Spinning particles in a Yang-Mills field

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.