  Gyrokinetic theory is a basis for treating magnetised plasma dynamics slower
than particle gyrofrequencies where the scale of the background is larger than
relevant gyroradii. The energy of field perturbations can be comparable to the
thermal energy but smaller than the energy of the background magnetic field.
Properly applied, it is a low-frequency gauge transform rather than a treatment
of particle orbits, and more a representation in terms of gyrocenters rather
than particles than an approximation. By making all transformations and
approximations in the field/particle Lagrangian one preserves exact energetic
consistency so that time symmetry ensures energy conservation and spatial
axisymmetry ensures toroidal angular momentum conservation. This method draws
on earlier experience with drift-kinetic models while showing the independence
of gyrokinetic representation from particularities of Lie transforms or
specific ordering limits, and that the essentials of low-frequency
magnetohydrodynamics, including the equilibrium, are recovered. It gives a
useful basis for total-f electromagnetic gyrokinetic or gyrofluid computation.
Various versions of the representation based upon choice of parallel velocity
space coordinate are illustrated.
