Generalized Euler-Poincaré equations on Lie groups and homogeneous spaces, orbit invariants and applications

Details Generalized Euler-Poincaré equations on Lie groups and homogeneous spaces, orbit invariants and applications Generalized Euler-Poincaré equations on Lie groups and homogeneous spaces, orbit invariants and applications

2010-08-25

arxiv.org/abs/1008.4377v1

Mathematics

Analysis of PDEs

Scientific paper

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-Poincar\'e 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-Poincar\'e equations that have not yet been considered in the literature as well as integrable equations like Camassa-Holm, Degasperis-Procesi, $\mu$CH and $\mu$DP equations, and the geodesic equations with respect to right invariant Sobolev metrics on the group of diffeomorphisms of the circle.

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