Author ORCID Identifier
0009-0003-6871-0703
Document Type
Dissertation
Date of Award
8-31-2025
Degree Name
Doctor of Philosophy in Mathematical Sciences - (Ph.D.)
Department
Mathematical Sciences
First Advisor
Roy Goodman
Second Advisor
Amitabha Koshal Bose
Third Advisor
David Shirokoff
Fourth Advisor
Jonathan Jaquette
Fifth Advisor
Renato C. Calleja
Abstract
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.
To achieve this, 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 the non-physical singularities associated with collinear vortex configurations and facilitates a global phase space analysis, including a detailed and novel investigation of bifurcations.
Within this reduced framework, all relative fixed points are systematically identified and their stability and bifurcation behaviors are classified. The reduction is further applied to the vortex-dipole scattering problem, extending previous analyses to treat a more general case.
In this process, classical approaches dating back to Grobli (1877), which relied on pairwise vortex distances, are extended and refined. While historically effective, those coordinates introduce singularities that obstruct a complete analysis. The present method overcomes these limitations, enabling a singularity-free and unified description of the full three-vortex dynamics.
A major challenge arises in the case of vanishing total circulation, where the Jacobi coordinate reduction fails. This is addressed by refining the reduction technique to handle the degenerate case and establishing a continuous connection to the non-vanishing regime. This unified treatment enables a complete description of the three-vortex dynamics across all circulation configurations.
Finally, the methodology is extended to the integrable four-vortex problem that results under conditions of vanishing total circulation and linear impulse, providing new insight into vortex dynamics in regimes where classical tools are limited.
Recommended Citation
Anurag, Atul, "The global phase space of the three-vortex interaction system and its application to vortex-dipole scattering" (2025). Dissertations. 1851.
https://digitalcommons.njit.edu/dissertations/1851
