Three body coulomb problem

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

$${\cal H = T + V} = \sum ^{N=3}_{i=1} \frac {1}{2m_{i}} \Del... ...+ \sum ^{N=3}_{i<j} V _{ij} (\vert\rho _{i} - \rho _{j}\vert),$$ (15)

where $\rho _{i}$ is the particle position vector [3]. The center of mass motion can be separated out, leaving a six dimensional problem in r ₁, r ₂ space.

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.