Geometric Reductions, Dynamics and Controls for Hamiltonian System with Symmetry

This is a survey article, from the viewpoint of the completeness of the Marsden- Weinstein reduction, to introduce briefly some recent developments of the symmetric reductions and Hamilton-Jacobi theory of the regular controlled Hamiltonian systems, the nonholonomic controlled Hamiltonian systems and the controlled magnetic Hamiltonian systems. These research reveal the deeply internal relationships of the geometrical structures of phase spaces, the nonholonomic constraint, the dynamical vector fields and the controls of these systems.