  We discuss several geometric PDEs and their relationship with Hydrodynamics
and classical Electrodynamics. We start from the Euler equations of ideal
incompressible fluids that, geometrically speaking, describe geodesics on
groups of measure preserving maps with respect to the $L^2$ metric. Then, we
introduce a geometric approximation of the Euler equation, which involves the
Monge-Amp\`ere equation and the Monge-Kantorovich optimal transportation
theory. This equation can be interpreted as a fully nonlinear correction of the
Vlasov-Poisson system that describes the motion of electrons in a uniform
neutralizing background through Coulomb interactions. Finally we briefly
discuss an equation for generalized extremal surfaces in the 5 dimensional
Minkowski space, related to the Born-Infeld equations, from which the
Vlasov-Maxwell system of classical Electrodynamics can be formally derived.
