I am trying to compute the moment map, for an action.
As I understand it if I have a configuration space which is given by a manifold $M$ and an action $\rho_g \colon M \to M$, it induces an action on the phase space
$(\rho_{g^{-1}})^* : T^*M \to T^*M$, $(\rho_{g^{-1}})^* : (q,p) \mapsto (gq, (\rho_{g^{-1}})^*(p))$.
Then we can compute the fundamental vector field.
Let $X\in \mathfrak{g}=T_eG$ be the an element of the Lie algebra, we then have
$X^\#_{(q,p)}=\frac{d}{dt}_{\vert_{t=0}} \rho_{exp(tX)^{-1}}^*(q,p)$.
I was trying do do an example to understand how it works.
The mathematical pendulum with coordinate $(\phi,\theta)$, and the action (with $G=S^1$)
$\rho_g : S^2 \to S^2$, $\rho_{\theta}:(\phi_0,\theta_0) \mapsto (\phi_0,\theta_0+\theta)$.
But in practice, I am confused as to how exactly calculat $(\rho_{-\theta})^*$ and $X^\#_{(q,p)}$.
