On 3/16/2013 11:34 AM, Warren D Smith wrote:
It is perhaps worth noting that I proved that Newtonian N-body evolution is undecidable.
To be more precise, if I give you the initial velocities, masses, positions of N point masses (for some finite N) on some input tapes (the tapes are infinite so can store real numbers, most-significant digits first) and I provide two concentric spheres, and two concentric time-intervals, and I ask you: "will (A) any of the bodies enter the inner sphere during the inner time interval, or (B) will every body stay outside of both spheres, throughout the outer time interval?" [Note: I have phrased this in this way so that there is a 3rd physically possible intermediate option (C), and if it occurs the decider is allowed to provide any answer.]
then Theorem: no Turing machine program can answer this question. (But the bodies themselves would answer it.)
Since it's a Turing machine it can only use a finite number of digits from the initial values tape within a finite number of steps. But the bodies themselves are assumed to act as hyper-computers using real numbers? A very nice result.
However, if Newton's laws are replaced by certain more-realistic laws (better approximating special and general relativity) then the problem becomes decidable. Also, I proved in one of my finest results that a quantum version of the problem is decidable, which is quite odd (quantum mechanics computationally easier than classical mechanics).
Are these available online? Brent Meeker
Thus, believers in computability could have told the physicists that Newton's laws had to be wrong (in a superior alternate history).
Another result of mine is that the laws (Navier Stokes) of hydrodynamics for water in a rigid container, also lead to undecidable problems, i.e. "will more than a liter of water end up in this sphere during the next hour?" with same three-way phrasing game as for Newton, is undecidable, OR certain other nasty things happen such as Navier Stokes solution nonexistence or nonuniqueness, or such as Navier Stokes solutions vastly contradicting experiment in certain simple situations. In any of these 3 permitted alternatives, computational fluid dynamics is basically killed -- there will never be an always-rigorous Navier-Stokes solver, and indeed the Navier-Stokes equations are physically wrong and we can devise innocent-looking situations where atomic simulations would simulate reality but Navier-Stokes could not...
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