Basics of regularization theory

We consider the dynamics of three point masses, where we assume that the mass of the third body is so small that it does not affect the motion of the primaries. In the framework of the restricted three–body problem, we investigate the collisional trajectories, which correspond to a singularity of the equations of motion. We investigate the regularizing techniques known as Levi–Civita, Kustaanheimo–Stiefel and Birkhoff transformations. The Levi–Civita regularization is adapted to the study of the planar restricted three–body problem, when considering a single collision with one of the primaries. The Kustaanheimo–Stiefel method concerns the same problem when the bodies are allowed to move in the 3–dimensional space. A simultaneous regularization with both primaries is achieved through the implementation of Birkhoff’s transformation.