Three Body Problem and Undecidability

I was reading in the Feynman lectures that for many gravitational problems no exact analytical solution was possible. It is the case that there really "exists no solution" or that someone simply hasn't found a way? Feynman demonstrated however, that the motions could be approximated to any degree of accuracy. It is curious to note that although the planets in principle move about in a determined way, we are unable to determine that way, even in principle....

 

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You dropped an important word in that post, James!

 

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The essential problem is that of the impossibility of writing a multiplicity of simultaneous mathematical equations which can be solved simultaneously.

Computers do not have Powers of Magicke which can do the mathematical impossible. Computers are programmed to provide solutions to such problems by rapidly solving first one body's equation for one moment, then another body's, and so on. The end result is critically dependant upon the sequence of operations. It is ubiquitous for every different sequence to provide a significantly different solution.

 

Wow, CANGAS, you said that as though you know what you're talking about!

It's a shame your conclusion is so completely wrong.

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I really do know what I am talking about. And it is not my conclusion but rather what I was taught in both college math and physics classes about a half a century ago.

If you ever had the faintest inkling of what math and physics really teach you would not make such absurd statements on such a regular basis.

Please provide substantial proof when you claim I am wrong. You have never done so on even one occaision. I have provided decisive proof of your grossly innacurate science statements on more than one occaision.

I do not come to these forums to be heckled and cajoled by amateur science wannabees like you and ( expletive deleted ). You are not much of a scientist or knowledgeable physicist, although you have learned enough of the buzz words to wow the truly ignorant.

Please stop trying to pick fights with me just because you are bored and too stupid to figure out anything else to do.

 

You have a real habit of spouting nonsense without support, CANGAS.
I don't expect you'll ever change.

 

No analytic solution for 3-body. Generally they present complicated phenomena like fractals. You know one electron is electromagnetics and many electrons are thermal physics!

Einstein field equation is based on two body and he believed that it applies to galaxies, to the whole universe. Now you can understand why people talk about dark matter!!!!!!!!!!

 

Pete is the true source of garbage in the claim of science.

Pete can, if he wishes, prove his claims;

Pete claims that the three body problem is solveable by writing multiple simultaneous equations. Pete, write the equations and post them here.

IN CASE YOU DIDN'T HEAR ME, PETE, WE WANT YOU TO WRITE AND POST THE SIMULTANEOUS EQUATIONS FOR THE THREE BODY PROBLEM.

 

Just curious. Please tell us what word he'd dropped. Lol

 

CANGAS has serious comprehension problems.
Nothing new here.

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"not"

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Not true. Problems that have no analytic solution are not solved with computers as you suggest. Look into finite time step approaches.

I.e. the forces (or what ever is controlling the system - for example chemical reaction rates changing with temperature, etc) acting on all components of the system are computed for one instant of time, t1. (not sequentially as you incorrectly suggest). The order of this calculation does not matter as none of the system compontes is alowed to change. Then, AFTER all the "drivers" have been calculated, one projects ahead the state of the ENTIRE system by a small finite "time step" to t2. I.e. the state of the system at t2 is computed for its state ate t1. This sequence of finite time steps is repeated until one reaches the end of the period being investigated. large time step give cheaper but less accurate results.

The smaller the time step (t2-t1) the more expensive is the calculation but it is more accurate, so long as the "rounding errors" associated with all digital machines are not important. Making finite time step calculation has become a science - very sophisticated means* have been developed to greatly improve the accuracy with a fixed time step. These means do require more computations for each time step, but far less computation that the 100s (or 1000s) of smaller time steps the fixed time step would need to be subdivided into to achieve the same accuracy.

I am afraid I must agree with Pete. - Normally you are the best source of error pretending to know what you are speaking of to be found here and especially dangerous as you do the job of pretending well. - Surely often misleading many.
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*The basic math problem in my book, Dark Visitor was a three body problem. Approaching unseen body (probably a black hole), the Sun and the Earth. Object was to find the change in the Earth’s orbit the gravitational impulse the passing Dark Visitor would give to the Earth (a permanent change in Earth's orbit eccentricity** that leads to a interesting new type of Ice Age - Only the Northern Hemisphere is very rapidly covered by thick ice.)

Because I wanted to keep the math so simple that even a high school student could understand, I used a very primitive, but easily understood refinement to the time steps approach. I.e. from the forces acting at t1, I calculated the state at t2, then calculated the forces that were acting at t2. Then I average the t1 and t2 forces to get a reasonably good estimate of the average forces that act DURING the interval between t1 and t2. With this average force, returning to t1 I and again calculated the state at t2. This procedure slightly more than doubles the computation time for each time step, but improves the accuracy by about an order of magnitude so less time or larger time steps could be used. - This was important as I give the code (written in spreadsheet form as one line to be “copied down” to fill many of the spreadsheets rows. - I wanted the reader to explore other gravitational impulses etc.)

More details at web page, under my name, including how to read book for free.
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**The new eccentricity is still small - less than Mars currently has, but because Earth is always just on the edge of an ice age not much is required. Book is actually a vehicle trying to interest non-scientific types in science by teaching them some (without letting that be obvious as all of it is embedded in the story) while hopefully scaring them enough to make them want to know more - presented as if the approaching Dark Visitor has just been detected by its very slight perturbation now present on weakly bound Pluto.

 

Hahaha. Indeed! Thanks.

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BillyT:

Problems with no proper mathematical solution can only be solved by some form of trial and error or successive approximation. Anyone with a realistic understanding of science and engineering has already understood that my statements have been a summary condemnation of any foolish claim that the three body problem has a rigorous solution, and my mention of "solving such problems" referred only to obtaining a workable although actually inaccurate, practical, or, engineering, solution.

Anyone who claims that my statements are not correct about computer operation in solving, in fact, any problem may indulge themself in posting real proof contrary to my statements. Computers are wonderful devices but they perform their work in the dumbest way possible: it is ALL just addition and subtraction being done at lightning speed; if mathematical theory does not give us a way to simultaneously solve too many simultaneous equations ( and two seems to be the universal "speed" limit ), then it does not matter how many sleight of hand tricks a business expert and a science fiction writer speak of: if many centuries of physics and mathematical theory not only fail to give a three body solution but indeed prove to us that it is impossible, I for one agree with five hundred year's worth of theorists, and contravene a businessman and a fiction author.

I know what I am talking about: roughly a half century ago, my college freshman physics prof assigned us an overnight problem of writing simultaneous equations for the electrostic interraction of three identical bodies. After about three hours of trying I gave up and expected to be the goat in the next day's class. Lo and behold, nobody else guessed the way to do it either, and then prof admitted that it was impossible and explained why.

Anyone, spelled ANYONE who had the same experience of being taught about the three body problem in freshman college physics class would not now be so foolish as to give more than a moment's thought to the matter. Because they would have gotten a complete understanding of it long time past.

Pete stays p**ssed off at me because it amuses him to play wannabee scientist even though he does not know enough about physics to avoid posting collosal blunders on a frequent basis, and I have no reason to keep quiet when he looks at a cat and calls it a dog. AND IT HAPPENS AS OFTEN AS NOT.

BillyT is a little more subtle and complicated, and I will not say very much about him, except for mentioning that he also plays wannabee, though on a different scale. Billy; we all want to invent something wonderful and become rich, but if we forget, as you did, that electric field strength drops on the inverse SQUARE, and claim, as you did, that it drops on the inverse CUBE, it is not CANGAS's fault. CANGAS was just the messenger that told you the truth.

Way back once upon a time in this thread, I explained that the three body problem has no exact mathematical solution and that our practical solutions are obtained by using computers to perform successive operations to approximate a workable solutiuon. Pete, in his usual ebbulient but unfounded fashion, boldly claimed that "my conclusions were all totally wrong"

My "conclusions" were:
1. The three body interraction cannot be rigorously mathematically solved, as I have been taught by five hundred years of physics and math.
2. Computers are used to perform fast successive approximations of the state of each of the bodies in order to arrive at a conclusion of their end states. Because computers are perfectly stupid, they act upon the instructions written by wise computer scientists: the decimal round off and the order in which each successive operation is executed always significantly influence the bottom line result.

Businessman wannabee scientist and fie on the fink that says businessman says science garbage.

Paperback writer wannabee rich and famous and fie on the fink that says writer says cube when it is really square.

So gang of two criticizes CANGAS because he sounds like knows what he is talking about. And gang of two says CANGAS is wrong, but gang of two is always too busy with something else to ever actually POST PROOF OF ERROR.

 

Still missing the point, as usual

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Numerical methods, cangas.
The n-body problem can be solved numerically to any desired degree of precision. In practice, the limitation is on how well the initial conditions are known.

 

CANGAS, as someone who's actually used computational methods to model a 250-body L-J interacting gas, might I suggest that you're wrong? Certainly an exact solution is not possible, but that's irrelevant. If we limited physics to the problems that had exact solutions, most areas of physics would never have got off the ground. Statistical physics, for example, uses mean field approximations to deduce properties of fluids, with success in the low density limit. Other approximations are used to get more accurate descriptions of high density fluids.

 

Almost all areas of physics except for textbook examples would never have got off the ground if an exact solution were required.

Certainly not rockets.

And solutions like these to the N-body problem would not exist: http://ssd.jpl.nasa.gov/?ephemerides

 

I assume that you mean you are referring to yourself when you say "missing the point as usual". And I can only agree that Pete is missing the point as usual.

In my initial post I stated that the three body problem is not subject to exact mathematical solution. Does Pete or anyone else disagree?

In my initial post I stated that the three body problem is solved ( implied in a practical engineering sense ) by numerical methods, virtually all of the time by using computers rather than using a roomful of trained chimpanzees, or, at last resort, Pete clones with abacusses. Does Pete or anyone else disagree?

Pete, there are both practical and theoretical limits to the degree of precision that can be acquired by numerical methods and you are insulting the intelligence of anyone in this forum when you plainly claim otherwise. This is very well known to anyone with even an amateur's understanding of physics and mathematics and computer science.

I have been very foolish to have been sucked into another one of your pointless dilettant science wannabee arguements and am at least wise enough to be much more cautious about such a thing in the future. You and all of your groupies may wallow in your ignorance all you please.

 

For 2 body interaction it is possible to write a mathematical solution, for example - a function that will give a position x for any point in time t. So, 2 body interaction has a proper mathematical solution
Obviously, 3 body interaction can only be described using step calculation and addition. The size of the step decides on the accuracy. It is impossible to write a function that will return a bodies position for any given point in time. The position x for t is calculated by calculating all the previous positions and interactions from the starting parameters via an additional algorhythm and usually perfomed by computers. The algorhytm is "played" until moment t is reached at which we check x to see how much was added to it since start.
Reducing the step size makes the order of the sequence less important, but only a infinitely small step and infinite precision floating point calculations will provide total accuracy.

Obviously as Billy T explained, the step calculations accuracy might be improved by all kinds of techniques, but nevertheless, it will always remain step calculation and addition so that really changes nothing regarding the subject at hand.

I don't see any reason for aggro in this topic