Integrable many-body problems in more than one dimension
Integrable many-body problems in more than one dimension can be trivially constructed by piecing together one-dimensional integrable models. For instance the three-dimensional n-body problems with the Hamiltonian \[ H=1/2\sum^{n}_{j=1}(p^ 2_ j+\omega^ 2r^ 2_ j)+g^ 2\sum^{n}_{j>k=1}V(r_ j-r_ k) \] and the pair potentials \[ V(r)=(1-\sin^ 2\phi \sin^ 22\theta)/(r \cos \phi \cos 2\theta)^ 2 \] or \[ V(r)=(1-\sin^ 2\phi \cos^ 22\theta)/(r \sin 2\phi \sin 2\theta)^ 2 \] are integrable (here r, \(\theta\) and \(\phi\) are of course the polar coordinates of the 3-vector r).
scientific article; zbMATH DE number 4039368