INTEGRABLE AND SOLVABLE MANY-BODY PROBLEMS IN THE PLANE VIA COMPLEXIFICATION

A simple prescription allows us to transform, by appropriate complexif ication, any one-dimensional many-body problem (describing ''motions o n a line'') with Newtonian (''acceleration equals forces'') equations of motion featuring forces which depend analytically on the positions, and possibly on the velocities, of the particles, and which are moreo ver scaling- or translation-invariant, into a two-dimensional many-bod y problem (describing ''motions in the plane'') with rotation-invarian t equations of motion. If the original (one-dimensional) model is Hami ltonian, the two-dimensional model is also Hamiltonian (in fact, bi-Ha miltonian). In this manner, starting from known integrable or solvable one-dimensional many-body problems, integrable or solvable many-body problems in the plane, generally featuring much richer dynamics, are o btained. Several examples are exhibited. Finally another, more direct, prescription is outlined, to transform by complexification almost any one-dimensional many-body problem into a rotation-invariant many-body problem in the plane. (C) 1998 American Institute of Physics. [S0022- 2488(98)02310-X].