How do singularities move in potential flow?

The equations of motion of point vortices embedded in incompressible flow go back to Kirchhoff. They are a paradigm of reduction of an infinite-dimensional dynamical system, namely the incompressible Euler equation, to a finite-dimensional system, and have been called a "classical applied mathematical playground". The equation of motion for a point vortex can be viewed as the statement that the translational velocity of the point vortex is obtained by removing the leading-order singularity due to the point vortex when computing its velocity. The approaches used to obtain this result are reviewed, along with their history and limitations. A formulation that can be extended to study the motion of higher singularities (e.g. dipoles) is then presented. Extensions to more complex physical situations are also discussed. (C) 2011 Elsevier B.V. All rights reserved.