It is well known that the three body problem does not have a general
analytical solution, and it has been the subject of considerable academic
efforts since the 17th century. The history as well as recent progress on
the oldest three body problem, the Moon-Earth-Sun system have been reviewed
by Gutzwiller [87].
The Coulomb three body problem, in particular, has been a difficult one to
solve accurately, due in part to the long range nature of the interaction,
in comparison to few nucleon systems in which the interaction is
short-ranged. It is an active area of research, as indicated by a number of
recent developments and refinements of theoretical techniques with the help
of ever-increasing computing resources. Efforts are being
made to extend the calculations to full four-body muonic
problems [92,93].
In general, the three body Hamiltonian can be written as
The exact form of the Hamiltonian depends on the choice of the co-ordinates r 1, r 2. In terms of kinematics, there is no unique choice of the co-ordinates, hence choosing suitable ones, which will give accurate results, is one of the challenges that theorists face.