The radiation reaction problem in a simple coupled model

The complete description of the interaction between an external electromagnetic field and a charged particle causing it to radiate is one of the most fundamental problems in classical electrodynamics. Here we provide a simple coupled model that describes via the Lagrangian of the physical system the full radiation reaction process resulting from the particle-field evolution equations obtained from the Lagrangian are studied as an initial value problem giving rise to the particle's radiation reaction equation which includes the complete interaction with the field. Using stability theory we analyze this equation and prove that every solution is global uniform asymptotically stable, evidencing that the particle coupled to the field radiates. The model proves to be interesting for it clearly illustrates the particle-field interaction process and shows the radiation's origin.