A Hamiltonian description of the vortex theory in two dimensions
Abstract
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 HelmholtzKirchhoff 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.
 Publication:

Archiv of Mechanics, Archiwum Mechaniki Stosowanej
 Pub Date:
 1984
 Bibcode:
 1984ArMeS..36..715S
 Keywords:

 Flow Theory;
 Hamiltonian Functions;
 Incompressible Fluids;
 Three Dimensional Flow;
 Two Dimensional Flow;
 Vortices;
 Canonical Forms;
 Differential Equations;
 Dipoles;
 Dirac Equation;
 Euler Equations Of Motion;
 Helmholtz Equations;
 Lie Groups;
 Monopoles;
 Fluid Mechanics and Heat Transfer