Why is the moment map constant on the orbits of the action of the Lie algebra

Question

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?

Answer 1

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.