Let $G$ be a Lie group and $(M, \omega)$ a symplectic manifold. An action of $G$ on $M$ is Hamiltonian if it is equipped with a co-moment map $\widetilde{\mu} : \mathfrak{g} \to C^\infty(M)$ which is a Lie algebra morphism and satisfies $X_{\widetilde{\mu}(\alpha)} = \alpha_M$ for any $\alpha \in \mathfrak{g}$, where $\alpha_M$ is the infinitesimal generator and $X_{\widetilde{\mu}(\alpha)}$ is the Hamiltonian vector field of $\widetilde{\mu}(\alpha)$.

This condition is sometimes relaxed by not requiring $\widetilde{\mu}$ to preserve brackets. In this setting, for instance, Noether's theorem still holds (if $f \in C^\infty(M)$ is $G$-invariant then the moment map $\mu$ is preserved along the flow of $X_f$).

For connected $G$, the condition that $\widetilde{\mu}$ preserves brackets is equivalent to $\mu : M \to \mathfrak{g}^*$ being equivariant (in general, the latter implies the former). This is important if one wants to do symplectic reduction.

What are other instances where this condition is useful? Why impose this condition instead of the stronger condition of $\mu$ being equivariant?