Perturbation Theory
en.wikipedia.org
en.wikipedia.org
The basic assumption of perturbation theory is that the frequency of the motion changes with the amplitude and as the amplitude increases the frequency shifts across resonances that change the motion.
In the case of the harmonic oscillator, the frequency doesn't change so this doesn't apply at all. Instead of working from that as the ε=0 limit, you work from an integrable system like
https://en.wikipedia.org/wiki/Toda_lattice
which models the changing frequency w/ amplitude and breaks the symmetry of the system.
The case of orbital motion in three dimensions (classically the same problem for celestial mechanics and an electron in a hydrogen atom) is also problematic because the frequency of the three motions (i) around the star, (ii) in-and-out away from the star, and (iii) up-and-down out of the equatorial plane are all the same so it is a degenerate case and takes consideration of how exactly the symmetry breaks.
I'd contrast that to quantum theory where the perturbation theory "just works" for the simple cases like the harmonic oscillator and orbital motion. (That's how you develop the theory of spectral lines for the hydrogen atom, as the electromagnetic field is modeled as a perturbation.)
Note: I studied chaos theory for my PhD a long time ago. Unfortunately very few people have been trained in this field and gotten professorships since 1970 or so, so the field is stillborn and people are still learning from old textbooks. People praise V.I. Arnold's book on classical mechanics but it was written before people knew about the integrability of the Toda Lattice and circa 1995 you could make it through a graduate level class on classical dynamics and not learn the right way to think about the "perturbed harmonic oscillator".
A few extra points:
(1) People don't get academic credit for improving the exposition of things, only for making discoveries that are "new"
(2) You often see Poincare sections with anomalies such as tori that are folded over, the problem here is that the energy surface is not flat but has a topology like a sphere or toroid so you see strange things when you project these, I found in some cases you could make a three-dimensional visualization of the energy surface and draw the Poincare section on that and it would be more clear what was going on.
(3) The KAM theorem is stronger with N=2 oscillators (two q and two p coordinates) than it is with N>2 oscillators. With N=2 the tori are solid walls that constrain chaotic motion whereas with N>3 they don't have enough dimensions to really hold the trajectories inside so there are probably trajectories that go from one regular area to a chaotic area and then another regular area. One consequence of this is that we can't really say the solar system is table. Arnold talked about this in a hypothetical way a long time ago but this 'Arnold Diffusion' is still an undiscovered country.
(4) Despite that, people from NASA have done some very nice work at mapping chaotic trajectories that make it possible to get from one planet to another with much less energy that you would otherwise. See
https://en.wikipedia.org/wiki/Interplanetary_Transport_Netwo...
(5) I'm particularly irked that when people draw Poincare sections for Hamiltonian systems, such as
https://i.stack.imgur.com/Wym1Z.png
you get these areas that look like television static that certainly have more structure than they appear. The proof that these areas are chaotic is based on the knowledge that, paradoxically, these are filled with stable periodic orbits and there is a fractal network of resonances that resonate with resonances and so forth that determines the motion in that area which is not visible because of the sensitivity to initial conditions and practical problems that come from doing the math. There were people talking in the 1980s that there ought to be a way to make better pictures with interval math but I don't think it's been done since then. Lately I've been interested in revisiting it not so much because I care about the science but because I'm always on the lookout for algorithms to draw interesting images.
L{y} = V(x)
or in quantum mechanics
L{y} = V(x)y (now L is i times nabla)
Then you expand the potential V(x) as a power series to the quadratic term, which gives you a harmonic oscillator.
[0] https://www.youtube.com/watch?v=LYNOGk3ZjFM&list=PLzcd6SoIsc...
You get done solving out a problem not only does math feel like magic, your paper looks like some kind of annotated spell.
My line of work has nothing to do with physics whatsoever. But I sometimes get the urge to purchase an overpriced textbook on Amazon with a solutions manual and revisit QM, for old times sake.
It's not even that I feel that time was wasted - I like my field and I don't think I was suited for the path I didn't take. But it's such a rich and interesting field (physics, maths, chemistry, the lot) it's so fun to talk about. And with that training at least I could engage meaningfully in those discussions when I was fresh out of uni.
Now, not so much. My amateur astronomer colleague knows more about cosmology than I do at this point.
I don't really miss lunch table discussions though. Typically if it's by members of the general public, they're sprouting some misunderstanding based on some magazine article they read, written by some journalist who didn't really grok the topic either, and I can't just be arsed to start lecturing people how it's not that simple and their understanding is all wrong. More agreeable to just talk about football, beer, or the weather. Or if it's a lunch table discussion between experts, then it's usually about some minute detail of their particular expertise and you need to be pretty knowledgeable about that particular subtopic in order to be able to contribute.
To get back on the topic of perturbation theory I remember how the mathematical physicists in our lectures always complained that the systems the professors would use it for were not well suited to use it in the first place but often times the results were kinda nice so they used it anyway. Not that I remember any details but I was so fascinated when I found perturbation theory used again in QFT and it really dawned on my how powerful of a tool it is.
Since finishing grad school I've taught myself the basics of General Relativity and Quantum Field Theory. You can do it too.
I went into physics thinking I would be working with state of the art quantum mechanics powered magic technology (watching Stargate and other scifi shows as a teenager heavily influenced this). Instead I ended up in finance.
Life in any kind of research is just too hard.
I think that the real problem with textbook education is that it doesn't distinguish or prioritize the parts that trip us up. Which is really bad for divergent thinkers (like with ADHD) who remember things as a series of understood concepts where each builds on the last, rather than through rote memorization.
I wish there was a better way of organizing textbooks more like a wiki, where the obvious parts (assumed to be known) would be in the footnotes, and the pins would make up the main body of the text. It would amount to the questions that students ask during lectures. That way whole semesters could be compressed down to perhaps 2 weeks of learning.
Is that Cliffs Notes? Writing this out, I just realized that I never used them! Maybe there's something better today?
Edit: maybe orphaned branch is a better term, but you get the idea! So I need tags for those branches or they're lost forever on the tip of the tongue.
[1] https://en.m.wikipedia.org/wiki/Structure_and_Interpretation...
see "appendix C" on p17 of the pdf in https://arxiv.org/abs/1611.01676
The basic idea is that we started with a basis of solutions to the linear equation (d=0), then expanded the nonlinear equation order-by-order in the small parameter d. The math was pretty messy since it was a system of linear equations (involving both E and B fields) but conceptually it was pretty straightforward. It was pretty cool, and definitely very satisfying when the resulting eigenvalues agreed nearly exactly with numerical simulation.
As long as the difference between the planned and measured doses was small (like, under 3%) then perturbation theory was valid for this use case, but if it got bigger, then the estimate was increasingly inaccurate. During development, software verification and validation took quite a while because of the complexities of the data sets combined with establishing the limits where perturbation theory was no longer valid. There was a lot of debate on where to set warning messages versus plain old disabling the output as invalid.
Training new users always seemed to bring up perturbation theory, because of this limit. Like, the first thing almost every physicist did was to create a huge dose difference to see how it would handle it. The software would pop up a message saying the difference was too big for perturbation theory to apply and so the output was disabled (this was a choice made for patient safety; better to display no data than bad data). Then the new user would ask why it wasn't working. And I would have to remind them about how perturbation theory worked...
Here's a few papers about it in case anyone is interested:
Wait, so the Theory of Epicycles is morally a Perturbation Theory? (disregarding matching evidence from relativistic effects yadda yadda).
In that sense —morally speaking— Epicycles at the time doesn't sound that like it was that crazy to do scientifically. Generally speaking, Epicycles would be on the same level as how perturbation theory is used in quantum mechanics.
It had one layer of epicycles, that was it; it did have additional complications, in the form of the eccentric and the equant, but these additional complications were not additional layers of epicycles that could be stacked arbitrarily.
(Copernicus may have used a second layer of epicycles in his heliocentric system, which eliminated the equant (but not, I believe, the eccentric)? Admittedly two does suggest the possibility of generalizing in a way that one doesn't, but I don't think that generalization was ever performed at the time before Kepler made the whole thing irrelevant.)
Had it been used, it basically would have been Fourier series, yeah! I don't know whether that counts as "perturbation theory" proper. But it's worth noting that the real development didn't happen that way, and the further corrections that were actually used were the eccentric and the equant, not further epicycles!
I note this, as most of the popularizations of solar system astronomical history I'd seen, generally derided epicycles as a bad thing.
They weren't. They were a sort of perturbation theory. One that generated other problems. But still did a somewhat better job than the (at the time) accepted theory.