Gyrokinetic resonant theory of low frequency electromagnetic perturbation

It's pointed out that the traditional gyrokinetic theory dealing with low frequency electromagnetic perturbation violates the near identity transformation, which is supposed to be obeyed by Lie perturbed transformation theory, if resonance happens between \omega and k\cdot v. A modification is given to overcome this problem by not requiring all components in the first order Lagrangian 1-form equaling zero. A numerical example is given as an application of the new theory.