Three New Classes of Solvable N-Body Problems of Goldfish Type with Many Arbitrary Coupling Constants

Abstract: Three new classes of N-body problems of goldfish type are identified, with N an arbitrary positive integer (N ě 2). These models are characterized by nonlinear Newtonian (“accelerations equal forces”) equations of motion describing N equal point-particles moving in the complex z-plane. These highly nonlinear equations feature many arbitrary coupling constants, yet they can be solved by algebraic operations. Some of these N-body problems are isochronous, their generic solutions being all completely periodic with an overall period T independent of the initial data (but quite a few of these solutions are actually periodic with smaller periods T{p with p a positive integer); other models are isochronous for an open region of initial data, while the motions for other initial data are not periodic, featuring instead scattering phenomena with some of the particles incoming from, or escaping to, infinity in the remote past or future.