Real description of classical Hamiltonian dynamics generated by a complex potential

Abstract

We show that the complex dynamics defined by an analytic potential v: C -> C admits a description in terms of the phase space R-4 equipped with an unconventional symplectic structure J. We obtain the general form of J and devise an equivalent real description of the system that is based on the conventional symplectic structure on R-4. In this description the real part of the complex Hamiltonian generates the dynamics and its complex part is an independent integral of motion rendering the system integrable.