Hmm if I'm understanding this correctly, (and recalling correctly,) there was a chapter at the end of Kleppner & Kolenkow that dealt with this in the n=2 case: Basically assume the solution is a power series and truncate it past x^2. Well, you know how to solve diffeq's like this, so do it and see the solutions. This was offered as a reason why so many things display periodic or seemingly periodic behavior. If the perturbations (x) are small enough to ignore the terms above x^3, then it's just like the spring equations with dampening/forcing.
The common examples that you see first in classical dynamics are ones for which the ordinary perturbation theory are problematic.
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
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".
You're almost remembering correctly. Usually you have a differential operator L (often the Laplacian) and a potential term V(x) (where x may be a vector) giving the equation
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.
One of the few college lectures I can remember 15 years later is the day our Physical Chemistry teacher taught us this technique to approximate solutions to quantum mechanical problems.
You get done solving out a problem not only does math feel like magic, your paper looks like some kind of annotated spell.
I have a similar feeling with undergraduate quantum mechanics. Developing first order solutions with perturbation theory felt like magic, yet somehow intuitive. That and spectral theory
Looking at the Wikipedia entry for perturbation theory in quantum mechanics makes me sad. I used to understand these concepts. I still recognize some things like V for potential energy and bra kets, but I've forgotten how they work.
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.
I get exactly the same pang of sadness. I even distinctly remember learning perturbation theory. But I use so little of what I learnt, and it's so long ago, that at lunch table discussions I can't even be the "ex-physicist that'll know the answer to that" any more. It fills me with great sadness.
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 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.
Same here, I learned differential equations for solving circuits and calculus in any number of variables or dimensions but it's all so foggy. I sometimes use Khan Academy to connect the dots. So the memories are there, I just don't have a pin where they start from. I've also noticed as I've gotten older that more pins have fallen away, so it's like I have countless trees of knowledge in detached HEAD states that I can't access until I read up on something and revive the pins.
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.
This is used in the programming book Structure and Interpretation of Classical Mechanics [1]. With some of the same authors as the famous Structure and Interpretation of Computer Programs (SICP).
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.
An interesting application of perturbation theory which I used to work with was a software application called 3DVH, which was used in radiation treatments (for cancer for example). Its inputs were the planned geometrical distribution of the radiation absorbed in the patient, and an array of measured dose points around a simulated patient, done before treatment. The output was a fairly accurate estimate of the actual dose distribution in the patient, taking into account the differences between the planned and measured doses. The reason you might want this is because if the difference between planned dose and reality was cutting it close to say, an organ or a major nerve, you'd really like to know which side of that margin your treatment was going to be on.
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:
> The perturbative expansion is created by adding successive corrections to the simplified problem.
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.
The thing to note here is that the theory of epicycles was never actually a thing; contrary to popular belief, geocentric astronomy did not fix its problems by adding additional layers of epicycles.
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!
Basically ... yes. This is one of my arguments about the scientific method in general. You start with something you can't really explain, that seems to match data a bit better than the previous theory. You keep refining it, and eventually someone (often someone else) derives an insight from this work.
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.
Not exactly. The interesting thing with perturbation theory as I remember it is that you have a series of infinitesimal perturbations, not just any old perturbations.
[+] [-] martinjacobd|3 years ago|reply
[+] [-] PaulHoule|3 years ago|reply
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".
[+] [-] scythe|3 years ago|reply
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.
[+] [-] user070223|3 years ago|reply
[0] https://www.youtube.com/watch?v=LYNOGk3ZjFM&list=PLzcd6SoIsc...
[+] [-] azepoi|3 years ago|reply
[+] [-] pkilgore|3 years ago|reply
You get done solving out a problem not only does math feel like magic, your paper looks like some kind of annotated spell.
[+] [-] _sword|3 years ago|reply
[+] [-] tuanx5|3 years ago|reply
[+] [-] GalenErso|3 years ago|reply
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.
[+] [-] tescocles|3 years ago|reply
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.
[+] [-] vjk800|3 years ago|reply
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.
[+] [-] zackmorris|3 years ago|reply
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.
[+] [-] Ambolia|3 years ago|reply
[1] https://en.m.wikipedia.org/wiki/Structure_and_Interpretation...
[+] [-] xdavidliu|3 years ago|reply
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.
[+] [-] FiatLuxDave|3 years ago|reply
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:
https://pubmed.ncbi.nlm.nih.gov/22225277/
https://pubmed.ncbi.nlm.nih.gov/22830756/
[+] [-] platz|3 years ago|reply
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.
[+] [-] Sniffnoy|3 years ago|reply
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!
[+] [-] hpcjoe|3 years ago|reply
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.
[+] [-] 6gvONxR4sf7o|3 years ago|reply
[+] [-] unknown|3 years ago|reply
[deleted]
[+] [-] eulenteufel|3 years ago|reply