A Hamiltonian description of the vortex theory in two dimensions

The construction of a Hamiltonian structure of the vortex theory using general methods of symplectic geometry and Lie groups is demonstrated. A Hamiltonian form of the Helmholtz-Kirchhoff equations is shown to be related to a Hamiltonian form of Euler's equations due to Arnold (1976). This form is a bilinear, symplectic form on an infinite dimensional manifold of vector fields; the manifold is a coadjoint orbit in terms of Lie groups. It is shown that a similar, finite dimensional structure exists for some singular vector fields which describe various systems of vortices, including monopoles, vortex pairs, and dipoles. For each case, a Hamiltonian system of ordinary differential equations in a canonical form is obtained.