  We revisit the geodesic approach to ideal hydrodynamics and present a related
geometric framework for Newton's equations on groups of diffeomorphisms and
spaces of probability densities. The latter setting is sufficiently general to
include equations of compressible and incompressible fluid dynamics,
magnetohydrodynamics, shallow water systems and equations of relativistic
fluids. We illustrate this with a survey of selected examples, as well as with
new results, using the tools of infinite-dimensional information geometry,
optimal transport, the Madelung transform, and the formalism of symplectic and
Poisson reduction.
