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

  • Darryl D. Holm
  • Jerrold E. Marsden
Part of the Progress in Mathematics book series (PM, volume 232)

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 (ofnor 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 H1 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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Copyright information

© Birkhäuser Boston 2005

Authors and Affiliations

  • Darryl D. Holm
    • 1
    • 2
  • Jerrold E. Marsden
    • 3
  1. 1.Computer and Computational Science DivisionLos Alamos National Laboratory MS D413Los AlamosUSA
  2. 2.Mathematics DepartmentImperial CollegeLondonUK
  3. 3.Control and Dynamical Systems 107-81California Institute of TechnologyPasadenaUSA

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