It has been shown by Gupta and Padmanabhan that the radiation reaction forceof the Abraham-Lorentz-Dirac equation can be obtained by a coordinatetransformation from the inertial frame of an accelerating charged particle tothat of the laboratory. We show that the problem may be formulated in a flatspace of five dimensions, with five corresponding gauge fields in the frameworkof the classical version of a fully gauge covariant form of the Stueckelberg-Feynman-Schwinger covariant mechanics (the zero mode fields of the 0,1,2,3components correspond to the Maxwell fields). Without additional constraints,the particles and fields are not confined to their mass shells. We show that inthe mass-shell limit, the generalized Lorentz force obtained by means of theretarded Green's functions for the five-dimensional field equations providesthe classical Abraham-Lorentz-Dirac radiation reaction terms (with renormalizedmass and charge). We also obtain general coupled equations for the orbit andthe off-shell dynamical mass during the evolution. The theory does not admitradiation if the particle remains identically on-shell. The structure of theequations implies that the mass-shell deviaiton is bounded when the externalfield is removed.