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We explain a dissipative version of hamiltonian mechanics, based on theinformation content of the deviation from hamiltonian dynamics. From thisformulation we deduce minimal dissipation principles, dynamical inclusions, orconstrained evolution with hamiltonian drift reformulations. Among applicationswe recover a dynamics generalization of Mielke et al quasistaticrate-independent processes.  This article gives a clear and unitary presentation of the theory ofhamiltonian inclusions with convex dissipation or symplecticBrezis-Ekeland-Nayroles principle, presented under various conventions first inarXiv:0810.1419, then in arXiv:1408.3102 and, for the appearance ofbipotentials in relation to the symplectic duality, in arXiv:1902.04598v1.

Dissipation and the information content of the deviation from hamiltonian dynamics