Nonintegrable Perturbations of Two Vortex Dynamics

Abstract

The governing equations of motion of two point vortices in an ideal fluid in the plane has a Hamiltonian formulation that is completely integrable, so the dynamics are regular in the sense that one has quasiperiodic solutions confined to invariant two-dimensional tori accompanied by periodic orbits. Moreover, it is well known that the same is true of the dynamics of two point vortices in an ideal fluid in a standard half-plane (with a straight line boundary). It is natural to ask if this is also the case for half-planes whose boundaries are perturbations of a straight line. We prove here that there are such Hamiltonian perturbations of two vortex dynamics in the half-plane that generate chaotic — and a fortiori nonintegrable — dynamics, thereby answering an open question of rather long standing. Our proof, like most demonstrations of this kind, is based on Melnikov’s method.