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.