Letters in Mathematical Physics

, Volume 97, Issue 1, pp 45–60 | Cite as

Generalized Euler–Poincaré Equations on Lie Groups and Homogeneous Spaces, Orbit Invariants and Applications

Article

Abstract

We develop the necessary tools, including a notion of logarithmic derivative for curves in homogeneous spaces, for deriving a general class of equations including Euler–Poincaré equations on Lie groups and homogeneous spaces. Orbit invariants play an important role in this context and we use these invariants to prove global existence and uniqueness results for a class of PDE. This class includes Euler–Poincaré equations that have not yet been considered in the literature as well as integrable equations like Camassa–Holm, Degasperis–Procesi, μCH and μDP equations, and the geodesic equations with respect to right-invariant Sobolev metrics on the group of diffeomorphisms of the circle.

Mathematics Subject Classification (2000)

35Q35 58D05 53C30 

Keywords

Euler–Poincaré equation homogeneous space Cauchy problem well posedness 

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Copyright information

© Springer 2011

Authors and Affiliations

  1. 1.Section de MathématiquesEPFLLausanneSwitzerland
  2. 2.Fields InstituteTorontoCanada
  3. 3.Department of MathematicsWest University of TimişoaraTimişoaraRomania

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