Lagrangian and Hamiltonian aspects of Josephson type media

Dynamical properties of Josephson media are studied within Lagrangian and H amiltonian formalisms with gauge type constraints. A geometrical interpreta tion of the first class gauge type constraints involved is suggested basing on the Marsden-Weinstein momentum map reduction. An equivalent operatorial approach is also considered in detail giving rise to certain two essential ly different Poisson structures upon orbits of the Abelian gauge group, the first Poisson structure being nondegenerate in contrast to the second one- degenerated. The consideration of the Poisson structure associated with Jos ephson media are essentially augmented by means of introducing new Josephso n-Vlasov kinetic type equations endowed with the canonical Lie-Poisson brac ket. The reduction of the corresponding Hamiltonian flow explains the physi cal nature of the a priori involved constraints, (C) Elsevier, Paris.