Let $\Phi:G\times M\ni(g,m)\mapsto\Phi_g(m)\in M$ be a smooth left action of a Lie group $G$ on a manifold $M$, and let us denote by $\Psi:G\times T^\ast M\ni(g,\alpha)\mapsto(T^\ast\Phi_g)\alpha\in T^\ast M$ its cotangent lifting.
It's easy to see that $\Phi$ is free (resp. proper) if and only if $\Psi$ is free (resp. proper).
So if $\Phi$ is free and proper then both the orbit spaces $M/G$ and $(T^\ast M)/G$ get a uniquely determined smooth manifold struscture such that the respective projection maps are submersions.
I was wondering if there exists some kind of relation between the smooth manifolds $T^\ast(M/G)$ and $(T^\ast M)/G$.
Until now I have realized the existence of a diffeomorphism between $T^\ast(M/G)$ and $J/G$, where $J$ is the $G$-invariant vector sub-bundle of $\tau_M^\ast:T^\ast M\to M$ which is the annihilator of the involutive distribution on $M$ generated by its fundamental vector fields w.r.t. the action $\Phi$ of $G$ on $M$.