Critical Solutions of Three Vortex Motion in the Parabolic Case

Abstract

Gröbli (1877) laid the foundation for the analysis of the motion of three point vortices in a plane by deriving governing equations for triangular configuration of the vortices. Synge (1949) took this formulation one step further to that of a similar triangle of unit perimeter, via trilinear coordinates. The final reduced problem is governed by an integrable twodimensional system of differential equations with solutions represented as planar trajectories.Another key to Synge's analysis was his classification of the problem into three distinct cases: elliptic, hyperbolic and parabolic corresponding, respectively, to the sum of products