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 L2 or H1 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.Comment: 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