This will be familiar to anyone who reads John Baez’s This Week’s Finds in Mathematical Physics, but I can’t help but show these lovely exact solutions to the gravitational N-body problem. This one is beautiful in its simplicity: twenty-one point masses moving around in a figure-8.
The N-body problem is one of the most famous, and easily stated, problems in mathematical physics: find exact solutions to point masses moving under their mutual Newtonian gravitational forces (i.e. the inverse-square law). For N=2 the complete set of solutions is straightforward and has been known for a long time — each body moves in a conic section (circle, ellipse, parabola or hyperbola) around the center of mass. In fact, Kepler found the solution even before Newton came up with the problem!
But let N=3 and chaos breaks loose, quite literally. For a long time people recognized that the motion of three gravitating bodies would be a difficult problem, but there were hopes to at least characterize the kinds of solutions that might exist (even if we couldn’t write down the solutions explicitly). It became a celebrated goal for mathematical physicists, and the very amusing story behind how it was resolved is related in Peter Galison’s book Einstein’s Clocks and Poincare’s Maps. In 1885, a mathematical competition was announced in honor of the 60th birthday of King Oscar II of Sweden, and the three-body problem was one of the questions. (Feel free to muse about the likelihood of the birthday of any contemporary world leader being celebrated by mathematical competitions.) Henri Poincare was a favorite to win the prize, and he submitted an essay that demonstrated the stability of planetary motions in the three-body problem (actually the “restricted” problem, in which one test body moves in the gravitational field generated by two others). In other words, without knowing the exact solutions, we could at least be confident that the orbits wouldn’t go crazy; more technically, solutions starting with very similar initial conditions would give very similar orbits. Poincare’s work was hailed as brilliant, and he was awarded the prize.
But as his essay was being prepared for publication in Acta Mathematica, a couple of tiny problems were pointed out by Edvard Phragmen, a Swedish mathematician who was an assistant editor at the journal. Gosta Mittag-Leffler, chief editor, forwarded Phragmen’s questions to Poincare, asking him to fix up these nagging issues before the prize essay appeared in print. Poincare went to work, but discovered to his consternation that one of the tiny problems was in fact a profoundly devastating possibility that he hadn’t really taken seriously. What he ended up proving was the opposite of his original claim — three-body orbits were not stable at all. 
Not only were orbits not periodic, they didn’t even approach some sort of asymptotic fixed points. Now that we have computers to run simulations, this kind of behavior is less surprising (example here from Steve McMillan — note how the final “binary” is not made of the same “stars” as the original one), but at the time it came as an utter shock. In his attempt to prove the stability of planetary orbits, Poincare ended up inventing chaos theory.
But the story doesn’t quite end there. Mittag-Leffler, convinced that Poincare would be able to tie up the loose threads in his prize essay, went ahead and printed it. By the time he heard from Poincare that no such tying-up would be forthcoming, the journal had already been mailed to mathematicians throughout Europe. Mittag-Leffler swung into action, telegraphing Berlin and Paris in an attempt to have all copies of the journal destroyed. He basically succeeded, but not without creating a minor scandal in elite mathematical circles across the Continent. (The Wikipedia entry on Poincare tells a much less interesting, and less accurate, version of the story.)
However, just because the general solution to the three-body (and more-body) problem is chaotic, doesn’t mean we can’t find special exact solutions in highly-symmetric conditions, and that’s just what Cris Moore and Michael Nauenberg have recently been doing. The image at the top really is an exact solution to twenty-one equal-mass objects moving in a figure-eight under their mutual gravitational attraction. They’re moving in a plane, of course, but that’s not strictly necessary; here’s a close relative of the figure-8, perturbed outside the plane.
From there you can just go nuts; here’s an example with twelve objects orbiting with cubic symmetry — four distinct periodic paths with three particles each.
Knowledge of this exact solution, plus $3.50, will get you a grande latte at Starbucks. Mathematicians have all the fun.
Wow! These are stunning. Just for my information, has anybody studied N-body problem with non-inverse-square law?
Now as I understand it, any chaotic system is one in which infinitesimal pertibation to an initial condition, the system diverges exponentially, making long term prediction difficult/impossible.
So on a related note, I know that the initial system that Einstein proposed with the cosmological constant was a valid solution, but apparently an instable one. Are any of these other solutions stable? Or would any small ‘bump’ cause the system to transition to chaos?
At any rate cool stuff,
NM
You know when I saw these diagrams, they immediately reminded me of one of Greg Egan’s applet shown here.
As I followed your link to Michael’s site I noticed his acknowledgement of Greg Egan’s help. Small world.
Forgive me for layman thoughts.
Equillibrium points? I wonder if “heat death(QGP)” is really a time of tunnelling? Oui! Non?
The best, of course, is the bBuckyball planetary system.
Would nuclear physics come under that heading?
Can you imagine living on one of the 21 planets in the figure 8?
It’d be nerve-wracking! Every six months you’d have a near collision with another planet.
The thing is, you [i]know[/i] they won’t collide. It’ll be just like a (safe) rollercoaster side.
Imagine the situation of air(space) travel for the people living in those planets. It would be most economic to travel from some planet to the other during those times of “near-colission”, or when they bunch-up around the bend. 😀
are these solutions perturbatively stable?
Typo: Chris Moore -> Cris Moore (no h). Feel free to delete this after you fix it.
Made very interesting reading. I am interested in the answer to Nami’s question too..regarding N-body problems with a non inverse square law.
Nathaniel Grossman’s delightful text: The Sheer Joy of Celestial Mechanics chapter 2, section 3 describes other exact solutions to a non-inverse square law. For example, if it is cubed you get solutions called “Cote’s spirals. It seems to me I’ve seen N-body shareware software that lets you play with the exponent of the inverse force, but I can’t locate it now (I would be interested to know, if you find it). Such a study would be simple-enough to program yourself however, look at the “particle-particle” numerical method of my old N-body Methods web page. Perhaps you can find existing studies on the ArXiV archive too.
I should clarify that the exact solutions described in Grossman’s book are for limited cases, while the N-body methods on the web page are numerical and inexact solutions. A good site for details about writing numerical N-body codes here.
My interest in the N-Body problem as stated initially at Baez site, was relative to the last box in Sean’s above post.
If one adds a simplistic directional value to the system, then it cannot surely evolve ? all the above solutions are not really ideal?, for instatnce the first “figure-of-eight” is background independant.
If one was to find 21 point masses evolving thus, anywhere other than in a computer generated program, I would be amazed!
Their mutual Newtonian gravitational attraction, is realistically to “clinical”? . Example, any Cosmic background Expansion paramiter would nullify the systems evolution?
Even a simplistic 2-body system cannot evolve as an isolated system indefinate ?
Again, the input of the constrained orbits in the last box is neat, and is based on the unchanging Mathematics of Isolated Systems, which are not “Relative” to change !
Still cool to look at though.
Re: Amara
I just found this simulator program that “looks” great. I haven’t yet downloaded it, but will be doing soon. I came across this while searching for the program you mentioned. I’d really love to a non-inverse square universe.
http://gravit.slowchop.com/
Steve– thanks, fixed.
Nami– I don’t know anything about work with non-inverse-square force laws, but then again I wouldn’t.
None of these solutions is at all stable! Or realistic. But they are interesting and fun.
Navneeth: That gravit program implements a Tree code and with the usual inverse square law force, so you would need to modify the source code first. Tree methods are a deep leap into N-body methods (a heads up, if you’re not aware). I couldn’t find the module that calculates the forces in my cursory look of the gravit source, but then C is not my strength, and I have never implemented a Tree-code before. (Particle-particle and symplectic schemes are my only N-body coding experience.)
This dynamic n-body curve may be a [complex?] Spatial-3D, Time-1D [or more?] version of the Lemniscate of Bernoulli. A 3 star system like Centauri may slightly differ from a black hole dominated galaxy, solar dominated planetary system or a planet dominated lunar system.
http://curvebank.calstatela.edu/lemniscate/lemniscate.htm
The latter three tend to resemble a logarithmic spiral that may relate to the ‘Mice Problem’
http://mathworld.wolfram.com/MiceProblem.html
Yet all the multiple body systems would appear to likely have some type of radial symmetry that may resemble a ‘Superellipse’.
http://mathworld.wolfram.com/Superellipse.html
In general relativity the two body problem can’t be solved exactly. And in QFT even the zero body problem (vacuum) can’t be solved. 🙂
Nicholas #2:
This is a slightly different issue than the one above: Einstein was looking for a universe that was neither expanding nor contracting. By adding a cosmological constant, and fine-tuning the mass in the universe, this can be done pretty easily within GR–simply write down the standard cosmology that prefers no direction nor any position (as modern cosmologists do when taking their first, most basic run through things), and then set all the time derivatives equal to zero. This ends up giving you a set of relationships betwene the CC, the density of the universe, and the pressure of the matter in the universe.
It is a perfectly legitimate solution in the presence of a CC. However, a slight bump of this system leads not to a transition to chaos, but rather will cause the universe to either contract or to expand. Therefore, the Einstein static universe ends up beng akin to blaancing an egg on the top of a basketball–not impossible, per se, but it is essentially impossible to do perfectly enough to get the system to stay like that.
But you wouldn’t really consider the system chaotic–it would still remain a well-defined robertson-walker-desitter universe and would still be relatively close to the original system in the parameter space, at least initially. Late observers will be able to deduce the initial conditions of the system. Chaotic systems are much more uppity and difficult to manage.
chris #6–
nuclear physics is inherently quantum mechanical, so you can’t get away with using the cold determinism of classical mechanics and hope to get answers corresponding to what experimentalists observe–you run into all sorts of odd probleems having to do with the indistinguishability of particles inside the nucleus, as well as the probabilistic nature of quantum mechanics.
See the bottom of this page for an animation of the Klemperer rosette of the puppeteer Fleet of Worlds in the Larry Niven novel Ringworld.
‘… the ‘inexorable laws of physics’ … were never really there … Newton could not predict the behaviour of three balls … In retrospect we can see that the determinism of pre-quantum physics kept itself from ideological bankruptcy only by keeping the three balls of the pawnbroker apart.’ — Tim Poston and Ian Stewart, Analog, November 1981.
It isn’t quantum physics that is the oddity, but actually classical physics! The normal teaching of Newtonian physics (at least at low levels) falsely claims/indoctrinates the persistent lie that it allows the positions of the planets to be exactly calculated (determinism) when it does not if you have 3+ bodies, which you do. Richard P. Feynman conceded this in his book QED:
‘when the space through which a photon moves becomes too small (such as the tiny holes in the screen) … we discover that … there are interferences created by the two holes, and so on. The same situation exists with electrons: when seen on a large scale, they travel like particles, on definite paths. But on a small scale, such as inside an atom, the space is so small that … interference becomes very important.’
The interference is due to many vacuum virtual charges:
‘All charges are surrounded by clouds of virtual photons, which spend part of their existence dissociated into fermion-antifermion pairs.’ — I. Levine, D. Koltick, et al., Physical Review Letters, v.78, 1997, no.3, p.424.
The duration and maximum range of these charges is easily estimated: take the energy-time form of Heisenberg’s uncertainty principle and put in the energy of an electron-positron pair and you find it can exist for ~10^-21 second; the maximum possible range is therefore this time multiplied by c, or 10^-12 metre. This is far enough to deflect electrons but not enough to be observed as vacuum radioactivity. Like Brownian motion, it introduces chaos on small scales, not lare ones:
‘… the Heisenberg formulae can be most naturally interpreted as statistical scatter relations, as I proposed [in the 1934 book ‘The Logic of Scientific Discovery’]. … There is, therefore, no reason whatever to accept either Heisenberg’s or Bohr’s subjectivist interpretation …’ — Sir Karl R. Popper, Objective Knowledge, Oxford University Press, 1979, p. 303.
The Schroedinger wave equation arises naturally from a sea of particles because we know that you get waves in particle-based fluids: http://feynman137.tripod.com/#b
There has been some exploration of the N-body problem for potentials other than the inverse square law – a 1/r^D-1 generlization follows from extending Gauss’s law to extra dimensions (standard grad problem is to consider “the problem” in arbitary dimensions).
Most orbits are unstable in D != 3, so the problem is not as interesting is the short answer. The long answer is that the problem has not been studied as hard.
Not a lot of people know that there is an exact perturbation expansion solution to the three body problem, but it has rather poor convergence…
You mention that, “The Wikipedia entry on Poincaré tells a much less interesting, and less accurate, version of the story.” May I ask why if the story is so woefully inadequate 1) you link to it and 2) you have yet to fix the article?