Perturbation Theory(en.wikipedia.org)
en.wikipedia.org
Perturbation Theory
https://en.wikipedia.org/wiki/Perturbation_theory
32 comments
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
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".
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".
Wow thank you for taking the time to write all of this out. I'm glad I said this as it has given me a few new lines of reasoning to look at and study
I think the field of chaos theory has branched into several other fields, so I would not necessarily agree that it is dead in the water. I personally have worked on projects within the last few years studying chaotic dynamical systems from the point of view of operator algebras.
Is there a modern reference you would recommend?
I haven't looked for stuff from the last 20 years but I will ask around. I've got a book at home that you could figure out the bit about the Toda lattice if you read between the lines and I will look it up this evening.
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.
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.
There has been a lot of research on the quantum consequences of classic unstable periodic orbits lately - look up “many-body scars”
Perelomov’s book covers Toda integrability I think.
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.
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.
Carl Bender lecture series is excellent [0]
[0] https://www.youtube.com/watch?v=LYNOGk3ZjFM&list=PLzcd6SoIsc...
[0] https://www.youtube.com/watch?v=LYNOGk3ZjFM&list=PLzcd6SoIsc...
Thanks for the pointer. Here is an alternative link with higher quality and downloadable slides
https://pirsa.org/C11025
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.
You get done solving out a problem not only does math feel like magic, your paper looks like some kind of annotated spell.
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.
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.
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.
Same. Although I had long ago decided that an academic career wasn't for me, I was always thinking that after I finish my PhD (computational condensed matter physics), I would somewhat keep up with my field just for fun. But between kids, work, other hobbies etc. I never really had the time nor energy to do that. So while I'm somewhat confident I could relatively quickly relearn much of what I have forgotten, it also feels that world is slipping further and further away for each moment. Not that I really dwell over it, there's a lot of interesting things in life and so little time, you can't just do everything.
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.
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.
I’m kind of relieved to know I’m not alone. Got my masters 4 years ago and immediately became a web dev after. I am greatful for the time and I think a lot of the resilience, logical thinking and general problem solving skills transferred well to my current work. Yet some part of me wishes I had stayed and gotten a PhD but I also remember I was overworked and had significantly higher imposter syndrome although my grades indicated that I was doing really well.
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.
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.
I'd felt the same and decided I actually could not live without it. So I am now 34 and studying for an undergraduate degree in life sciences (just finished the notoriously challenging organic chemistry 1&2) while adding some graduate-level physics classes on top. Really, really refreshing and good experience so far, though I don't know if I have the stomach to try going for a PhD again (would rather become independently wealthy and fund my own research, heh).
I encourage you both to do this if you are interested. It isn't that hard and, in fact, many universities have put their entire quantum mechanics courses online if you want a refresher.
Since finishing grad school I've taught myself the basics of General Relativity and Quantum Field Theory. You can do it too.
Since finishing grad school I've taught myself the basics of General Relativity and Quantum Field Theory. You can do it too.
The same for me.
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 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.
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.
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).
[1] https://en.m.wikipedia.org/wiki/Structure_and_Interpretation...
[1] https://en.m.wikipedia.org/wiki/Structure_and_Interpretation...
Back in 2016 me and some collaborators used perturbation theory to calculate the stable modes in a nonlinear PDE that describes lasers.
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.
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.
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:
https://pubmed.ncbi.nlm.nih.gov/22225277/
https://pubmed.ncbi.nlm.nih.gov/22830756/
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/
> 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.
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!
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!
interesting history, thanks! I was mainly noting the "recursion"/series similarities, yes. but didn't know the complications from epicycles were more varied in nature.
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.
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.
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