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An exact solution is given to the classical electromagnetic (EM)radiation-reaction (RR) problem, originally posed by Lorentz. This refers tothe dynamics of classical non-rotating and quasi-rigid finite size particlessubject to an external prescribed EM field. A variational formulation of theproblem is presented. It is shown that a covariant representation for the EMpotential of the self-field generated by the extended charge can be uniquelydetermined, consistent with the principles of classical electrodynamics andrelativity. By construction, the retarded self 4-potential does not possess anydivergence, contrary to the case of point charges. As a fundamentalconsequence, based on Hamilton variational principle, an exact representationis obtained for the relativistic equation describing the dynamics of afinite-size charged particle (RR equation), which is shown to be realized by asecond-order delay-type ODE. Such equation is proved to apply also to thetreatment of Lorentzian particles, i.e., point-masses with finite-size chargedistributions, and to recover the usual LAD equation in a suitable asymptoticapproximation. Remarkably, the RR equation admits both standard Lagrangian andconservative forms, expressed respectively in terms of a non-local effectiveLagrangian and a stress-energy tensor. Finally, consistent with the Newtonprinciple of determinacy, it is proved that the corresponding initial-valueproblem admits a local existence and uniqueness theorem, namely it defines aclassical dynamical system.

Exact solution of the EM radiation-reaction problem for classical finite-size and Lorentzian charged particles