  The exact analytical description of non relativistic charge motion in general
magnetic fields is, apparently, a simple problem but, it has not been solved up
to now apart for rare cases. The key feature of the present derivation is to
adopt a non perturbative magnetic field description to find new solutions of
motion. Among all solutions, two are particularly important: guiding particle
and gyro-particle solutions. The guiding particle has been characterized to be
minimally coupled to the magnetic field, the gyro-particle has been defined to
be maximally coupled to the magnetic field and, also, to move on a closed
orbit. The generic charged particle motion is shown to be expressed as the sum
of such particular solutions. This non perturbative approach corresponds to the
description of the particle motion in the gyro-center and/or guiding center
reference frame obtained at all the orders of the modern gyro-center
transformation. The Boltzmann equation is analyzed with the described exact
guiding center coordinates. The obtained gyrokinetic equation is solved for the
Boltzmann equation at marginal stability conditions.
