Radiation reaction

Radiation reaction is the influence on a source's motion of its own electromagnetic field, including the balance of radiated energy and momentum with changes of the near field. For sufficiently weak radiation loss, an averaged power balance can give the leading damping law without solving the full self-field dynamics. Replacing a constant-frequency radiation formula by its instantaneous slowly varying value requires an additional time-scale assumption; it is not an exact treatment of arbitrarily rapid motion changes.

For $|\epsilon \omega |\ll 1$, use the zeroth-order equation $\ddot{x}=f-x$ inside the small third-derivative term, givingFor positive $\epsilon$ this reduced equation has a stable retarded response with no high-frequency runaway. Its response $G_{\mathrm{red}}=(1-i\epsilon \omega )/(1-{\omega }^2-i\epsilon \omega )$ agrees with the unreduced oscillator to first order at fixed nonresonant frequency. Order reduction deliberately removes the singular fast mode outside the assumed low-frequency regime; it is an approximation rather than an exact alternative boundary condition for the third-order equation.

For positive $\epsilon$, the real-axis frequency response of the radiation-reaction oscillator pole structure excludes future runaway growth but retains an upper-half-plane pole. Closing its inverse Fourier transform upward at $t<0$ gives a nonzero response before the force. The fast pole contributes approximately $\epsilon e^{t/\epsilon }$ for negative $t$. A strictly causal retarded prescription instead includes an exponentially growing term for positive $t$ and requires a Laplace transform contour. The pre-response is a consequence of imposing boundedness at future infinity, not an unavoidable property of all initial-value solutions of the differential equation.

For $\ddot{x}+x-\epsilon \stackrel{...}{x}=f$ and the $e^{-i\omega t}$ convention, the frequency response is $G(\omega )=[1-{\omega }^2-i\epsilon {\omega }^3{]}^{-1}$. At small nonzero real $\epsilon$, its poles areFor positive $\epsilon$ the slow oscillations decay but the fast mode runs away; for negative $\epsilon$ the fast mode decays but the slow oscillations grow. Thus no nonzero sign makes all modes stable. A causal initial-value response and a bounded response free of runaways impose different boundary conditions.