Vortex dynamics: A variational approach using the principle of least action

The study of vortex dynamics using a variational formulation has an extensive history and a rich literature. The standard Hamiltonian function that describes the dynamics of interacting point vortices of constant strength is the Kirchhoff-Routh (KR) function. This function was not obtained from basic definitions of classical mechanics (i.e., in terms of kinetic and potential energies), but it was rather devised to match the already known differential equations of motion for constant-strength point vortices given by the Bio-Savart law. Instead, we develop a variational formulation for vortex dynamics based on the principle of least action. As an application, we consider two-dimensional massive vortices of constant strength. Interestingly, the obtained equations of motion are second-order differential equations defining vortex accelerations, not velocities. The resulting dynamics are more complex than those obtained from the KR formulation. For example, a pair of equal-strength, counter-rotating vortices could be initialized with different velocities, resulting in interesting patterns. Also, the developed model easily admits external body forces. When an electrodynamic force is considered, the interaction between it and the hydrodynamic vortex force leads to a rich, counterintuitive behavior that could not be handled by the KR formulation.