mass collinear four-body problem and have a partial answer for initial conditions leading to SBC. After introducing a Levi-Civita type transformation, we analyze the new transformed differential system of SBC and solve for all possible solutions. The problem is studied in two cases: decoupled case and coupled case. In the decoupled case where SBC is treated as two separated binary collisions, the initial conditions leading to SBC satisfy several simple algebraic identities. This result gives insights to the coupled case, which is SBC in the equal-mass collinear four-body problem. Furthermore, we show from a different perspective that solutions passing through SBC must be analytic in the transformed system and the initial condition set leading to SBC has a measure 0.