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.