  We consider the Vlasov-Maxwell equations with one spatial direction and two
momenta, one in the longitudinal direction and one in the transverse direction.
By solving the Jacobi identity, we derive reduced Hamiltonian fluid models for
the density, the fluid momenta and the second order moments, related to the
pressure tensor. We also provide the Casimir invariants of the reduced Poisson
bracket. We show that the linearization of the equations of motion around
homogeneous equilibria reproduces some essential feature of the kinetic model,
the Weibel instability.
