Geodesic Equations on Diffeomorphism Groups

TL;DR

This paper explores the derivation of geodesic equations on diffeomorphism groups related to hydrodynamical systems, emphasizing their geometric structure and connection to Euler's equations.

Contribution

It provides a formal derivation of hydrodynamical systems as geodesic equations on diffeomorphism groups with specific invariant metrics.

Findings

Unified geometric framework for hydrodynamical equations

Derivation of geodesic equations from Euler's equation

Insights into the structure of 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.