  We show that a smooth, small enough Cauchy datum launches a unique classical
solution of the relativistic Vlasov-Darwin (RVD) system globally in time. A
similar result is claimed in Comm. Math. Sci. 6, 749-764 (2008) following the
work in Int. Mat. Res. Not. 57191, 1-31 (2006). Our proof does not require
estimates derived from the conservation of the total energy, nor those
previously given on the transverse component of the electric field. These
estimates are crucial in the references cited above. Instead, we exploit the
formulation of the RVD system in terms of the generalized space and momentum
variables. By doing so, we produce a simple a-priori estimate on the transverse
component of the electric field. We widen the functional space required for the
Cauchy datum to extend the solution globally in time, and we improve decay
estimates given in Comm. Math. Sci. 6, 749-764 (2008) on the electromagnetic
field and its space derivatives. Our method extends the constructive proof
presented in "Collisionless kinetic equations from astrophysics: The
Vlasov-Poisson system. Handbook of Differential Equations: Evolutionary
Equations, Vol.3, Elsevier, 2007" to solve the Cauchy problem for the
Vlasov-Poisson system with a small initial datum.
