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This paper presents two different electrostatic gyrokinetic models derivedthrough two different perturbative methods. One of the two models is just theconventional electrostatic gyrokinetic model, the derivation of which isrepeated using the Lie transform perturbative method. One term is rectified inthe derivation of the conventional model. To derive the other model, we use anew method, which is based on the covariant transform formula of thedifferential 1-form. The new method doesn't split the coordinate transform intothe guiding-center transform and the gyrocenter transform. It carries out thecoordinate transform up to the order equaling that of the amplitude of theperturbative wave. Compared with the conventional model, the finite Larmorradius terms are completely removed from the orbit equations of the new one,making it simpler for the numerical application.

Two electrostatic gyrokinetic models derived by two different perturbative methods