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 non-relativistic regime via each of these 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.
Second-order gravitational self-force in a highly regular gauge
