I am trying to understand the significance of coisotropic reduction in a mechanical way.
Let $(M,\omega)$ be a symplectic manifold and $i: N \hookrightarrow M$ a coisotropic submanifold i.e. $T_qN^{\perp} \subseteq T_qN \,\,\, \forall q \in N.$ The distribution $TN^\perp$ is regular and involutive and thus comes from a maximal regular foliation $\mathcal{F}$. Suppose $N/\mathcal{F}$, the space of all leaves, has a smooth manifold structure such that $\pi: N \rightarrow N/\mathcal{F}$ (the canonical projection) defines a submersion. Then, the Weinstein Theorem guarantees the existence of an unique symplectic form $\omega_N$ in $N/\mathcal{F}$ such that $\pi^*\omega_N = i^*\omega.$
In the case of reduction arising from symmetries from a Lie group $G$ (an action $\Phi: G \times M \rightarrow M$), an invariant Hamiltonian $H$ projects to a Hamiltonian $\widetilde H$ in the orbit space (that also inherits an unique symplectic structure). In this case, the dynamics of $X_H$ can be completely recovered from the dynamics of $X_{\widetilde H}$.
Is this also true for coisotropic reduction? That is, given a Hamiltonian $H$ (I am guessing that we need to ask $X_H$ to be tangent to $N$ so the dynamics are constrained), does it project to a Hamiltonian $\widetilde H$ in $N/\mathcal{F}$ such that the mechanics from $X_H$ can always be recovered from the mechanics of $X_{\widetilde H}$?