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Comment by martinjacobd | Hacker News Reader

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.

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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".

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

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.

Perelomov’s book covers Toda integrability I think.

There has been a lot of research on the quantum consequences of classic unstable periodic orbits lately - look up “many-body scars”

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.

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.

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