Modification of Lie's transform perturbation theory for charged particle motion in a magnetic field

It is pointed out that the conventional Lie transform perturbation theory for guiding center motion of charged particles in a magnetic field needs to be modified ordering inconsistency. There are two reasons. First, difference between temporal variation gyrophase and other phase space coordinates taken into account. Second, it also important note parametric limit derivative function not equivalent function. When these facts account, near identity transformation rule one form related Lagrangian modified. With rule, drift can described first order, instead second order beyond through tedious expansion process as formulation. This resolves discrepancy direct treatments particle field.