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....

No analytic solution does not mean no solution at all. As Feynman says, numerical solutions can easily be calculated (e.g. using a computer).

You dropped an important word in that post, James!

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. :rolleyes:

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

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

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!!!!!!!!!!

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

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.

You dropped an important word in that post, James!

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

CANGAS has serious comprehension problems.
Nothing new here. :rolleyes:

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

"not" :)

...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.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.
------------------------------------
*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.
------------------------------
**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.

"not" :)

Hahaha. Indeed! Thanks. :)

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.
------------------------------------
*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.
------------------------------
**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.


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 :rolleyes:.

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.

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.

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

Still missing the point, as usual :rolleyes:.

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.

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

As an innocent victim, I also have been surprised by the great amount of aggressive animosity unjustifiably heaped upon me, although my personal analysis of the personalities involved has explained it to my satisfaction.

BillyT's method is a valiant effort in terms of a first step toward arriving at an approximation method which could provide workable practical engineering solutions. However, it shares the failure of not being an exact and mathematically rigorous answer to the chronic and seemingly impossible multiple body problem.

But ... isn't the definition of 'analytic solution' rather arbitrary? Even if your solution is x² + y² = r², you'll most likely have to use iterative methods to solve the squareroots :)

Analytic seems generally to mean 'using only commonly-used functions'. Most people can't analytically integrate e^{x^2}, but plug it into Mathematica and it will tell you the answer is a generalised Gamma function.

So technically, if a system of equations has no analytic solution, can't you define 'g' to be a function fitting the system and then say it has analytic solution g? Pointless, but just hypothetically speaking...

As an innocent victim, I also have been surprised by the great amount of aggressive animosity unjustifiably heaped upon me.

Poor innocent victim? Sorry, I don't buy it.

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.

Who is picking the fight here? You, CANGAS, are the one spewing out vitriol. I could post many more examples from this thread and from others. Don't claim to be the innocent victim.

BTW, here are the simultaneous differential equations that completely describe the N-body problem.

Let

m_i be the mass of body i
r_i(t) be the position of i at time t
v_i(t) be the velocity of i at time t

The time derivatives of the states (positions and velocities) of the i bodies at time t are

d/dt(r_i(t)) = v_i(t)
d/dt(v_i(t)) = -sum G*m_j/||r_i(t)-r_j(t)||³ * (r_i(t)-r_j(t))
where the sum for body i is over all bodies j ~= i

Given an set of known conditions r_i(t₀) and v_i(t₀), at some time t₀, the above differential equations can be used to numerically find the states (positions and velocities) of the i bodies at any other time t.

If you knew anything about numerical integration, you wouldn't spout the nonsense you do about the impossibility of solving the N-body problem. Combining numerical integration combined with extended numerical precision gives the ability to solve an initial value problem such as the above to any degree of accuracy desired.

Trivial freshman-level problems have analytic solutions. Most real-world problems do not have analytic solutions. Big deal. They can still be solved, and are solved, every day.

Two bodies are actually a single system. Three bodies are not a single system. This is the difference. A single system can be solved as the relative values between the bodies are one set of values. For example, two bodies have a single relative distance value. Three bodies have 3 relative distance values. A triangle.
Distance between two bodies in a two body system is function of time.
Distance between each two bodies in a three body system is a function of time and distance from the third body.
So, to calculate distance of two bodies in a three body system you need to know the distance from the third body - which is the same function. The functions reference each other circularly. So, this system is impossible to solve simultaneously as each distance function of any two bodies references the other functions.
So, the only way to solve them is to calculate one distance using the "old" distance from the third body. This type of solving is inherently incorrect. But if the calculation is performed cyclicaly in small steps/iterations the "old" distance becomes closer and closer to the "current" distance that should have been used so the calculation becomes more and more accurate. This kind of cyclical calculation is usually perfomed by computers obviously.

So, a solvable analytical solution is impossible due to circular references between the equations.

Nobody has claimed that the N-body problem has an analyic solution. It does have a numeric solution. The two body problem is a toy problem presented in introductory physics courses but never encountered in the real world. Real-world problems hardly ever have analytic solutions. That does not mean they are insoluble.

This thread looks like a shitty semantic argument over the word "solution". Cangas has said that computer methods can be used but necessarily do not produce exact answers. He's then told he's out to lunch and that you can find numercial solutions to as high a degree of accuracy as you like. I'm left wondering what the difference is? I think he even explicitely stated such methods give practical results.

Everyone seems to be in agreement that given an accurate enough initial conditions and enough computing power and time we can find the state of our N-bodies at a given time to within whatever error bounds we wanted. The furthur in time you want to go and the more accurately you want the positions, the more accurately you needed to know the initial conditions and the more computational time required. In practice, useful results can be obtained this way, though it will have it's limits. Everyone also seems to be in agreement that there is no "analytical solution" to the general 3 or higher body problem, whatever this phrase means. So what is this argument about? Whether the numerical approach warrants being called a "solution" without the "numerical" or "approximate" qualifiers? Who really cares what you call it?

I think it would be more usefull to go into the point Zephyr has raised about the phrase "analytic solution" being ambiguous. I'm only familiar with the 'no general solution to the N-body problem' in vague terms and can't answer exactly what is meant by this off the top of my head, though I'm sure there is nothing at all ambiguous about the type of solutions ruled out, the ambiguity is in the vague retellings of the tale. So, if anyone is familiar with the details of the results in this direction, please illuminate us, me at least. This will probably go some distance in clearing up alyosha's initial post.

I agree that this is a shitty argument about semantics. The statement "the three body problem has no general solution" boils down to the same as "the integral of exp(-x*x) cannot be solved with elementary functions".

Some history: The n-body problem was one of the most important fields of study in the 18th and 19th centuries, back in the days when mathematicians and physicists were one and the same. Some of the more notable people who studied this problem include Bernoulli, Euler, Lagrange, and Poincare.

Bottom line of all this work: The general n-body problem cannot be solved in terms of elementary functions. Sundman found a slowly converging infinite series that solves the general problem almost everwhere (the series fails to converge on a set of measure zero). Explicit solutions in terms of elementary functions do exist for some special cases such as the Lagrange points for the restricted three-body problem.

Bottom line of all this work: The general n-body problem cannot be solved in terms of elementary functions.

Finite combinations of them? Or is something like a power series solutions ruled out as well? One converging everywhere that is, namely is an improvement to Sundman's result in this direction ruled out as well?

Explicit solutions in terms of elementary functions do exist for some special cases such as the Lagrange points for the restricted three-body problem.

Some more on specific setups with explicit solutions, including a few pictures (I don't think you need to be logged into the ams site to read it)

http://www.ams.org/notices/200105/fea-montgomery.pdf

I am at fault in this thread, I am afraid. I apologise for picking a fight with CANGAS.

As I said (in an unnecessarily nasty way) it was the conclusion to his first post that I disagreed with:

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.

CANGAS didn't say that computer methods don't provide exact solutions - he essentially said that computer methods provide inconsistent solution. Unfortunately, he misinterpreted my position (not unusual, but in this case the fault was mine) and began arguing over something that there was no argument about.

I am at fault in this thread, I am afraid. I apologise for picking a fight with CANGAS.

As I said (in an unnecessarily nasty way) it was the conclusion to his first post that I disagreed with:



CANGAS didn't say that computer methods don't provide exact solutions - he essentially said that computer methods provide inconsistent solution. Unfortunately, he misinterpreted my position (not unusual, but in this case the fault was mine) and began arguing over something that there was no argument about.


It is so very sweet of Pete to finally recogize that Pete had gone off on a tangent for no reason and has started a fight with CANGAS that was totally unjustified.

It was not the first time and I will be surprised as Hell if it is the last time that Pete does such damage to the integrity of this forum and such damage to the sense of comraderie that SHOULD PREVAIL AMONG WE WHO REALLY DESIRE TO DISCUSS AND DISCOVER SCIENCE AND PHYSICS TRUTH.

WHILE APOLOGIES ARE ALWAYS APPRECIATED, THE HEART OF THE MATTER IS WHETHER SOMEONE WILL, IN FUTURE CASES, BEHAVE WITH A REASONABLE SENSE OF MUTUAL RESPECT AND CIVILITY.

Well, I guess Pete and CANGAS just showed who is the better man with their last two posts.

CANGAS, if you truly want to know about the supposed insolubility of the N-body problem I suggest you read up on Sundman's work, read up on Lagrange points, and read the paper shoe just posted.

Well, I guess Pete and CANGAS just showed who is the better man with their last two posts.

CANGAS, if you truly want to know about the supposed insolubility of the N-body problem I suggest you read up on Sundman's work, read up on Lagrange points, and read the paper shoe just posted.

Write and post simultaneous equations for three bodies and I will trip over my own feet in my eagerness to get to and read your revelation.

You don't even have to carve your three body simultaneous equations on tablets of stone.

Write and post simultaneous equations for three bodies and I will trip over my own feet in my eagerness to get to and read your revelation.

You don't even have to carve your three body simultaneous equations on tablets of stone.

Look at the link I just gave, it has some exact solutions to specific 3-body (and higher) systems. This in no way contradicts the "no general solution" result.

Look at the link I just gave, it has some exact solutions to specific 3-body (and higher) systems. This in no way contradicts the "no general solution" result.


Worrying about the three body problem is not my day job.

I should have expanded upon my post and said "post YOUR three body simultaneous equations AND SHOW US your WORK-OUT OF A SAMPLE SOLUTION, AS SIMPLE AS YOU WISH TO MAKE IT, STEP BY STEP, WITH REAL NUMBERS, and if the results make sense, then I will applaud you and try to nominate you for the Nobel.".

I am off this case and I don't really care how it turns out.

Worrying about the three body problem is not my day job.

I should have expanded upon my post and said "post YOUR three body simultaneous equations AND SHOW US your WORK-OUT OF A SAMPLE SOLUTION, AS SIMPLE AS YOU WISH TO MAKE IT, STEP BY STEP, WITH REAL NUMBERS, and if the results make sense, then I will applaud you and try to nominate you for the Nobel.".

I am off this case and I don't really care how it turns out.

Okie, you've lost me, I have no clue what you're going on about now. Is there some usage of the word "solution" in this thread that hasn't been cleared up yet? D H, myself, or anyone else, aren't claiming anything that would contradict the "no general solution" result, so I don't understand why you want to carry on a shouting match.

Write and post simultaneous equations for three bodies and I will trip over my own feet in my eagerness to get to and read your revelation.

Here is a special-case solution for three point masses. Call the masses of these three objects m₁, m₂, and m₃.

1. Form an equilateral triangle with three point masses with the bases of the triangles equal to some length r.

2. Give each an initial velocity equal to that of a circular orbit around the common center of mass with rotation rate w, where
<img src="http://www.forkosh.com/mimetex.cgi?\omega = \sqrt{\frac{G\,(m_1+m_2+m_3)}{r^3}} ">

The total gravitational acceleration of each of the three bodies toward the other two bodies is always directed from the body in question toward the common center of mass and has magnitude w² times the distance to the center of mass.

In other words, gravitational attraction maintains the circular motion with this special set-up. The triangle rotates as a unit.

This is one of Lagrange's solutions.

Edited to add

CANGAS, there is no need to nominate me for the Nobel prize. These special case solutions have been around for a long time. Many were, in fact, discovered by the very same people who found that no general case solution exists.

...(1) 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....
I know what I am talking about...
BillyT is a little more subtle and complicated, and I will not say very much about him, except for mentioning that he also plays (2) wannabee, though on a different scale. Billy; we all want to (3)invent something wonderful and become rich, but if we forget, (4) 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....
Businessman (2)wannabee scientist ...
(1) From prior experience with you, I knew I would be personally attacked, instead of thanked, for correcting your misunderstanding of how computers are used for three-body problems; but as I did not want your ignorant, pretensuos "explanation" to mislead others as ignorant as you are about this, I explained (in post 13 second paragraph)* why your statement in post 4, reproduced below now, is false.
...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.
(2) I am not a "wannabee." Perhaps I am a "hasbeen" - After getting my Ph.D in physics, I worked as a physicist for a few months shy of 30 years for ONE employer, who was very well pleased with me. - Made me "principle staff" - the highest existing staff level, etc. and allow me to do pretty much whatever I liked most of the time. (Occasionally I was asked to serve on some "tiger teams" when some serious problem came up.) I spent most of my time working with Doctors at Johns Hopkins Hospital or on energy projects of my own design that the US Navy and the DOE gladly funded.

(3) I have approximately 15 patents, several of which are secret, one I can not even tell the title of. I am rich, (working on my third self-made million now) but not from sale of patents. (Sort of an answer to: "If you are so smart, why aren't you rich." As student, I was very poor, waited tables from my meals etc and always on full-need scholarships thru the Ph. D, although in those years I did teach undergraduate physics also.)
If you are referring to my book, Dark Visitor I have made negative income from it as I always tell how to read for free (see how at web site under my name). I wrote it as a science recruiting tool as I am concerned with the loss of technical leadership in the West to Asia, paid to publish it, etc. (All the science in it is hidden in the story as target reader would never knowing open a “science book” on his/her way to a law degree etc.)

(4) I have never said electric field drops off as the cube. It would do so if it is from a dipole (separated equal opposite charges) more distant than where the "near field terms" are important. Provide reference to where I did so - or stop your false slur.

Have you no shame as well as lack manors to thank for corrections?

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*From my post 13 second paragraph:
"... 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 components is allowed 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 from its state at t1. ..."

{speaking of the n-body probllem} Or is something like a power series solutions ruled out as well? One converging everywhere that is, namely is an improvement to Sundman's result in this direction ruled out as well?You are far more advanced than me, but none the less I want to join your comments here and note that many of the so called "analytic" solutions, such as physic's favorates: the simple harmonic oscillator and wave equation, used the sin function in their solutions.

But what is the "sin function?" -

If answer is "an infinite power series," then the accurate solution is "expressible," but not precisely "evaluatable," at least for most times (angles etc).

If answer is "something that has been tabulated" then modern digital computers can is some ways be considered "analytic" also but there is still a huge important distinction:

If the computer "solution" is of the "time step" nature, then the errors accumulate with time, but if the solution is "analytic" they do not. For me this is the really distinguishing difference.

As you are much better informed in this field than me, perhaps you will comment.

"analytic solution" usually means expressible in terms of "known functions". What you count as a known function will depend on your situation. Maybe you include Bessel functions, Gamma, etc, maybe not. Usually anything you include can be numerically approximated very well. "Very well" is another vague term you'd probably quantify with the speed of the algorithm needed to get within a certain bound as a function of the size of the argument.

Rather vague, but there's no universal definition. This is why I had asked what specicifically was excluded from the no general solution to the N-body problem. Anytime you have some sort of "impossible to solve" theorem, you can be sure they are working with a very precise definition of what solutions they are ruling out (like with integration in finite terms or no general solution to quintic or higher polynomials)

"analytic solution" usually means expressible in terms of "known functions". This is why I had asked what specicifically was excluded from the no general solution to the N-body problem.

From Marion, J.B., "Classical Dynamics of Particles and Systems: Second Edition", Academic Press, New York, 1970
The addition of a third body to the system, however in general renders the problem insoluble in finite terms by means of any elementary function.

In other words, the solution is not an elementary function. The problem is soluble, just not using elementary functions. Sundman showed that an integral power series representation must exist in terms of the inverses of the cube roots of the radial distances must exist. A 1915 review of his work is here: http://articles.adsabs.harvard.edu/cgi-bin/nph-iarticle_query?1915Obs....38..429.&amp;data_type=PDF_H IGH&amp;type=PRINTER&amp;filetype=.pdf

The series converges very slowly due to the uncountable number of poles in the problem domain (any path with a collision sometime in the future creates a pole in the expansion). I work on spacecraft trajectory planning, guidance, navigation and control. Nobody that I know uses Sundman's approach.

BTW, CANGAS, Sundmann was awarded the de Pontécoulant's Prize by the French Academy of Science for his work in solving the N-body problem.

Sundman showed that an integral power series representation must exist in terms of the inverses of the cube roots of the radial distances must exist. A 1915 review of his work is here: http://articles.adsabs.harvard.edu/cgi-bin/nph-iarticle_query?1915Obs....38..429.&amp;data_type=PDF_H IGH&amp;type=PRINTER&amp;filetype=.pdf

The series converges very slowly due to the uncountable number of poles in the problem domain (any path with a collision sometime in the future creates a pole in the expansion).

Thanks for the link. When you said earlier that Sundman's series failed to converge on a set of measure zero, I gather this set is just when you have a collision?

The series fails to converge if some projection of the trajectory into the future involves a collision or if the initial state has zero total angular momentum. Since the masses are modeled as point masses, the measure of the paths involving collisions has measure zero. The measure of the initial states with total angular momentum equal to zero is also a set of measure zero.