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Hamiltonian G-Spaces with Regular Momenta

Perry, Anthony D. (1995) Hamiltonian G-Spaces with Regular Momenta. California Institute of Technology , Pasadena, CA. (Unpublished)

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Let G be a compact connected non-Abelian Lie group and let (P, w, G, J) be a Hamiltonian G-space. Call this space a G-space with regular momenta if J(P) ⊂ g*reg, here g*reg⊂g* denotes the regular points of the co-adjoint action of G. Here problems involving a G-space with regular momenta are reduced to problems in an associated lower dimensional Hamiltonian T-space, where T ⊂ G is a maximal torus. For example two such G-spaces are shown to be equivalent if and only if they have equivalent associated T-spaces. We also give a new construction of a normal form due to Marle (1983), for integrable G-spaces with regular momenta. We show that this construction, which is a kind of non-Abelian generalization of action-angle coordinates, can be reduced to constructing conventional action-angle coordinates in the associated T-space. In particular the normal form applies globally if the action-angle coordinates can be constructed globally. We illustrate our results in concrete examples from mechanics, including the rigid body. We also indicate applications to Hamiltonian perturbation theory.

Item Type:Report or Paper (Technical Report)
Additional Information:The author is grateful for frequent discussions and general guidance provided by Jerrold Marsden, as well as Tudor Ratiu. These experts share their ideas generously, and progress in this research has been frequently catalyzed by their comments. The author is also grateful to Alan Weinstein for pointing out the papers by Marle (1983) and Dazord and Delzant (1987).
Group:Control and Dynamical Systems Technical Reports
Record Number:CaltechCDSTR:1995.033
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Usage Policy:You are granted permission for individual, educational, research and non-commercial reproduction, distribution, display and performance of this work in any format.
ID Code:28109
Deposited By: Imported from CaltechCDSTR
Deposited On:18 Oct 2006
Last Modified:03 Oct 2019 03:29

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