Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

The Hamiltonian Tube Of A Cotangent-Lifted Action

Miguel Rodriguez-Olmos, Miguel Teixidó-Román

TL;DR

This work develops an explicit, fibered Hamiltonian tube construction for cotangent-lifted actions on $T^*Q$, refining the classical MGS normal form to respect the cotangent bundle’s fibration and making the model explicit up to solving a differential equation on the group $G$. The authors introduce simple and restricted $G$-tubes as building blocks, together with a $\Gamma$-shift, to produce a general tube around any orbit, including the nonzero momentum case via a fibered shift. They reduce the problem to a zero-momentum case on a cotangent model $T^*(G\times_H S)$ and then compose it with the $\Gamma$-shift to handle general momentum, proving local symplectomorphism and injectivity in carefully chosen neighborhoods. A fibered Bates–Lerman lemma is established, enabling a fibered description of momentum level sets in cotangent bundles, and the theory is illustrated with explicit tubes for $SO(3)$ and $SL(2,\mathbb{R})$, plus a detailed $SO(3)$ example acting on $T^*\mathbb{R}^3$. The results provide explicit coordinates for cotangent-lifted actions, with potential impact on geometric mechanics and quantization by enabling tractable, fiber-compatible local models and singular reduction analyses.

Abstract

The Marle-Guillemin-Sternberg (MGS) form is local model for a neighborhood of an orbit of a Hamiltonian Lie group action on a symplectic manifold. One of the main features of the MGS form is that it puts simultaneously in normal form the existing symplectic structure and momentum map. The main drawback of the MGS form is that it does not have an explicit expression. We will obtain a MGS form for cotangent- lifted actions on cotangent bundles that, in addition to its defining features, respects the additional fibered structure present. This model generalizes previous results obtained by T. Schmah for orbits with fully-isotropic momentum. In addition, our construction is explicit up to the integration of a differential equation on $G$. This equation can be easily solved for the groups $SO(3)$ or $SL(2)$, thus giving explicit symplectic coordinates for arbitrary canonical actions of these groups on any cotangent bundle.

The Hamiltonian Tube Of A Cotangent-Lifted Action

TL;DR

This work develops an explicit, fibered Hamiltonian tube construction for cotangent-lifted actions on , refining the classical MGS normal form to respect the cotangent bundle’s fibration and making the model explicit up to solving a differential equation on the group . The authors introduce simple and restricted -tubes as building blocks, together with a -shift, to produce a general tube around any orbit, including the nonzero momentum case via a fibered shift. They reduce the problem to a zero-momentum case on a cotangent model and then compose it with the -shift to handle general momentum, proving local symplectomorphism and injectivity in carefully chosen neighborhoods. A fibered Bates–Lerman lemma is established, enabling a fibered description of momentum level sets in cotangent bundles, and the theory is illustrated with explicit tubes for and , plus a detailed example acting on . The results provide explicit coordinates for cotangent-lifted actions, with potential impact on geometric mechanics and quantization by enabling tractable, fiber-compatible local models and singular reduction analyses.

Abstract

The Marle-Guillemin-Sternberg (MGS) form is local model for a neighborhood of an orbit of a Hamiltonian Lie group action on a symplectic manifold. One of the main features of the MGS form is that it puts simultaneously in normal form the existing symplectic structure and momentum map. The main drawback of the MGS form is that it does not have an explicit expression. We will obtain a MGS form for cotangent- lifted actions on cotangent bundles that, in addition to its defining features, respects the additional fibered structure present. This model generalizes previous results obtained by T. Schmah for orbits with fully-isotropic momentum. In addition, our construction is explicit up to the integration of a differential equation on . This equation can be easily solved for the groups or , thus giving explicit symplectic coordinates for arbitrary canonical actions of these groups on any cotangent bundle.

Paper Structure

This paper contains 26 sections, 17 theorems, 129 equations.

Table of Contents

  1. Introduction
  2. Acknowledgements.
  3. Preliminaries
  4. Proper actions and slices
  5. Hamiltonian actions and Hamiltonian tubes
  6. The MGS model as a reduced space
  7. The symplectic slice for cotangent-lifted actions
  8. A Lie algebra decomposition
  9. The symplectic slice
  10. $G$-tubes
  11. Simple $G$-tubes
  12. Restricted $G$-tubes
  13. Cotangent bundle Hamiltonian tubes
  14. Cotangent-lifted Palais model
  15. The $\alpha=0$ case
  16. ...and 11 more sections

Key Result

Theorem 2.1

Let $(M,\omega)$ be a symplectic manifold endowed with a proper Hamiltonian action of a Lie group $G$ with momentum map $\mathbf{J}:M\rightarrow \mathfrak{g}^*$. Let $z\in M$, $\mu=\mathbf{J}(z)$, and choose a $G_z$-invariant splitting $\mathfrak{g}_\mu=\mathfrak{g}_z\oplus \mathfrak{m}$. Let $N$ be where $u_i=(T_eL_g\xi_i;\dot \nu_i,\dot v_i) \in T_{(g,\nu,v)}(G\times \mathfrak{m}^*\times N)$ and

Theorems & Definitions (33)

  • Theorem 2.1: Hamiltonian Tube Theorem
  • Proposition 3.1
  • proof
  • Lemma 3.2
  • Remark 3.3
  • Theorem 3.4: Theorem 6.1 of MR2421706
  • Proposition 4.1
  • Definition 4.2
  • Proposition 4.3: Existence of simple $G$-tubes
  • proof
  • ...and 23 more