Symplectic subcluster methods and periodic vortex motion on a sphere

This dissertation deals with the equations of motion for N point vortices moving on an S2 sphere (the PVS system), and obtains numerically the bifurcations, the periodic orbits, the equilibrium solutions and the streamline patterns arising from the motion of at least four vortices on the sphere. Often eight, twelve, fourteen, twenty, twenty-four, forty-eight and even sixty vortices are involved. A new symplectic algorithm is proposed involving the use of subcluster integrative techniques as well as the results from other integrators as a means for comparison with the symplectic method.; The Platonic solids are shown to be the fixed and the relative equilibrium discretizations of the Euler equations. The frequency of the Platonic solid body rotation is derived and shown to be equal to kGamma/2pi R2 where the constant k depends on the geometry of the solid. The PVS system undergoes a bifurcation with the Platonic equilibria giving birth to a multitude of periodic orbits. The dynamical stability of such periodic orbits is analyzed using Floquet theory.; A new technique is described to obtain families of equilibria based upon the principle of superposition. A vortex buckyball is then 'grown' and the vortex molecules resulting from various Archimedean solids are shown to vibrate (at a frequency which depends on the number of the vortices looping together).; Conditions for the fixed equilibria for a separate class of prismatic and antiprismatic solutions are derived from the minimum energy criterion. Scaling laws are obtained for interacting vortex rings. Lastly, the PVS system is shown to be related to some other physical systems.