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

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, $\mu$CH and $\mu$DP equations, and the geodesic equations with respect to right invariant Sobolev metrics on the group of diffeomorphisms of the circle.

References in corpus (5)

• Integrable evolution equations on spaces of tensor densities and their peakon solutions
• The Degasperis-Procesi equation as a non-metric Euler equation
• Geodesic flows on semidirect-product Lie groups: geometry of singular measure-valued solutions
• Geodesic Equations on Diffeomorphism Groups
• Momentum Maps and Measure-valued Solutions (Peakons, Filaments and Sheets) for the EPDiff Equation

Cited by in corpus (9)

• Zermelo Navigation and a Speed Limit to Quantum Information Processing
• Right-invariant Sobolev metrics of fractional order on the diffeomorphism group of the circle
• The geometry of a vorticity model equation
• Geometric investigations of a vorticity model equation
• Geometric hydrodynamics and infinite-dimensional Newton's equations
• A view of the peakon world through the lens of approximation theory
• Can we run to infinity? The diameter of the diffeomorphism group with respect to right-invariant Sobolev metrics
• Un-reduction in field theory, with applications
• Lagrangian Reduction on Homogeneous Spaces with Advected Parameters