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Abstract mechanical connection and Abelian reconstruction for almost Kähler manifolds

Pekarsky, Sergey and Marsden, Jerrold E. (2001) Abstract mechanical connection and Abelian reconstruction for almost Kähler manifolds. Journal of Applied Mathematics, 1 (1). pp. 1-28. ISSN 1110-757X.

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When the phase space P of a Hamiltonian G-system (P,ω,G,J,H) has an almost Kähler structure a preferred connection, called abstract mechanical connection, can be defined by declaring horizontal spaces at each point to be metric orthogonal to the tangent to the group orbit. Explicit formulas for the corresponding connection one-form. A are derived in terms of the momentum map, symplectic and complex structures. Such connection can play the role of the reconstruction connection (see A. Blaom, Reconstruction phases via Poisson reduction. Diff. Geom. Appl., 12:231{252, 2000.), thus signicantly simplifying computations of the corresponding dynamic and geometric phases for an Abelian group G. These ideas are illustrated using the example of the resonant three-wave interaction. Explicit formulas for the connection one-form and the phases are given together with some new results on the symmetry reduction of the Poisson structure.

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Additional Information:© 2001 Hindawi Publishing Corporation. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Received 12 September 2000 and in revised form 20 February 2001. Preprint: Summer 1998. Last revised: February 24, 2001. We would like to extend our gratitude to Anthony D. Blaom for his time and invaluable help in the preparation of this manuscript; his input was crucial in developing some key ideas discussed here. We also would like to thank Tudor Ratiu and Sameer Jalnapurkar for helpful comments.
Issue or Number:1
Classification Code:2000 Mathematics Subject Classification: 37J15, 53D20, 32Q15
Record Number:CaltechAUTHORS:20100913-103504526
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Official Citation:Sergey Pekarsky and Jerrold E. Marsden, “Abstract mechanical connection and abelian reconstruction for almost Kähler manifolds,” Journal of Applied Mathematics, vol. 1, no. 1, pp. 1-28, 2001. doi:10.1155/S1110757X01000043
Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:19903
Deposited By: Ruth Sustaita
Deposited On:15 Sep 2010 22:40
Last Modified:03 Oct 2019 02:03

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