Reduced dynamics for momentum maps with cocycles (Q2778347)

Due to the Noether's theorem one can associate with any symplectic action of the finite-dimensional Lie group \(G\) on the finite-dimensional symplectic manifold \((M,\omega)\) a local momentum map which in turn rise to a well-defined infinitesimal 2-cocycle. This allows to be defined another action (if the cocycle is nontrivial) which is infinitesimally equivariant. In this last situation the authors find a way to construct a ``local model'' of the symplectic manifold and to split the Hamiltonian equations for a \(G\)-invariant Hamiltonians making use of the local coordinates in the vicinity of the \(G\)-orbit. As examples of all above the authors refer to vortex dynamics and charged particle motion in magnetic fields. The example which they study in some details, is the action of the special Euclidean group SE(2) on the plane that possesses a nontrivial 2-cocycle which is related to various regimes of dynamics of vortices in the plane.