  We introduce and study the basic notion of polarized Poisson manifolds
generalizing the classical case of Poisson manifolds and extend this last
notion for the ${k-}$% symplectic stuctures. And also, we show that for any
polarized Hamiltonian map, the associated Nambu's dynamical system and
polarized Hamiltonian system are connected by relations characterizing the
mechanical aspect of the $k-$symplectic geometry.
