Novel isochronous N-body problems featuring N arbitrary rational coupling constants

Abstract

A novel class of N-body problems is identified, with N an arbitrary positive integer (N ≥ 2). These models are characterized by Newtonian (“accelerations equal forces”) equations of motion describing N equal point-particles moving in the complex z-plane. These highly nonlinear equations feature N arbitrary coupling constants, yet they can be solved by algebraic operations and if all the N coupling constants are real and rational the corresponding N-body problem is isochronous: its generic solutions are all completely periodic with an overall period T independent of the initial data (but many solutions feature subperiods T/p with p integer). It is moreover shown that these models are Hamiltonian.