We know that

(19.7) |

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
total rate:

(19.8) |

(the Larmor formula). These are the two pieces we've thus far treated independently, neglecting the one to obtain the other.

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:

(19.9) |

(19.10) |

Thus (rewriting Newton's second law in terms of this force):

(19.11) |

defines the

- We would like energy to be conserved (as indicated above), so that the energy that appears in the radiation field is balanced by the work done by the radiation reaction force (relative to the total work done by an external force that makes the charge accelerate).
- We would like this force to vanish when the external force vanishes, so that particles do not spontaneously accelerate away to infinity without an external agent acting on them.
- We would like the radiated power to be proportional to , since the power and its space derivatives is proporotional to and since the force magnitude should be dependent of the sign of the charge.
- Finally, we want the force to involve the ``characteristic time'' (whereever it needs a parameter with the dimensions of time) since no other time-scaled parameters are available.

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
interval:

(19.12) |

(19.13) |

(19.14) |

*One* (sufficient but not necessary) way to ensure that this
equation be satisfied is to let

(19.15) |

(19.16) |

This is called the

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

(19.17) |

(19.18) |

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:

(19.19) |

or

(19.20) |

This latter equation has no runaway solutions or acausal behavior as long as is differentiable in space and time.

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...