Ideal vortex motion in two dimensions: Symmetries and conservation laws

Abstract

Each of the conservation laws of the motion of a system of vortices in a two-dimensional ideal fluid are shown to be uniquely associated with one of the symmetry transformations of the system, one of which can be a scale transformation. In the case of an infinite fluid with no boundaries the five symmetry transformations are: (a) two independent translations in space, (b) spatial rotations, (c) translation in time, and (d) the scale transformation t′ =e2ηt, z′k=eηzk, where η is real and zk are the positions of the vortices in the z plane. Two cases of reduced symmetry are examined: the upper half-plane and the interior of the unit circle.