Geometric hydrodynamics and infinite-dimensional Newton’s equations

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