electromagnetism - Radiation Resistance

When a charge is accelerated, it radiates and loses kinetic energy. This can be modeled by having another force act on the charge, which is proportional to the derivative of the acceleration. So if this force is included the equations of motion, we end up with 3rd order equations. Is this allowed? Why don't we need to fix another constant to determine the motion? Is it because we usually neglect this force because of its small magnitude?

Also, the electromagnetic Lagrangian only contains up to first derivatives (of the 4-potential in this case). How can this result in 3rd order equations of motion?

Answer

This is a hard problem! I don't think that a completely satisfactory theory of classical electrodynamics, with point charges, including radiation reaction, is known. In fact, if I had to conjecture, I'd bet that there is no such theory -- that is, classical theory simply doesn't play nicely with pointlike charges.

The first thing to point out is that if we don't have truly pointlike charges, there's no problem. If you're willing to model all of your charges as spheres of some small but nonzero radius, or anything like that, then the whole theory is perfectly self-consistent. The radiation reaction force on one of these spheres is just caused by the force on each little element of the sphere, due to all the other elements. The whole theory, consisting of just Maxwell's equations, the Lorentz force law, and some assumption about the additional stresses required to hold the sphere together against its own repulsion, is self-consistent, conserves energy-momentum, and generally behaves the way you want it to.

The details of this theory depend on precisely what you assume about the arrangement of charge on the sphere, how it responds to stresses (e.g., does it always remain spherical in its own instantaneous rest frame?), etc. But in cases where the radius of the sphere is small compared to other length scales in the problem, you'd expect those details not to matter much, and indeed that is the case.

In these models, the equation of motion for one of these not-quite-point charges turns out to be an integro-differential equation, in which the acceleration of the charge at any time $t$ is determined by an integral over the charge's position for times $t'$ in the range $t-2a/c

Griffiths's textbook has a nice "toy model" discussion of this, which is good for building intuition. For all the details, the best places to go are a book and a bunch of articles (e.g., this one) by F. Rohrlich.

Once you've got a good theory for spheres of radius $a$, the natural thing to do is to take the limit as $a\to 0$. When you do, you get a third-order differential equation of motion for the (now pointlike) charge. This same third-order equation can be derived in a bunch of different ways, going all the way back to the early 20th century. It's often called the Lorentz-Dirac equation, although Abraham's name gets thrown in too sometimes.

The fact that the equation is third-order is actually not as bad as it seems. Kijowski makes a nice argument that you should have expected a third-order equation all along! To formulate this setup as an initial-value problem, you need to specify the position and velocity of the charge, and the electromagnetic field. The latter actually determines the charge's acceleration at that moment. To be specific, the specific way the field diverges as you approach the charge depends on the acceleration. Since the acceleration, as well as the position and velocity, are determined by the initial conditions, it makes sense that the equation of motion is third order.

But even if you're OK with the fact that the equation of motion is third-order, the Lorentz-Dirac equation is pathological. It contains "runaway" solutions, in which the charge's acceleration grows exponentially forever after any external force is removed. You can artificially remove these runaway solutions, but the solutions left behind suffer from a problem of "preacceleration": the motion of the charge at time $t$ depends on what external forces will be applied at all later times.

So there's a finite-sized-particle theory that works fine, but it goes bad as the size goes to zero. In fact, it actually goes bad even earlier: typically, the finite-sized theory starts to have pathological solutions when the size of the particle gets to be less than the classical particle radius $q^2/(4\pi\epsilon_0mc^2)$. I strongly suspect (but I can't prove) that what this is telling us is that classical electrodynamics simply doesn't work with particles smaller than this size.

That's a surprising situation, but it doesn't contradict anything fundamental that I can think of. In particular, we have no right to expect there to be a "correct" theory of classical point particles, if "correct" is taken to mean "in agreement with physical reality." After all, physical reality is quantum!