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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.

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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.

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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.
Funding AgencyGrant Number
Department of EnergyW-7405-ENG-36
Los Alamos National LaboratoryUNSPECIFIED
California Institute of TechnologyUNSPECIFIED
Air ForceF49620-02-1-0176
Series Name:Progress in Mathematics
Issue or Number:232
Record Number:CaltechAUTHORS:20100810-153020247
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Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:19380
Deposited By: Ruth Sustaita
Deposited On:13 Aug 2010 17:50
Last Modified:08 Nov 2021 23:51

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