Resonant terms on gyrocenter coordinates introduced by resonant electromagnetic perturbations

It's pointed out that the treatment of the resonant electromagnetic perturbation with the Lie transform method adopted in the gyrokinetic theory generates some nonphysical terms in the trajectory equations. By utilizing a modified application of this transform method, the resonant terms in the trajectory equations satisfying each resonant condition are found out up to the second harmonic resonance. Through separating the fast-dynamic terms from the slow-dynamic ones, the slow-dynamic trajectory equations including the resonant terms are derived for various resonant conditions. The slow-dynamic evolution equation of gyroangle reveals that the real gyrating frequency of the charged particle around the magnetic field line under driving by the resonant wave has a shift from the cyclotron frequency without the wave driving. To study the effect caused by the frequency shift, a simple example for the fundamental-frequency cyclotron resonance is presented.