Holm, Darryl D. and Marsden, Jerrold E. and Ratiu, Tudor S. (1986) The Hamiltonian structure of continuum mechanics in material, inverse material, spatial and convective representations. In: Hamiltonian structure and Lyapunov stability for ideal continuum dynamics. Séminaire de mathématiques supérieures. No.100. Presses de l'Universite de Montreal , pp. 11-124. ISBN 9782760607712 http://resolver.caltech.edu/CaltechAUTHORS:20100921-072443815
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Ideal continuum models (fluids, plasmas, elasticity, etc.) can be studied using a variety of representations, each of which has a Hamiltonian structure. This paper shows how groups (typified by the group of particle relabelling symmetries) and the inversion operator which swaps the reference and current particle positions generate maps between the representations. These maps, derived using the theory of momentum maps and reduction, are all Poisson (or canonical) maps which carry the brackets in one representation to those in another. the results are developed abstractly in the framework of reduction of a pair of principal bundles by left and right group actions. Examples are given treating the motion of an incompressible fluid with surface tension, the heavy top, and ideal compressible (barotropic) flow.
|Item Type:||Book Section|
|Additional Information:||© 1986, Supported by DOE Contract W-740S-ENG-36. Supported by DOE Contract DE-AT03-85ER-12097. Supported by Sloan foundation fellowship.|
|Usage Policy:||No commercial reproduction, distribution, display or performance rights in this work are provided.|
|Deposited By:||Ruth Sustaita|
|Deposited On:||22 Sep 2010 18:26|
|Last Modified:||01 May 2015 17:47|
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