arXiv subjects
Publications and source records attributed to Simon Hochgerner.
We develop a bundle picture for the case that the configuration manifold has only a single isotropy type, and give a formula for the reduced symplectic form in this setting. Furthermore, as an application of this bundle picture we consider Calogero-Moser systems with spin associated to polar representations of compact Lie groups.
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.
We consider the Poisson reduced space $(T^*Q)/K$ with respect to a cotangent lifted action. It is assumed that $K$ is a compact Lie group which acts by isometries on the Riemannian manifold $Q$ and that the action on $Q$ is of single isotropy type. Realizing $(T^*Q)/K$ as a Weinstein space we determine the induced Poisson structure and its symplectic leaves. We thus extend the Weinstein construction for principal fiber bundles to the case of surjective Riemannian submersions $Q\to Q/K$.