Exact solution of the planar motion of three arbitrary point vortices

TL;DR

This paper provides an exact analytical solution for the planar motion of three arbitrary point vortices, classifying all possible motions, stability conditions, and special configurations, including biperiodic, expanding, and collision trajectories.

Contribution

It offers the first exact solution for three vortex dynamics, detailing stability, classifications, and special motions, extending the understanding of vortex interactions.

Findings

Classified all possible vortex motions and configurations.

Identified conditions for stability and special trajectories.

Connected vortex dynamics to adiabatic invariants and Onsager's quantities.

Abstract

We give an exact quantitative solution for the motion of three vortices of any strength, which Poincar\'e showed to be integrable. The absolute motion of one vortex is generally biperiodic: in uniformly rotating axes, the motion is periodic. There are two kinds of relative equilibrium configuration: two equilateral triangles and one or three colinear configurations, their stability conditions split the strengths space into three domains in which the sets of trajectories are topologically distinct. According to the values of the strengths and the initial positions, all possible %RC motions are classified. Two sets of strengths lead to generic motions other than biperiodic. First, when the angular momentum vanishes, besides the biperiodic regime there exists an expansion spiral motion and even a triple collision in a finite time, but the latter motion is nongeneric. Second, when two strengths are opposite, the system also exhibits the elastic diffusion of a vortex doublet by the third vortex. For given values of the invariants, the volume of the phase space of this Hamiltonian system is proportional to the period of the reduced motion, a well known result of the theory of adiabatic invariants. We then formally examine the behaviour of the quantities that Onsager defined only for a large number of interacting vortices.