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In preceding work on plasma gyro-kinetics magnetic coordinates were introduced prior to making the transformation to the guiding centre variables. It is the transformation from the particle variables to the guiding centre variables which permits finite gyroradius effects to be retained in lowest order. The present treatment avoids the substantial mathematical complications inherent in these prior treatments by introducing the transformation to the guiding centre variables and performing the guiding centre gyrophase average before specifying the magnetic coordinates to be employed. In this way the unperturbed, gyro-averaged Vlasov operator which retains finite gyro-effects is obtained in the most convenient manner for arbitrary unperturbed magnetic fields.
Peter J. Catto (Sat,) studied this question.