Hamiltonian Coupling of Electromagnetic Field and Matter

Reversible part of evolution equations of physical systems is often generated by a Poisson bracket. We discuss geometric means of construction of Poisson brackets and their mutual coupling (direct, semidirect and matched-pair products) as well as projections of Poisson brackets to less detailed Poisson brackets. This way the Hamiltonian coupling between transport of mixtures and electrodynamics is elucidated.