The dynamics of three vortices revisited

Abstract

The dynamics of three vortices was studied by Synge using the length of the sides of the triangle formed by three vortices as prime variables. The critical states at which the lengths of the sides remain fixed throughout the motion were found to be either equilateral triangles or collinear configurations. The equilateral configurations were shown to be stable or unstable depending on whether the sum of the products of strengths K was greater or less than zero, respectively. In the case K=0, a one-parameter family of solutions of contracting and another of expanding similar triangles were found. In this paper, it is shown that for this special case, the family of contracting similar solutions is always unstable while the family of expanding ones is stable. The critical states for collinear configurations in the general case are then studied where K is greater than or less than 0. It is shown that there are either six or four critical states depending on the strengths of the vortices. When there are six collinear critical states, three of them are always stable, one is not while the remaining two are unstable (stable) if the equilateral triangle configuration is stable (unstable). When there are only four collinear critical states, they are all stable while the equilateral triangle configuration is always unstable. Since there are two equilateral triangle configurations, clockwise and counterclockwise arrangements of the three vortices, the sum of the indices of all the critical states is equal to +2 regardless of whether K is greater than or less than 0. An integral invariant in trilinear coordinates is derived. When all the critical points and the integral invariant are known, the global behavior of the trajectories is obtained.