  Suppose that a Lie group $G$ acts properly on a configuration manifold $Q$.
We study the symplectic quotient of $T^*Q$ with respect to the cotangent lifted
$G$-action at an arbitrary coadjoint orbit level $\mathcal{O}$. In particular,
if $Q=Q_{(H)}$ is of single orbit type we show that the symplectic quotient of
$T^*Q$ at $\mathcal{O}$ can be constructed through a minimal coupling procedure
involving the smaller cotangent bundle $T^*Q_H$, the symplectic quotient of
$\mathcal{O}$ at 0 with respect to the $H$-action, and the diagonal Hamiltonian
$N(H)/H$-action on these symplectic spaces. A prescribed connection on $Q_H\to
Q_H/(N(H)/H)$ then yields a computationally effective way of explicitly
realizing the symplectic structure on each stratum of the symplectic quotient
of $T^*Q$. In an example this result is combined with the projection method to
produce a stratified Hamiltonian system with very well hidden symmetries.
