The Euler-Arnold equation is essentially a coordinate invariant geodesic equation that can be written down on any Lie group, even those of infinite dimension, with a specific kind of inner product on its Lie algebra. I'll review and expand upon Boramey's introduction to this topic last semester. Time permitting I'll talk about curvature calculations on the group of volume preserving diffeomorphisms of a manifold with applications to weather prediction, as well as magnetic extensions of Lie groups, which allow one to phrase the system of ideal magnetohydrodynamics equations as geodesic equations on an infinite dimensional Lie group.
An Introduction to the Euler-Arnold Equation