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.
