The chapter begins with a discussion of circle actions and their relation to 2-sphere bundles. It continues with a section on general Hamiltonian group actions and moment maps, then proceeds to discuss various explicit examples in both finite and infinite dimensions, and introduces the Marsden–Weinstein quotient, together with new examples that explain its relation to the construction of generating functions for Lagrangians. Further sections give a proof of the Atiyah–Guillemin–Sternberg convexity theorem about the image of the moment map in the case of torus actions, and use equivariant cohomology to prove the Duistermaat–Heckman localization formula for circle actions. It closes with an overview of geometric invariant theory which grows out of the interplay between the actions of a real Lie group and its complexification.
Hamiltonian group actions on exact symplectic manifolds with proper momentum maps are standard
