We take the viewpoint that the physically acceptable solutions of the
Lorentz--Dirac equation for radiation back-reaction are actually determined by
a second order equation of motion, the self-force being given as a function of
spacetime location and velocity. We propose three different methods to obtain
this self-force function. For two example systems, we determine the second
order equation of motion exactly in the nonrelativistic regime via each of
these three methods, the three methods leading to the same result. We reveal
that, for both systems considered, back-reaction induces a damping proportional
to velocity and, in addition, it decreases the effect of the external force.Comment: 13 page
