PhD Dissertation: The global phase space of the three-vortex interaction system and its application to vortex-dipole scattering

This dissertation presents a global reduction of the classical three-vortex problem that is free from coordinate singularities, enabling a comprehensive analysis of the system’s dynamics across all circulation regimes. A two-step symplectic reduction procedure is developed. The first step introduces Jacobi coordinates adapted to the symplectic structure of the vortex system, and the second applies a Lie-Poisson reduction to the resulting system. This formulation eliminates non-physical singularities associated with collinear vortex configurations and enables a global phase space analysis, including a detailed study of bifurcations. Within this framework, all relative fixed points are classified along with their stability and bifurcation behavior. The reduction is also applied to the vortex-dipole scattering problem, extending classical results to a more general setting. Classical approaches dating back to Grobli (1877), based on pairwise vortex distances, are refined and extended. Although historically effective, those coordinates introduce singularities that prevent a complete global analysis. The present formulation overcomes these limitations, providing a unified singularity-free description of three-vortex dynamics. A key difficulty arises when the total circulation vanishes, where the Jacobi reduction degenerates. This case is treated via a modified reduction scheme that connects continuously to the non-vanishing regime, yielding a unified description across all circulation configurations. Finally, the methodology is extended to a special integrable four-vortex case under vanishing total circulation and linear impulse, offering new insights into regimes where classical techniques are insufficient.

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