Geodesic Equations on Diffeomorphism Groups

Abstract

We bring together those systems of hydrodynamical type that can be written as geodesic equations on diffeomorphism groups or on extensions of diffeomorphism groups with right invariant $L^2$ or $H^1$ metrics. We present their formal derivation starting from Euler's equation, the first order equation satisfied by the right logarithmic derivative of a geodesic in Lie groups with right invariant metrics.

This is a contribution to the Proc. of the Seventh International Conference ''Symmetry in Nonlinear Mathematical Physics'' (June 24-30, 2007, Kyiv, Ukraine), published in SIGMA (Symmetry, Integrability and Geometry: Methods and Applications) at http://www.emis.de/journals/SIGMA/

References in corpus (9)

• An integrable shallow water equation with peaked solitons
• The Camassa-Holm equation as a geodesic flow on the diffeomorphism group
• Abelian extensions of infinite-dimensional Lie groups
• Geometry and curvature of diffeomorphism groups with $H^1$ metric and mean hydrodynamics
• Extensions of (super) Lie algebras
• Momentum Maps and Measure-valued Solutions (Peakons, Filaments and Sheets) for the EPDiff Equation
• Magnetic hydrodynamics with asymmetric stress tensor
• Geodesics and curvature of semidirect product groups
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• Curvature and stability of quasi-geostrophic motion
• Differential-geometrical approach to the dynamics of dissipationless incompressible Hall magnetohydrodynamics: II. Geodesic formulation and Riemannian curvature analysis of hydrodynamic and magnetohydrodynamic stabilities
• Zeitlin's model for axisymmetric 3-D Euler equations
• Surface tension of multi-phase flow with multiple junctions governed by the variational principle
• Lagrangian Reduction on Homogeneous Spaces with Advected Parameters
• A two-component geodesic equation on a space of constant positive curvature
• Stochastic Euler-Poincaré reduction for central extension