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Accelerating charges lose energy through radiation; this energy loss manifests as a damping force in the equation of motion. For oscillating motion, the radiation damping force acts like a friction proportional to the time derivative of acceleration, with coefficient depending on the classical electron radius.
From the Larmor formula, you know that an accelerating charge radiates power P = q²a²/(6πε₀c³). This energy doesn't appear from nothing — it must come from the charge's kinetic energy. As the charge radiates, it must slow down. But here is the fundamental puzzle: how exactly does the field the charge creates act back on the charge itself? This is the radiation reaction problem — one of the oldest conceptually troubling issues in classical electrodynamics.
The answer comes from energy conservation over a complete oscillation cycle. The total energy radiated per cycle must equal the work done by some damping force F_rad acting against the motion. Working backward from the Larmor formula through integration by parts, this force turns out to be F_rad = (q²/6πε₀c³)(da/dt), where da/dt is the time derivative of acceleration — sometimes called jerk. This is the Abraham-Lorentz force. Its coefficient q²/(6πε₀c³) can be written as (2/3)(r_e/c)m_e, where r_e = q²/(4πε₀m_ec²) is the classical electron radius from your prerequisite — the length scale at which classical self-energy of the electron equals its rest mass energy.
The physical picture is that the electromagnetic field created by the charge propagates outward at finite speed c, so the charge experiences a small retarded force from its own near field. For simple oscillatory motion at frequency ω, the Abraham-Lorentz force reduces to an effective friction: F_damp ∝ −ωv⃗, which damps the oscillation at a rate Γ ∝ ω². This radiation damping gives spectral lines a natural linewidth — atoms in excited states don't radiate forever; they decay with a characteristic time τ ≈ r_e/c ~ 10⁻²³ s times (λ/a)² ~ 10⁻⁸ s, which matches observed atomic lifetimes.
The Abraham-Lorentz equation has deeply problematic features: it is third-order in time (requiring initial acceleration as well as position and velocity), and it admits runaway solutions where the charge accelerates without any applied force, gaining energy from its own radiation field. These pathologies signal that classical electrodynamics breaks down near the scale of the classical electron radius. A consistent treatment requires quantum electrodynamics, where the radiation reaction is reinterpreted in terms of photon emission and the electron's self-energy is handled through renormalization. For practical purposes, however — antenna design, spectral linewidths, plasma physics — the classical radiation damping framework provides a correct and essential account of energy balance in radiating systems.