  We interpret the Lorentz force equation as a geodesic equation associated
with a non-linear connection. Using a geometric averaging procedure, we prove
that for narrow and smooth one-particle distribution functions whose supports
are invariant under the flow of the Lorentz equation, a bunch of charged point
particles can be described by a charged cold fluid model in the
ultra-relativistic regime. The method used to prove this result does not
require additional hypotheses on the higher moments of the distribution. This
is accomplished by estimating the expressions that include the differential
operators appearing in the charged cold fluid model equation. Under the
specified conditions of narrowness and ultra-relativistic dynamics, it turns
out that these differential expressions are close to zero, justifying the use
of the charged cold fluid model. The method presented in the work can also be
applied to justify the use of warm plasmas and other models.
