  In this Letter the method of Lund is applied to formulate a variational
principle for the motion of charged vortices in an effective non-linear
Schr\"{o}dinger field theory describing finite size two-dimensional quantum
Hall samples under the influence of an arbitrary perpendicular magnetic field.
Freezing out variations in the modulus of the effective field yields a $U(1)$
sigma-model. A duality transformation on the sigma-model reduces the problem to
finding the Green function for a related electrostatics problem. This duality
illuminates the plasma analogy to the Laughlin wave function.
