Relative equilibria of point vortices and the fundamental theorem of algebra

Abstract

Relative equilibria of identical point vortices may be associated with a generating polynomial that has the vortex positions as its roots. A formula is derived that relates the first and second derivatives of this polynomial evaluated at a vortex position. Using this formula, along with the fundamental theorem of algebra, one can sometimes write a general polynomial equation. In this way, results about relative equilibria of point vortices may be proved in a compact and elegant way. For example, the classical result of Stieltjes, that if the vortices are on a line they must be situated at the zeros of the N th Hermite polynomial, follows easily. It is also shown that if in a relative equilibrium the vortices are all situated on a circle, they must form a regular N -gon. Several other results are proved using this approach. An ordinary differential equation for the generating polynomial when the vortices are situated on two perpendicular lines is derived. The method is extended to vortex systems where all the vortices have the same magnitude but may be of either sign. Derivations of the equation of Tkachenko for completely stationary configurations and its extension to translating relative equilibria are given.