If you're curious about the historical details of the various "solutions" of the three-body problem (prior to 1996), then there's an excellent book for the layman, titled Poincare and the Three Body Problem by June Barrow-Green.
I work on other problems in three body dynamics, but I'm very pleased to see this article show up here on HN!
One thing I could expand on a little is these singularities in Sundman's series. These singularities are nothing more than collisions! It's these collisions and near-collisions that can make the three body problem so hard. Gravity is a 1/r^2 force, so the forces get very strong and change very rapidly when the two objects come close to each other. This is not only hard to handle analytically, but numerically as well!
In this field there is a well known transformation known as the Kustaanheimo-Stiefel transformation (although the full name is not so well known---everyone just refers to it as the KS transformation or KS regularization). What this transformation does is it moves you from a regular Cartesian or polar coordinate system (which has the force diverge when r --> 0) to a coordinate system in which this singularity is moved to infinity. After a KS transformation a computer can integrate even extremely elliptical orbits or collisional orbits. The only downside to this is the algorithmic complexity. In an ordinary Cartesian system, if you were to perform some numerical integration the complexity goes like O(N^2), with N being the number of objects in your system. The complexity of computing an orbit after KS regularization is O(N^3), however. (My understanding of this is that KS regularization works by turning position vectors into quaternions---which then can basically be treated as matrices. Numerically evolving the system then amounts to performing matrix multiplication, which goes as O(N^3). My intuition may be flawed, however!) KS regularization therefore can't help if you have more than ~10 objects. In practice in my research I haven't found it useful even at N=3.
One last thing to mention is that even though these orbits can in principle be solved numerically, they can't in practice. Three body systems are chaotic and the machine precision of your computer leads to round off errors that cause your calculation to diverge from the true solution remarkably quickly. There are special-purpose programs that can calculate orbits to arbitrary precision, but they run really, really slowly (as you might imagine), so they're not useful for more general studies of three body systems. Fortunately, back in 1991, Quinlan & Tremaine showed that there exist so-called "shadow orbits" that track the orbit numerically computed by your computer. These shadow orbits are real orbits for some initial conditions that come arbitrarily close to the numerically calculated orbits. This leads some credence to numerical few body studies. Unfortunately, what's still unknown is whether these shadow orbits have the same statistical properties as the general set of orbits. Not enough research has been done on this question because it's still open and the accuracy of my research depends on this property being true! But ultimately, this question is probably linked to the ergodic hypothesis which is central to the foundations of statistical mechanics---so it's an important question for all physicists, not just me!
> Gravity is a 1/r^2 force, so the forces get very strong and change very rapidly when the two objects come close to each other. This is not only hard to handle analytically, but numerically as well!
> the force diverge when r --> 0
the [physical] force isn't diverge - no 2 real physical objects can have infinite gravity force between them however close you bring the objects together. It is only the [simplified] mathematical model of the force that does diverge.
In general with regard to slow converging "solutions" my professor was saying that we can just iterate over the rational numbers (countable set) and thus we can always hit any desired neighborhood of the precise solution in finite amount of time :)
> One last thing to mention is that even though these orbits can in principle be solved numerically, they can't in practice.
This is true in the general case, but there are numerous examples of N-body systems that are both of real-world interest and remarkably stable. The Solar System, for instance. The reason is ultimately anthropic, of course - if it weren't stable, we probably wouldn't be here observing it.
Many systems (including the Solar System) are stable over short time periods, but the stability of the Solar System is actually unknown over longer time periods. As an example, one of the numerical experiments Scott Tremaine has done has been to simulate the solar system, but to perturb the initial conditions of the planets by 1mm (that's one millimeter!). Over ~100 million years, in ~1% of systems Mercury actually crashes into Venus! [1]
Thanks for that insightful explanation! I was wondering: could you explain in a nutshell what the KAM theorem is all about? I've heard about it many times, but I still have no idea what 'invariant tori' are...
Sure! In classical mechanics it's nice when you have a problem for which the solution is integrable. Oftentimes these solutions will also be periodic. One very useful example of this is an elliptical orbit---the object will return to the same position in the orbit at equal intervals. Unfortunately, there aren't very many problems that are simple enough to have periodic solutions in the real world. While Mars's orbit is approximately periodic, perturbations from Jupiter cause it to vary a little bit, so it's not strictly periodic. Now, we know from observations that the orbit of Mars is almost an ellipse, so these perturbations from Jupiter don't blow up and cause Mars to go careening through the Solar System. Intuitively, it makes sense that this should be possible because if you have some infinitesimally small perturbation, the orbit should remain approximately the same as the unperturbed orbit.
The question the KAM theorem answers is "Under what kinds of perturbations do the perturbed orbits remain approximately the same as the unperturbed orbits?" The answer is that if the perturbations are (1) small, (2) smooth, and (3) non-resonant, then the perturbed orbits will be close to the unperturbed orbits. Now, condition (1) doesn't tell you exactly how small those perturbations need to be---only that they exist if they're small enough. Condition (2) will be met for any realistic system---it just prevents you from choosing some pathological perturbation function like the Weierstrass function. Condition (3) basically states that if the perturbation is periodic, then the ratio between its frequency and the frequency of the orbit must be an irrational number. If you had, for example, a 1:1 resonance, so the two frequencies are the same, then the perturbation will be applied at the same spot in every orbit. It will then build up over time until it will eventually have a large effect and the perturbation will no longer be small.
The KAM theorem goes further, too. If you look at the unperturbed orbit in phase space, it will just trace out a nice, simple ellipse (or, in general, some closed loop). Since the perturbed orbit will be close to the unperturbed orbit, the KAM theorem also tells you what shape it will have in phase space. By energetic arguments, the KAM theorem tells you that the perturbed orbit is restricted to a torus centered around the unperturbed orbit. This is what's called the invariant torus. Not only is the perturbed orbit restricted to this torus, but given enough time, the perturbed orbit fills out the entire torus. The perturbed orbit will eventually come arbitrarily close to any point on the torus. (I believe this is due to the non-resonance condition---that is, if the perturbed orbit was strictly periodic then there would be a resonance between the perturbation and the orbit---but I'm not sure on this point.)
These orbits are known as quasiperiodic because they are confined to a torus centered around a periodic orbit, but they never strictly repeat themselves.
Fond memories of when I took a course on celestial mechanics. Weirdly enough, the theoretical framework was taught by a more "applied" researcher, whereas the "problem solving" classes were taught by one local leading researcher in the field. I still remember determining degrees of singularities on collisions and regularisations near collisions as an "easy problem to get up to speed"!
http://www.worldcat.org/search?qt=worldcat_org_bks&q=Poincar...