Hi all, I should start by saying I'm an amateur at mathematics so please excuse me if the question seems a bit daft. But I'm just reading a book about chaos and complexity theory (John Gribbin: Simply Complexity) and it brought up a problem that I've certainly heard of before; the 'three body problem'. Where (for example) 3 planets are exerting forces on each other simultaneously and hence moving accordingly. I understand that these problems become very complicated very quickly and so any problem with n+ bodies involved may become practically impossible to solve. But the book seems to imply (unless I've misunderstood), that they are literally impossible to solve - even if all the variables are known to complete accuracy. A particular example is a problem where one (let's say) billiard ball hits two others simultaneously. Gribbin says determinism breaks down in this example, and even a computer (or deity or whatever you prefer) with full knowledge of all the variables would not be able to solve this. Is this the case?! If so, why? Or have I just misunderstood (and again, if so, why)? Look forward to hearing back from someone who can help explain this to me. Thanks
I think his augment is based on the chaos idea of not being able to specify the initial conditions precisely. Computers have limited precision. Hence, any inputs after n iterations would begin to go astray and differ from the actual system you're trying to model.
Suppose you have a billiard ball collision where the sequence of events dramatically affects the outcome. If the cue ball hits ball A and then ball B, the result is different from that obtained when it hits ball B then ball A. As the time between the two collisions becomes shorter and shorter the required accuracy of your initial measurements and of your calculations becomes greater and greater. If the collisions are actually simultaneous (which is an event with zero probability) then the laws of classical mechanics do not predict a unique outcome. But that is not physical. Before you can get to the point where the collisions are simultaneous and laws of physics fail to predict an outcome you must first have approximated the billiard balls as ideal perfectly rigid spheres which act on each other with perfectly impulsive forces. In reality they are not rigid, the forces are not impulsive and collisions take non-zero time. If you remove that problem by making the model more accurate then you eventually run into the realm where quantum mechanical effects intrude. That runs squarely into the problem that perfect measurements of position and momentum are impossible. [But it is likely that you will have run into real world measurement problems and real world limitations on computational resources long before such theoretical problems become important]
n body problems with n > 2 cannot be solved other than numerically. This is in contrast with 2 body problem where closed form solutions (conic sections) can be derived explicitly.
The billiards ball problem is easy. Those multi-body rigid body collisions means the system is not Lipschitz continuous. A system that fails to satisfy the Lipschitz continuity condition doesn't have a unique solution by the Picard–Lindelöf_theorem. Those rigid body collisions are an idealization, so the derivatives aren't quite singular. However, they are nonetheless so sharp that chaos ensues rather quickly. This is an unstable chaos. If you could repeat the same setup as close as possible many times over, you will find the balls ending in very different states. If there are pockets in the pool table, it's game over. Sinking a ball is a singularity that violates Lipschitz continuity in a very big way. The n-body problem typically pertains to gravitation, and hopefully a gravitational system where collisions don't occur. (Predictability is a lost cause when planets collide.) There's another kink that can make a n-body system fail to satisfy the Lipschitz continuity, and that's planetary ejections. There are a number of resonances between the orbits of the planets. One of them is between Jupiter, Venus, and Mercury. Mercury's eccentricity is gradually increasing because of this. In a few billion years, Mercury's orbit may well cross Venus's orbit. Mercury and Venus may collide, or they might do a series of gravitational slingshots. That might put make both planets have an eccentricity high enough to cross Earth's orbit, and any of the three innermost planets could take out Mars. The ensuing chaos might wipe out the inner solar system. Planets might be ejected, or they might collide and combine, or they might collide and obliterate one another. Or it might not happen at all. A change in initial position much smaller than the size of an atom can mean the difference between the inner solar system being wiped out in three or four billion years, or everything remaining hunky-dory (until the Sun burns out it's hydrogen and turns into a red giant).
Thanks guys! Really interesting stuff there. Sounds like there's a few things I need to read up on to really get a clear idea, but thanks for pointing me in the right direction. I must admit, I think I'm thinking of the problem in a more philosophical/pure maths kind of a way. I can certainly understand why quantum effects would prevent two billiard balls ever being struck literally simultaneously. I think the main thing I'm trying to understand is the purely hypothetical scenario of a collision between idealised perfectly rigid spheres. Or in the example of the planets representing the n body problem, even though I understand that the basic idea that it requires a practically impossible degree of accuracy in the measurements to set as the initial conditions in the equations. And hence the calculation will become less and less reliable after successive iterations. What I was really wondering was, if it were possible, to know these details to 100% accuracy, if you fed those completely accurate initial conditions into the set of differential equations that should in theory dictate the relevant positions after any given time interval, would this not (purely in theory) allow you to work out exactly how these bodies would act indefinitely into the future? My understanding of chaos (again, just a laymen's understanding) was that it dealt with situations that where slight changes in initial conditions would have dramatic effects. But that didn't mean it was fundamentally impossible to to predict the way these scenarios would unfold if one did (somehow) know the initial conditions precisely.
From my recollection of chaos in dealing with swinging building dynamics. Linear solutions could predict the angular envelope of the swing and the periodic motion whereas as in non-linear solutions you could predict the angular envelope of the swing (which was always larger than the linear swing) but the actual motion was never fully predictable. In a totally non-linear random solution even the envelope was not predictable. For orbiting systems, the same analogy would apply in a 2-body solution you'd know the actual orbit and for 3-body chaotic soultions you'd know the inner and outer range of an unpredictable orbit.