Kinematic variational principle for motion of vortex rings

We show how the ideas of topology and variational principle, opened up by Euler, facilitate the calculation of motion of vortex rings. Kelvin-Benjamin's principle, as generalised to three dimensions, states that a steady distribution of vorticity, relative to a moving frame, is the state that maximizes the total kinetic energy, under the constraint of constant hydrodynamic impulse, on an iso-vortical sheet. By adapting this principle, combined with an asymptotic solution of the Euler equations, we make an extension of Fraenkel-Saffman's formula for the translation velocity of an axisymmetric vortex ring to third order in a small parameter, the ratio of the core radius to the ring radius. Saffman's formula for a viscous vortex ring is also extended to third order.