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We study the asymptotic behavior of solutions to the Vlasov equation in thepresence of a strong external magnetic field. In particular we provide amathematically rigorous derivation of the guiding-center approximation in thegeneral three dimensional setting under the action of large inhomogeneousmagnetic fields. First order corrections are computed and justified as well,including electric cross field, magnetic gradient and magnetic curvaturedrifts. We also treat long time behaviors on two specific examples, the twodimensional case in carte-sian coordinates and a poloidal axi-symmetricgeometry, the former for expository purposes. Algebraic manipulations thatunderlie concrete computations make the most of the linearity of the stiffestpart of the system of characteristics instead of relying on any particularvariational structure.

Asymptotics of the three dimensional Vlasov equation in the large magnetic field limit