I want to study about Symplectic group actions and moment map, especially Hamiltonian Group Actions. Can you help me with some concretely example of Hamiltonian group actions ? Where can I find some exemple?

I think these http://www.math.ist.utl.pt/~acannas/Books/lsg.pdf are very good notes, and freely available. 


Consider the representation of $U(1)$ on $\mathbb{C}^n$ defined by $$t\cdot (z_1,\ldots,z_n)=(tz_1,\ldots,tz_n),$$ where $t\in U(1)$. This action is Hamiltonian. 

