We know that
However, at the same time it is being acted on by the external force
(and is accelerating), it is also radiating power away at the
However, in order for Newton's law to correctly lead to the conservation
of energy, the work done by the external force must equal the increase
in kinetic energy plus the energy radiated into the field. Energy
conservation for this system states that:
Thus (rewriting Newton's second law in terms of this force):
Let's start with the first of these. We want the energy radiated by
some ``bound'' charge (one undergoing periodic motion in some orbit,
say) to equal the work done by the radiation reaction force in the
previous equation. Let's start by examining just the reaction force and
the radiated power, then, and set the total work done by the one to
equal the total energy radiated in the other, over a suitable time
One (sufficient but not necessary) way to ensure that this
equation be satisfied is to let
Note that this is not necessarily the only way to satisfy the integral constraint above. Another way to satisfy it is to require that the difference be orthogonal to . Even this is too specific, though. The only thing that is required is that the total integral be zero, and short of decomposing the velocity trajectory in an orthogonal system and perhaps using the calculus of variations, it is not possible to make positive statements about the necessary form of .
This ``sufficient'' solution is not without problems of its own, problems that seem unlikely to go away if we choose some other ``sufficient'' criterion. This is apparent from the observation that they all lead to an equation of motion that is third order in time. Now, it may not seem to you (yet) that that is a disaster, but it is.
Suppose that the external force is zero at some instant of time . Then
Recalling that at and , we see that this can only be true if (or we can relax this condition and pick up an additional boundary condition and work much harder to arrive at the same conclusion). Dirac had a simply lovely time with the third order equation. Before attacking it, though, let us obtain a solution that doesn't have the problems associated with it in a different (more up-front) way.
Let us note that the radiation reaction force in almost all cases will be very small compared to the external force. The external force, in addition, will generally be ``slowly varying'', at least on a timescale compared to seconds. If we assume that is smooth (continuously differentiable in time), slowly varying, and small enough that we can use what amounts to perturbation theory to determine and obtain a second order equation of motion.
Under these circumstances, we can assume that
, so that:
We will defer the discussion of the covariant, structure free generalization of the Abraham-Lorentz derivation until later. This is because it involves the use of the field stress tensor, as does Dirac's original paper -- we will discuss them at the same time.
What are these runaway solutions of the first (Abraham-Lorentz) equation of motion? Could they return to plague us when the force is not small and turns on quickly? Let's see...