In this paper we illustrate some recent work [1], [2] on
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.