  We study Brenier's variational models for incompressible Euler equations.
These models give rise to a relaxation of the Arnold distance in the space of
measure-preserving maps and, more generally, measure-preserving plans. We
analyze the properties of the relaxed distance, we show a close link between
the Lagrangian and the Eulerian model, and we derive necessary and sufficient
optimality conditions for minimizers. These conditions take into account a
modified Lagrangian induced by the pressure field. Moreover, adapting some
ideas of Shnirelman, we show that, even for non-deterministic final conditions,
generalized flows can be approximated in energy by flows associated to
measure-preserving maps.
