  A gyrokinetic reduction is based on a specific ordering of the different
small parameters characterizing the background magnetic field and the
fluctuating electromagnetic fields. In this tutorial, we consider the following
ordering of the small parameters: $\epsilon\_B=\epsilon\_\delta^2$ where
$\epsilon\_B$ is the small parameter associated with spatial inhomogeneities of
the background magnetic field and $\epsilon\_\delta$ characterizes the small
amplitude of the fluctuating fields. In particular, we do not make any
assumption on the amplitude of the background magnetic field. Given this choice
of ordering, we describe a self-contained and systematic derivation which is
particularly well suited for the gyrokinetic reduction, following a two-step
procedure. We follow the approach developed in [Sugama, Physics of Plasmas 7,
466 (2000)]:In a first step, using a translation in velocity, we embed the
transformation performed on the symplectic part of the gyrocentre reduction in
the guiding-centre one. In a second step, using a canonical Lie transform, we
eliminate the gyroangle dependence from the Hamiltonian. As a consequence, we
explicitly derive the fully electromagnetic gyrokinetic equations at the second
order in $\epsilon\_\delta$.
