Equilibria, Lattices, and Chaotic Dynamics of Point Vortices.

This thesis deals with three separate and somewhat unrelated problems involving two-dimensional, inviscid Hamiltonian point vortices. The first chapter presents a method for analytically determining stationary configurations for point vortices, for systems which may have arbitrary numbers and strengths of vortices. For the equilibrium configurations, the existence of an arbitrary parameter in the solutions is shown, which may have arbitrary numbers and strengths of vortices. For the equilibrium configurations, the existence of an arbitrary parameter in the solutions is shown, which continuously changes the symmetry and has as yet no known physical interpretation. By adding an external flow field, it is also possible to determine configurations which move rigidly or produce more complicated flow lines; examples are given of uniformly translating configurations and quadrupole field configurations. Finally, a connection is made between the structure of the simplest equilibrium configurations and the solutions of the Korteweg -de Vries equation, which raises the possibility of transforming between solution sets. Chapter two presents a technique for exactly calculating the energy of a two-dimensional lattice of vortices, for a Bravais lattice with arbitrary cell parameters and arbitrary numbers of vortices in the primitive cell. The Hamiltonian and dynamical equations are also calculated, and these are used to implement a numerical relaxation technique to determine minimum energy configurations corresponding to stable or metastable lattices. Several new simple lattice structures are presented, and their stability is discussed. Also, an investigation is made of the energy of an imperfect lattice as a function of the density of vacancies at lattice sites. Several suggestions are made for further applications. Chapter three presents an investigation of the integrability of the motion of a vortex system in two types of bounded domains, the ellipse and the rectangle. The Green's functions for these two domains are calculated using conformal mapping techniques, and these are used to construct the Hamiltonian and dynamical equations. Numerical experiments are then performed to determine the regularity of the motion of a two-vortex system in both boundary geometries, and in both cases these systems are stochastic, indicating the absence of a second global integral for either boundary. The behaviour of the vortex trajectories are also discussed in the context of KAM theory for near-integrable Hamiltonian systems. Finally, a numerical experiment is performed to investigate some of the structure of the phase space, by perturbing a period-one torus and examining the behaviour around the resulting hyperbolic fixed point.