Celletti, A., Universita' di Roma "Tor Vergata"
Basics of regularization theory
(745K, pdf)

ABSTRACT.  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.