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Momentum maps and measure-valued solutions (peakons, filaments, and sheets) for the EPDiff equation

Holm, Darryl D. and Marsden, Jerrold E. (2004) Momentum maps and measure-valued solutions (peakons, filaments, and sheets) for the EPDiff equation. In: The Breadth of Symplectic and Poisson Geometry. Progress in Mathematics. No.232. Birkhäuser Boston , pp. 203-235. ISBN 0817635653 http://resolver.caltech.edu/CaltechAUTHORS:20100810-153020247

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Abstract

This paper is concerned with the dynamics of measure-valued solutions of the EPDiff equations, standing for the Euler-Poincaré equations associated with the diffeomorphism group (of ℝn or of an n-dimensional manifold M). It focuses on Lagrangians that are quadratic in the velocity fields and their first derivatives, that is, on geodesic motion on the diffeomorphism group with respect to a right invariant Sobolev H 1 metric. The corresponding Euler-Poincaré (EP) equations are the EPDiff equations, which coincide with the averaged template matching equations (ATME) from computer vision and agree with the Camassa-Holm (CH) equations for shallow water waves in one dimension. The corresponding equations for the volume-preserving diffeomorphism group are the LAE (Lagrangian averaged Euler) equations for incompressible fluids.


Item Type:Book Section
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http://dx.doi.org/10.1007/0-8176-4419-9_8DOIUNSPECIFIED
http://www.springerlink.com/content/r793488781644r9n/PublisherUNSPECIFIED
Additional Information:© 2004 Springer. The first author is grateful for support by US DOE under contract W-7405-ENG-36 for Los Alamos National Laboratory, and Office of Science ASCAR/AMS/MICS. The research of the second author was partially supported by the California Institute of Technology, by the National Science Foundation through NSF grant DMS-0204474, and by Air Force contract F49620-02-1-0176. We are very grateful toAlanWeinstein for his collaboration, help and inspiring discussions over the years.We thank Martin Staley for letting us illustrate some important points using his computations. We also thank Anthony Bloch, Simon Donaldson, Jonathan Munn, Tudor Ratiu, and Richard Thomas for valuable advice and comments. To Alan Weinstein on the occasion of his 60th birthday.
Funders:
Funding AgencyGrant Number
Department of EnergyW-7405-ENG-36
Los Alamos National LaboratoryUNSPECIFIED
Office of Science ASCAR/AMS/MICSUNSPECIFIED
California Institute of TechnologyUNSPECIFIED
NSFDMS-0204474
Air ForceF49620-02-1-0176
Record Number:CaltechAUTHORS:20100810-153020247
Persistent URL:http://resolver.caltech.edu/CaltechAUTHORS:20100810-153020247
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ID Code:19380
Collection:CaltechAUTHORS
Deposited By: Ruth Sustaita
Deposited On:13 Aug 2010 17:50
Last Modified:26 Dec 2012 12:18

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