Motion of a system of vortex rings in an incompressible fluid

A Hamiltonian formalism is developed for the motion of a system of coaxial vortex rings in an unbounded ideal incompressible fluid. An additional motion invariant is determined which is the momentum of the surrounding fluid. In the case of two vortex rings, the equations of motion are fully integrable, which explains the mutual slip-through of the rings, described qualitatively by Helmholtz. The effect of viscosity on the initial stage of motion is assessed.