Why is the moment map constant on the orbits of the action of the Lie algebra Given the action of a Lie group on a symplectic manifold, the moment map gives a mapping $\mu: M \rightarrow \mathfrak{g}^*$ to the dual of the Lie algebra $\mathfrak{g}^*$ defined by $d(\langle \mu,\eta\rangle)=i_{X_\eta}\omega$, where $X_\eta$ is the vector field generated by the action of $\eta \in \mathfrak{g}$ and $\langle \mu,\eta\rangle$ is just the pairing between elements of the Lie algebra and it's dual. I can compute simple examples but what I cannot see, intuitively, why the moment map is constant on the orbits of the action - which is of course why the moment map is important and useful in the first place. Can someone say an inspired sentence that will give me that "aha!" moment?

Since the flow is Hamiltonian we have $d(\langle \mu, \eta \rangle)=i_{X_\eta}\omega=dH_\eta$ for some function $H_\nu$. The orbits of the Hamiltonian vector field $X_\eta$, occur on the levels sets of $H_\eta$ and so on the level sets of $\langle \mu, \eta \rangle$, for which $\mu$ is constant. 
