  We propose a novel derivation of the gyrokinetic field-particle Lagrangian
for non-collisional ion-electron plasmas in a magnetic background with strong
variations (maximal ordering). Our approach follows the two-step reduction
process, where the guiding-center coordinate transformation is followed by the
gyrocenter coordinate transformation in the single-particle phase space. For
the first time both steps are addressed within a unique methodology, based on
near-identity coordinate transformations constructed as polynomial transforms.
These are well-defined transformations composed of a finite number of terms
that are linear and algebraic with respect to the generating functions. The
derivation is carried out in a fully non-dimensional framework, based on
parameters governing the magnetic fusion experiments ASDEX Upgrade and ITER.
Our method leads to a gyrokinetic Vlasov-Maxwell model for ions and electrons,
derived without the use of Lie perturbation methods. It is found that, based on
the employed ordering, curvature terms such as the gyro-gauge term and the
Ba\~nos drift appear at first order in the ion Hamiltonian, whereas ion
polarization terms appear only at second order. By contrast, curvature terms
are absent from the first-order electron Hamiltonian, where instead magnetic
flutter plays a role.
