Mutually Transitive Actions and Symplectic Reduction: Lie-Weinstein dual pairs and applications to the N-vortex problem on the plane.

This thesis investigates several theoretical aspects of Lie–Weinstein dual pairs, with particular emphasis on their relationship with pairs of mutually transitive actions. Within this framework, a general result, originally due to P. Skerritt and C. Vizman (Dual pairs for matrix groups, J. Geom. Mech. 11 (2019)), concerning the realization of symplectic quotients as coadjoint orbits is established. Following Ohsawa (Symplectic reduction and the Lie-Poisson shape dynamics of N point vortices on the plane, Nonlinearity 32 (2019)), this result is then applied to obtain the symplectically reduced equations for the N-vortex problem in the plane in the case of non-vanishing total circulation.