  The gyrokinetic paradigm in the long wavelength regime is reviewed from the
perspective of variational averaging (VA). The VA-method represents a third
pillar for averaging kinetic equations with highly-oscillatory characteristics,
besides classical averaging or Chapman-Enskog expansions. VA operates on the
level of the Lagrangian function and preserves the Hamiltonian structure of the
characteristics at all orders. We discuss the methodology of VA in detail by
means of charged-particle motion in a strong magnetic field. The application of
VA to a broader class of highly-oscillatory problems can be envisioned. For the
charged particle, we prove the existence of a coordinate map in phase space
that leads to a gyrokinetic Lagrangian at any order of the expansion, for
general external fields. We compute this map up to third order, independent of
the electromagnetic gauge. Moreover, an error bound for the solution of the
derived gyrokinetic equation with respect to the solution of the Vlasov
equation is provided, allowing to estimate the quality of the VA-approximation
in this particular case.
