Hamiltonian Group Actions

Abstract

In this chapter, we will define Hamiltonian flows, Hamiltonian actions and moment maps. The layout of the chapter is as follows. In Sect. 2.1, we recall the original example of a Hamiltonian flow, namely, Hamilton’s equations. In Sect. 2.2, we will start by understanding what Hamiltonian vector fields and Hamiltonian functions are. In Sect. 2.3, we will introduce a bracket on the set of smooth functions on a symplectic manifold which will satisfy the Jacobi identity and will make the former into a Lie algebra. We will see some examples of such vector fields on $S^2$ and the 2-torus. In the final section (Sect. 2.4), we will define a moment map and will list some conditions which will guarantee the existence of moment maps and other conditions which guarantee their uniqueness.