My intuition, which is nowhere near a proof: All 2-body collisions/singularities can be replaced by perfectly elastic bounces, which don't lose any information. Three-body collisions/singularities cannot be "regularized" at all, and hence would appear to "lose information". Since all non-collisions are approximations (more or less) to singularities, it would appear that something much more complex is going on to either insert new information, or delete existing information. I think that one way to attack this problem is via information flow. At 12:28 PM 7/5/03 -0700, R. William Gosper wrote:
Henry>Answer: triple systems _are_ highly unstable, and _do_ fly apart.
But not always. There was a recent claim that three equal masses can "braid" forever, even with perturbations as "large" as 10^-5.
As you point out, the "stable" ones are hierarchical -- there's such a vast scale difference between the masses, that the small perturbations take a long time to build up to something meaningful.
What I wish I could try is two equal masses in tight circles simulating a double mass making a circle of r/2 (r large) with the third mass at r, with the signs of "both" (all three) angular velocities equal. Even assuming small initial noncircularities and nonplanarities, do you think this system will eventually decay? I guess we have to make the Newtonian approximation, else even the two body problem decays.*
How many bodies are necessary/sufficient to disambiguate time? E.g., if we observe sixteen swarm around for a while, whereupon fifteen crystallize into a motionless triangle while the sixteenth departs like a shot, our suspicions might be aroused, especially if it was the white one.
Reconsider now the low angular momentum case, which we see usually flies apart fairly quickly. This seems to disambiguate the sign of time, even though the physics is reversible. I.e., if you time-reverse a "fly-apart" event, two stars in tight orbit are cruising along when this rogue star comes out of nowhere and "collides inelastically" with the pair, converting its kinetic energy to the gravitational potential of the bound triple system. A likely story. Yet that is exactly what happens in the "almost fly-apart" scenario where the single and the pair don't quite escape, and eventually recombine. Furthermore, if you run the "likely story" for a while, it will usually (eventually) fly back apart as in the forward time case. So the apparent time disambiguation seems to be an illusion due to insufficient duration of observation. ?
There's another problem with stars that isn't true of point masses -- you can bring point masses arbitrarily close together, but when you do this with stars, they get pulled apart like taffy.
And if there's no lower limit on how tightly two can orbit, there is no limit on the energy with which the third can be ejected. (Moments after I typed this, the applet obliging exhibited its most violent ejection in my experience--two bodies nearly superposed, with no visible orbital motion, shot off to the left, the third twice as fast to the right, leaving afterimagess about a quarter inch apart. Minor puzzle: Why was there no noticeable vertical vibration in the tight pair proportional to their perceptible horizontal separation? I can think of three explanations: 1. A stroboscopic coincidence; 2. Only one body moves during a given time step; 3. The bodies were displayed superimposed, but at slightly different times, tricking my motion-tracking into seeing an illusory spread. (I failed to check the afterimages before their dimness surpassed mine.) --rwg
*Who did this to Pope? Nature and nature's laws lay hid in night, God said, "Let Newton be," and all was light.
It did not last; the devil howling "Ho! Let Einstein be!" restored the status quo.