  A crucial issue in relativistic plasma, particularly relevant in the
astrophysical context, is the description of highly magnetized plasmas based on
a covariant formulation of gyrokinetic dynamics. An interesting case in
question is that in which the background electric field (produced either by the
same plasma of by other sources) results suitably small (or vanishing) with
respect to the magnetic field, while at the same time short-wavelength EM
perturbations can be present. The purpose of this work is to extend the
relativistic gyrokinetic theory developed by Beklemishev \textit{et al.}
[1999-2005] to include, in particular, also the treatment of such a case. We
intend to show that this requires the development of a perturbative expansion
involving simultaneously both the particle 4-position vector and the
corresponding \textit{4-velocity vector}. For this purpose a synchronous form
of the relativistic Hamilton variational principle is adopted.
