Talk:N-body problem

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[edit] Name of the page

Why not rename the page "The n-body problem" to avoid the naming convention problem? [anon 1/6/06]

"The two-body problem is simple; its solution is that each body travels along a conic section and their common focus is the center of mass."

This is far from clear. Should it perhaps be: "... each body travels along a conic section whose focus [or: one of whose foci] is the two bodies' common center of mass"?

S.

Perhaps "...each body travels along a conic section which has a focus at the centre of mass of the system". -- Khendon 14:24 Oct 28, 2002 (UTC)


I have added the definition of the n-body problem to the page. Someone should expand the mathematical content a bit further (and perhaps illustrate my rather technical definition). I think the Many-body_problem is useless and the little content on the page should be merged into this page. MathMartin 14:47, 16 May 2004 (UTC)

It's a big decision, since many-body applies to quantum, typically, and N-body to celestial mechanics. Of course many-body should have better content.

Charles Matthews 14:53, 16 May 2004 (UTC)

Ok, then I will not merge the pages. I did not know there was a semantic difference between the many-body-problem and the n-body-problem. MathMartin 14:59, 16 May 2004 (UTC)

I think that the restricted three body problem was invented by Euler, he also gave the first known particular solution for the three body problem. Antonio 18:27, 10 Feb 2007 (UTC)

[edit] Mathematical Formulation

The parameter γ is not defined.

Done Oub 17:27, 8 May 2006 (UTC):

The exponent in the denominator should be a 2; the gravitational force decays with the square of the distance. Unless you intended to describe the N-body problem in 4-dimensions of space :) .

Well, no the formula is
m_j \ddot q_j = \gamma \sum\limits_{k\neq j }^{n} \frac{m_j m_k(q_j-q_k)}{|q_j-q_k|^3}, j=1,\ldots,n \qquad \qquad \qquad (1)
not
m_j \ddot q_j = \gamma \sum\limits_{k\neq j }^{n} \frac{m_j m_k}{|q_j-q_k|^3}, j=1,\ldots,n \qquad \qquad \qquad (1)
in other words it is correct.:) Oub 16:45, 21 May 2007 (UTC):
Ok, now I see. I expected to see a square in the denominator because I was thinking about the inverse square law, and I neglected to recognize that the additional power was to normalize the difference vector in the numerator. Perhaps it would be clearer to state the formula using the normalized difference vector in the numerator, and the square of the distance in the denominator (I would write it out myself, but I don't know to put a "hat" over the vector). Which brings me to another point - shouldn't the order of q_j and q_k be reversed in the subtraction? It looks like this formula defines a repulsive force - that is, unless gamma is negative (there is no description for gamma, despite your having responded otherwise to the initial related comment).

[edit] Sundmans Theorem

Plz someone give me Sundman's series !!! I want to know them!! I cant believe they are not here either ive been looking on the web a lot and nowere to find (read hate >-( ) and put it on here too. Thanks in advance tommy plz send a copy of the series to my email : tommy1729@hotmail.com bye.

I send you an email: are you still interested?Oub 14:57, 21 April 2006 (UTC):

[edit] Cannot Be Solved vs. Impossible to Solve

I've found a bit of disagreement in people I've talked to about the three-body problem. All of them know that no analytical solution is known at the moment, but they disagree as to if an (as yet unknown) solution exists. It's an abscence of proof/proof of abscence thing -- Some simply shrug and say we don't know yet, and others are insistent that it has been proven that there never will be an nalytical solution (like it has been proven that pi is trancendental). Unfortunately, none of them can say where they have heard that. I guess what I'm asking is for more information as to the mathematical status of the problem. -- 15:30, 28 March 2006 (UTC)

You are right. As a matter of fact Sundmans Theorem provides a global solution of the 3 body problem. Still even many textbooks claim that no such solution can exist. This wrong believe goes back to results of Bruns and Poincaré about the non existence of certain integrals. I think that even Poincaré claimed that from his result it would follow that certain perturbation series, for example the Linstedt series would diverge. That proved wrong see Sundman.

Oub 14:57, 21 April 2006 (UTC):


The fact that the Problem becomes analytical when submitted to a certain transformation of the independent variable, does not necessarily mean that you can always find its solution; at least, not by identifying coefficients as in the Frobenius method.

You can find, at most, the coefficients of an asymptotic expression (whether Taylor, Laurent or Puiseux) up to any fixed order, but not up to all orders, unless you fix initial conditions for the solution and its derivative... and your choice allows you to find the pattern. Chaugnar Faugn (talk) 00:38, 6 May 2008 (UTC)

Re: Chaugnar Do you mind indenting your responses? They are easier to read this way. I am not sure I entirely understand your point. But first are we talking about n=3 or n>3, because in a way Wang results is weaker than Sundmans, since no collisions are included. A part form that I would like to understand what your claim is. I don't have both theorems at hand, but basically you get for a large set of initial data (global) solutions which are analytical for all values of t. So it is a convergent power series solution. That was precisely what the Oscar prize was about. Are you saying this is not a real solution or what? Please clarify. Thanks Oub (talk) 14:33, 6 May 2008 (UTC):

[edit] Animated GIF

The animated GIF is way too large -- 2MB! Is there some wikipedia standard for animated things? Flash would be a much better idea (or SVG, though that wouldn't work for many people). ehudshapira 00:11, 14 July 2006 (UTC)

I thought this too, at first. I found that it's explained here, however, that (with regards to images) "You don't have to worry about server disk space and the loadtime of the Wikipedia pages that refer to them, since the software automatically generates and caches smaller (as specified in the articles) versions." Caillan 09:32, 20 August 2006 (UTC)
Later edit: Opps, I see now that 2MB is the size of the image in the article, not just the size of the file in the Commons. I've got a reduced version the image I'll talk to the contributer about replacing the current one. Caillan 09:41, 20 August 2006 (UTC)
It crashes my browser (because it eats up too much memory). Anyways "large" animated GIFs are very poorly handled by browsers... better formats for "large" animations would be QuickTime or RealVideo. --Doc aberdeen 12:03, 27 September 2006 (UTC)


I suspect it of crashing my whole computer. It adds about 200meg to firefox's memory usage.

[edit] Euler

I heard a quote about the three body problem supposedly by Euler: "it was the only problem that made my head ache". —Preceding unsigned comment added by 72.72.107.170 (talk) 16:47, 17 April 2008 (UTC)

That is a story about Newton rather than Euler, surely? Fathead99 (talk) 13:26, 3 June 2008 (UTC)