Slices for lifted tangent and cotangent actions

A cotangent bundle slice theorem

This article concerns cotangent-lifted Lie group actions; our goal is to find local and ``semi-global'' normal forms for these and associated structures. Our main result is a constructive cotangent bundle slice theorem that extends the Hamiltonian slice theorem of Marle, Guillemin and Sternberg. The result applies to all proper cotangent-lifted actions, around points with fully-isotropic momentum values. We also present a ``tangent-level'' commuting reduction result and use it to characterise the symplectic normal space of any cotangent-lifted action. In two special cases, we arrive at splitti…

余切提升作用的辛法空间

The Symplectic Normal Space of a Cotangent-Lifted Action

本研究针对李群等距光滑真作用提升下的光滑黎曼流形余切丛，刻画了任意点的辛法空间，证明其可分解为线性空间余切丛与余伴随轨道约化辛线性空间的直和，且该刻画可仅用底流形群作用与余伴随表示表述，还探讨了相关特例。

余切提升作用的Hamiltonian管

The Hamiltonian Tube Of A Cotangent-Lifted Action

针对余切提升作用，构造了尊重纤维结构的显式MGS形式，推广了Schmah的结果，并对SO(3)和SL(2)群给出显式辛坐标。

Singular Poisson reduction of cotangent bundles

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 $Q\to Q/K$.

Cotangent models for integrable systems

We associate cotangent models to a neighbourhood of a Liouville torus in symplectic and Poisson manifolds focusing on a special class called $b$-Poisson/$b$-symplectic manifolds. The semilocal equivalence with such models uses the corresponding action-angle coordinate theorems in these settings: the theorem of Liouville-Mineur-Arnold [A74] for symplectic manifolds and an action-angle theorem for regular Liouville tori in Poisson manifolds [LMV11]. Our models comprise regular Liouville tori of Poisson manifolds but also consider the Liouville tori on the singular locus of a $b$-Poisson manifold…

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 pres…