sg.symplectic geometry - cotangent bundle symplectic reduction and fibre bundles

This is more a comment towards Gourishankar than an answer to the original question.
It was part of my thesis, (UCB, about 1986), so, apologies, I chime in. For simplicity,
I take the case $G$ Abelian, and $H = $ trivial.
To map $J^{-1} (mu)$ equivariantly to $J^{-1}(0)$ subtract $mu cdot A$ where $A$ is any $G$-connection for $pi: X to X/G$. $J^{-1} (0)/G = T^* (X/G)$
canonically, independent of connection. The map `momentum shift map
of subtracing $mu cdot A$ from co-vectors is not symplectic,
relative to the standard structure, but it becomes symplectic if you subtract
$mu pi^* F_A$, where $F_A = curv(A)$, from the standard structure. So the reduced space at $mu$ is $T^*(X/G)$ with the standard structure minus the ``magnetic term''
$mu F_A$.

For non-Abelian $G$ ($H$ still trivial), it is easier to explain things in Poisson terms.
$T^* X/G$ is a Poisson manifold whose symplectic leaves
are the reduced spaces in question. The momentum
shift trick turns it into $T^* (X/G) oplus Ad^* (X)$ where $Ad^* (X) to X/G$
is the co-adjoint bundle associated to $X to X/G$ -- its fibers are the
dual Lie algebras for $G$. This direct sum bundle admits coordinates $s_i, p_i, xi_a$ where $s_i, p_i$ are canonical coordinates on $T^*(X/G)$ induced by coordinates $s_i$ on $X/G$
and where $xi_a$ are fiber-linear coordinates on the co-adjoint bundle induced
by a choice of local section of $pi$. Then the main tricky part of the bracket is that
the bracket of $p_i$ with $p_j$ is $Sigma xi_a F^a _{ij}$, $F$ being the curvature of the connection
relative to the choice of local section. The symplectic leaves = reduced spaces
are of the form $T^*(X/G) oplus $(co-adjoint orbit bundle).