Xavier RIVAS GUIJARRO

Bartosz M. Zawora (University of Warsaw)

Symplectic Reduction: The Marsden-Weinstein Theorem and Applications

This thesis explores the geometric foundations of reduction in Hamiltonian mechanics, a powerful method for simplifying dynamical systems with symmetry. The project will guide the student through the necessary prerequisites in symplectic geometry, specifically focusing on Hamiltonian group actions and the equivariant moment map. The core objective is to rigorously prove the Marsden-Weinstein Reduction Theorem, demonstrating how the phase space can be reduced to a lower-dimensional symplectic manifold ($M_\mu = J^{-1}(\mu) / G_\mu$). Furthermore, the student will show that the Hamiltonian dynamics descend naturally to this reduced space. Finally, this theoretical framework will be applied to a concrete physical system to explicitly derive the reduced equations of motion.

04-FEB-2026

symplectic geometry, Hamiltonian mechanics, Marsden-Weinstein reduction, momentum map, symmetry group

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