  We focus our attention on some relevant aspects of the beam-plasma
instability in order to refine some features of the linear and non-linear
dynamics. After a re-analysis of the Poisson equation and of the assumption
dealing with the background plasma in the form of a linear dielectric, we study
the non-perturbative properties of the linear dispersion relation, showing the
necessity for a better characterization of the mode growth rate in those flat
regions of the distribution function where the Landau formula is no longer
predictive. We then upgrade the original N-body approach, in order to include a
return current in the background plasma. This correction term is responsible
for smaller saturation levels and growth rates of the Langmuir modes, as result
of the energy density transferred to the plasma via the return current.
Finally, we include friction effects, as those due to the collective influence
of all the plasma charges on the motion of the beam particles. The resulting
force induces a progressive resonance detuning, because particles are losing
energy and decreasing their velocity. This friction phenomenon gives rise to a
deformation of the distribution function, associated with a significant growth
of the less energetic particle population. The merit of this work is to show
how a fine analysis of the beam-plasma instability outlines a number of
subtleties about the linear, intermediate and late dynamics which can be of
relevance when such a system is addressed as a paradigm to describe relevant
nonlinear wave-particle phenomena.
