Four -body problem with collision singularity

In this dissertation, regularization of simultaneous binary collision, existence of a Schubart-like periodic orbit, existence of a planar symmetric periodic orbit with multiple simultaneous binary collisions, and their linear stabilities are studied. The detailed background of those problems is introduced in chapter 1. The singularities of simultaneous binary collision in the collinear four-body problem is regularized in chapter 2. We use canonical transformations to collectively analytically continue the singularities of the simultaneous binary collision solutions in both the decoupled case and the coupled case. All the solutions are found and more importantly, we find a crucial first integral which describes the relationship between the decoupled solutions and the coupled solutions. In chapter 3, we show the existence of a Schubart-like orbit, a periodic orbit with singularities 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 a "turning point" technique and a continuity argument on differential equations of the regularized Hamiltonian. Analytical methods are used in chapter 4 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 can be extended to any even number of bodies. Multiple simultaneous binary collisions are a key feature of the orbits generated. In chapter 5, 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. For the collinear problem, this verifies the earlier numerical results of Sweatman for linear stability.