Integrodifferential Equations of Motion

We seek solutions to the third order AL equation of motion that evolve into the ``natural'' ones when the driving force is turned off. In other words, radiation reaction must, by hypothesis, only damp the system and not drive it. Clearly even this requirement makes no sense when time reversal symmetry is considered. Once we fall into the trap of choosing retarded interaction only, we are sunk and anything we do to fix it will be a band-aid.

Let us introduce an ``integrating factor'' into the equations of motion. If we assume (quite generally) that

$$\dot{\mbox{\boldmath$v$}}(t) = e^{t/\tau_r} \mbox{\boldmath$u$}(t)$$ (19.21)

where $\mbox{\boldmath$u$}(t)$ is to be determined, then the equations of motion simplify to

$$m \dot{\mbox{\boldmath$u$}} = - \frac{1}{\tau_r} e^{- t/\tau} \mbox{\boldmath$F$}(t) .$$ (19.22)

We can formally integrate this second equation, obtaining

$$m \dot{\mbox{\boldmath$v$}}(t) = \frac{e^{t/\tau_r}}{\tau_r} \int_t^C e^{-t'/\tau_r} \mbox{\boldmath$F$}(t') dt'$$ (19.23)

The constant of integration is determined by our requirement that no runaway solutions exist! Note well that it is a constraint that lives in the future of the particle. In order to use this to find $\mbox{\boldmath$v$}(t)$, we must know the force $\mbox{\boldmath$F$}(t)$ for some time (of order $\tau _r$) in the future! After this, the integrand is ``cut off'' by the decaying exponential.

This suggests that we can extend the integral to $C = \infty$ without difficulty. In the limit $\tau_r \to 0$, we recover Newton's law, as we should. To see this, let

$$s = \frac{1}{\tau_r} (t' - t)$$ (19.24)

so that

$$m \dot{\mbox{\boldmath$v$}}(t) = \int_0^\infty e^{-s} \mbox{\boldmath$F$}(t + \tau_r s) ds .$$ (19.25)

The force is assumed to be slowly varying with respect to $\tau$ (or none of this makes sense, just as was the case above) so that a Taylor series expansion converges:

$$\mbox{\boldmath$F$}(t + \tau s) = \sum_{n = 0}^\infty \frac{(\tau_r s)^2}{n!} a\frac{d^n \mbox{\boldmath$F$}(t)}{dt^n}$$ (19.26)

which, upon substitution and integration over $s$, yields

$$m \dot{\mbox{\boldmath$v$}} = \sum_{n = 0}^\infty \tau_r^n \frac{d^n\mbox{\boldmath$F$}}{dt^n} .$$ (19.27)

In the limit $\tau \to 0$ only the lowest order term survives. This is Newton's law without radiation reaction. The higher order terms are successive radiative corrections and matter only to the extent that the force varies in time. Note that this force obeys a ``Lenz's Law'' sort of behavior; when the applied force is changed (say, increased) there is an additional ``force'' in the direction of the change that acts on the particle. A particle moving in a circle has a force that changes direction but not magnitude. This change is (think about it) tangent to the motion and in the opposite direction. It acts to slow the charged particle down. Hmmmmmm.

There are two extremely annoying aspects to this otherwise noble solution. First, as we have repeatedly noted, it requires a knowledge of $\mbox{\boldmath$F$}(t)$ in the future of the particle to obtain its acceleration now. Truthfully, this isn't really a problem - obviously this is absolutely equivalent to saying that $\mbox{\boldmath$F$}(t)$ can be expanded in a Taylor series (is an analytic function). Second, (and even worse) it responds to a force that is completely in its future with an acceleration now. It ``knows'' that a force is going to act on it before that force gets there.

Figure: $\mbox{\boldmath $F$}(t)$, $\dot{\mbox{\boldmath $v$}}(t)$ and $\mbox{\boldmath $v$}(t)$ on a timescale of $\tau _r$. Note that the particle ``preaccelerates'' before ``the force gets there'', whatever that means.

Mind you, not long before the force gets there. About $10^{-24}$ seconds before (for reasonable forces). Classically this is very bad, but quantum theory fuzzes physics over a much larger time scale. This is viewed by many physicists as an excuse for not working out a consistently causal classical theory. You can make up your own mind about that, but note well that even if the integrodifferential equation had involved past values of the force you should have been equally bothered - either one makes Newton's law nonlocal in time!

Note well that we've already seen (nonlocal) integrodifferential equations in time in a somewhat similar context! Remember our derivation of of dispersion relations, in particular Kramers-Kronig? We had a kernel there that effectively sampled times in the future or past of a system's motion. This worked because we could integrate over frequencies with a constraint of analyticity - our fields were presumed fourier decomposable. Fourier transforms are, of course, infinitely continuously differentiable as long as we avoid sharp changes like (pure) heaviside function forces or field changes, and yes, they explicity provide a knowledge of the quantities in the future and past of their current values.

I personally think that this is yet another aspect of the mistake made by requiring that our description of electrodynamics always proceed from the past into the future with a retarded interaction. As we have seen, this is silly - one could equally well use only advanced interactions or a mix of the two and the solutions obtained for a given boundary value problem will be identical, where the ``boundary'' is now a four-volume and hence requires future conditions to be specified as well as the past conditions on a spatial three-surface bounding the four-volume.