N Vortices in the Plane

Abstract

In this chapter we consider the equations for N-point vortices in the unbounded plane. First we give the general Hamiltonian formulation showing all conserved quantities. A by product of this discussion is the simple proof that the system is integrable for three or fewer vortices of any strength. The equations are then written in terms of their intervortical distances. In Section 2.2 we specialize to the case N = 3, characterizing all relative equilibria and collapsing states. Using the trilinear coordinates of Synge (1949) and Aref (1979), we then describe the phase plane formulation for more general states. In Section 2.3 we focus on the 4-vortex case. First we use the coordinates introduced in Khanin (1982) to reduce the Hamiltonian to one with two degrees of freedom. An extra feature of the reduction method is that it prepares the system for an application of the KAM theorem which we describe. We then use the Melnikov method to prove nonintegra-bility for a special 4-vortex configuration, following ideas of Ziglin (1980), Koiller and Carvalho (1985, 1989) and, more recently, Castilla, Moauro, Ne-grini, and Oliva (1993). Section 2.4 offers a brief summary of some classical and recent work that is particularly interesting and worth reading. Thus, taken together, we show in this chapter that the typical N-vortex problem contains regions of phase space that support periodic, quasiperiodic, and chaotic orbits, all coexisting.