Abstract
The Hamiltonian formulation of classical mechanics has been key in establishing the dynamic and geometric properties of physical systems. This formalism experienced a resurgence during the last century in the form of symplectic geometry and, more generally, Poisson geometry. In this description, the interaction between conserved quantities and symmetries is encoded in the moment map, which generalizes the well-known classical moments. After reviewing these constructions, we introduce symplectic Lie algebroids. These objects lie between symplectic and Poisson structures and encode physical systems with singularities. We conclude the talk by describing an extension of the minimal coupling procedure to this setting.