Integrable Hamiltonian N-body problems of goldfish type featuring N arbitrary functions. Issue 1 (2nd January 2017) - Calogero, Francesco
Abstract : A simple application of a neat formula relating the time evolution of the N zeros of a (monic) time-dependent polynomial of degree N in the complex variable w to the time evolution of its N coefficients allows to identify integrable Hamiltonian N -body problems in the plane featuring N arbitrary functions, the equations of motions of which are of Newtonian type: accelerations equal forces nonlinearly dependent on the coordinates of the N particle. The motions generally take place in the complex z -plane, or, equivalently, in the Cartesian xy -plane with z = x +i y . It is also easy to identify qualitative features of special subclasses of these models, for instance cases in which all the motions starting from an arbitrary real set of initial data are confined and multiply periodic . It is also indicated how to generate from these models hierarchies of analogous models with analogous properties.