  In a previous paper, we proposed a symplectic version of
Brezis-Ekeland-Nayroles principle based on the concepts of Hamiltonian
inclusions and symplectic polar functions. We illustrated it by application to
the standard plasticity in small deformations. The object of this work is to
generalize the previous formalism to dissipative media in large deformations
and Eulerian description. This aim is reached in three steps. Firstly, we
develop a Lagrangian formalism for the reversible media based on the calculus
of variation by jet theory. Next, we propose a corresponding Hamiltonian
formalism for such media. Finally, we deduce from it a symplectic minimum
principle for dissipative media and we show how to obtain a minimum principle
for unstationary compressible and incompressible Navier-Stokes equation.
