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

摘要

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.