Nonlinear Science © 1994 Springer-Verlag New York Inc. On the Motion of Two-Dimensional Vortices with Mass

Summary. The Helmholtz-Kirchhoff ODEs governing the planar motion of N point vortices in an ideal, incompressible fluid are extended to the case where the fluid has impurities. In this case the resulting ODEs have an additional inertia-type term, so the point vortices are termed massive. Using an electromagnetic analogy, these equations also determine the behavior of columns of charges in an external magnetic field. Using the symmetries, we reduce the four degrees of freedom system of two "massive " vortices to two degrees of freedom. We exhibit an integrable case and a nonintegrable one, according to choices of parameters. Nonintegrability is verified using a recent result obtained independently b Lerman and by Mielke, Holmes, and O'Reilly. Finally, we discuss the behavior of solutions as the masses of the vortices tend to zero, using for initial conditions a point of the trajectory of the Helmholtz-Kirchhoff equations. Key words: vortex dynamics, dynamics of particles in fluids, Hamiltonian systems, singular perturbations, transversal homoclinic orbits