Generalization of Leapfrogging Orbits of Point Vortices

Point vortex motion arises in the study of concentrated vorticity in an ideal, incompressible fluid described by Euler’s equations. The two-dimensional Euler equations of fluid mechanics, a partial differential equation (PDE) system, support a solution where the vorticity is concentrated at a single point. Helmholtz derived a system of ordinary differential equations (ODEs) that describe the motion of a set of interacting vortices that behave as discrete particles, which approximates the fluid motion in the case that the vorticity is concentrated in very small regions. This system of equations has continued to provide interesting questions for over 150 years. We will discuss the special class of relative periodic orbits known as the leapfrogging orbits. The relative periodic orbit of a four-point vortex problem, consisting of two positive and two negative point vortices of equal absolute circulation arranged as coaxial vortex pairs, is known as the leapfrogging orbit. This dissertation presents generalizations of the leapfrogging motion of point vortices and vortex rings, including their stability and dynamics. More specifically, we study the leapfrogging motion of \(2N\) vortices, with half having positive circulation and half having negative circulation.