  We study the finite Larmor radius regime for the Vlasov-Poisson system. The
magnetic field is assumed to be uniform. We investigate this non linear problem
in the two dimensional setting. We derive the limit model by appealing to
gyro-average methods cf. \cite{BosAsyAna}, \cite{BosTraEquSin}. We indicate the
explicit expression of the effective advection field, entering the Vlasov
equation, after substituting the self-consistent electric field, obtained by
the resolution of the averaged (with respect to the cyclotronic time scale)
Poisson equation. We emphasize the Hamiltonian structure of the limit model and
present its properties~: conservationss of the mass, kinetic energy, electric
energy, etc.
