Symplectic bipotentials

In a previous paper, we proposed a symplectic version of the Brezis–Ekeland–Nayroles principle based on the concepts of Hamiltonian inclusions and symplectic polar functions. We applied it to the standard plasticity. The object of this work is to extend the previous formalism to non-associated plasticity. For this aim, we generalize the concept of bipotential to dynamical systems. The keystone idea is to define a symplectic bipotential. We present a method to build it from a bipotential. Next, we generalize the symplectic Brezis–Ekeland–Nayroles principle to the non-associated dissipative laws. We apply it to the non-associated plasticity and to the unilateral contact law with Coulomb’s dry friction.