Vortex Dynamics using the Principle of Least Action*

Variational formulation of vortex dynamics has a long history with a very rich literature. The standard Hamiltonian that describes the dynamics of interacting constant-strength point vortices is the Kirchhoff-Routh (KR) function. This function was not derived from basic definitions in classical mechanics (it is not the system kinetic and potential energies summation). Rather, it was devised such that its hamiltonian dynamics match the already known first-order differential equations of motion for constant-strength point vortices given by the Bio-Savart law. While this approach is widely accepted, it is an ad-hoc one that does not allow for extension to, say, time-varying vortices. Here we develop a new variational formulation of vortex dynamics using the principle of least action, and the Lagrangian density is given by the pressure. A system of non-deforming free vortex patches of constant-strength is considered as a case study. Setting the first variation of the action integral with respect to the position coordinates of the vortices results, for the first time, into second-order ODEs, defining vortex acceleration not velocity. Interestingly, for constant-strength point vortices (the limit to vanishing core), the current formulation reduces to KR dynamics. However, for vortices with finite cores, the resulting dynamics are richer than that derived from KR. For example, a pair of equal-strength counter-rotating vortices starting with the same velocity, will continuously attract and repel each other in contrast to the constant-relative-distance motion predicted by the KR dynamics. And if they have different initial velocities, many interesting patterns may occur, which can't be even handled by the standard KR approach. Finally, the fact that this new variational formulation is derived from mechanics first principles, allows straightforward extension to arbitrary time-varying vortices.