Reduction of symplectic manifolds with symmetry
Marsden, J. E. and A. Weinstein
Reports on Math. Phys., 5, (1974), 121-130
We give a unified framework for the construction of symplectic manifolds from systems with symmetries. Several physical and mathematical examples are given; for instance, we obtain Konstant's result on the symplectic structure of the orbits under the coadjoint representation of a Lie group. The framework also allows us to give a simple derivation of Smale's criterion for relative equilibria. We apply our scheme to various systems, including rotationally invariant systems, the rigid body, fluid flow, and general relativity.