We will compare the gyro-kinetic reductions obtained by different approaches in the framework of the magnetic confinement: Hamiltonian techniques and homogenization methods. Basically we will concentrate on the asymptotic behavior of the Vlasov-Poisson equations or the Vlasov-Maxwell equations, when the external magnetic field becomes strong. In particular, we will investigate all the contributions due to the curvature of the magnetic lines, in the three dimensional case. Other regimes will be discussed, including collisions, or several particle species, with disparate masses. We intend to obtain second order asymptotic approximations, and to validate them by numerical simulations.