Symmetry and Symplectic Reduction

The use of symmetries in the quantitative and qualitative study of dynamical systems has a long history that goes back to the founders of mechanics. In most cases, the symmetries of a system are used to implement a procedure generically known under the name of "reduction" that restricts the study of its dynamics to a system of smaller dimension. This procedure is also used in a purely geometric context to construct new nontrivial manifolds having various additional structures.