When does an oscillating charged particle stop radiating? (classic case)
Although self-force is one of the major problem in physics, I think with a few assumptions it's possible to deal with it, at least in Newtonian mechanics. Let's consider a periodic motion, where we have made a long hole in the Earth such that our charged particle can go through it and oscillates there for a long time (But not forever! It's not a perpetual motion machine). We just want to know how long does it oscillate and therefore radiate. See the picture below:
The red circle is a charged particle with mass $m$ and charge $q$. We know that gravity will force the particle to oscillate, because it applies a backward force (check shell theorem, this is a solid sphere with a hole, not a hollow one, or see this video), $F_g=-kr$, where $k$ is a constant and $r$ indicates the position of particle, that's, if charged particle goes to center of Earth (or $r=0$) there will be no force on it. Let's also assume that initial velocity of particle is zero. We know that when particle arrive at $r=0$ its velocity is not zero, so it continues its motion to the other end of hole, when it arrives there, it goes back and this continues for a long time. Along the way, it radiates too, because it's oscillating after all. It can not radiate forever or else we have a system that gives us energy for free, out of nothing and even better, forever! Eventually, it should stop at the center of the earth, but when?! after how much radiation? Ok, someone might want to use equivalence principle to say that there will be no radiation at all, please don't do that. 483
From mathmatical point of view, this problem is already hard, not to mention particle oscillates so it's not really a free falling particle (even it's, let's put it aside for now). Ok we have <PERSON>'s second law,
$$F_{net}=F_g+F_{rad}\rightarrow mr''=-kr+nr'''$$
Where $n$ is a constant for radiation (it depends on $q$ and $m$). And $F_{rad}$ is the scary self-force. Even with <PERSON>, solving this equation with respect to initial values is hard. It seems more logical to use <PERSON> formula, $$p=c*(r'')^2$$ where $p$ is total power radiated by a non relativistic point charge, $c$ is constant depends on particle's charge. We can use Larmor formula because motion is periodic. But $r''$ or acceleration is not constant, and it is not changing slowly enough to put it outside of integral with respect to time. I wanted to subtract it from initial energy of particle, but I can't do that.
TL;DR
Prove that at finite amount of time, an oscillating charged particle will stop, and it radiates finite amount of energy. Only rigorous solution with mathmatical details will be accepted (but I vote up whatever answer I like). 766