Talk:N-body problem

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)

Mathematical Formulation

I don't know much about mathematics and/or the N-body problem, but I think there is a mistake in the paragraph above the formula. In the sentence "Given initial ... order system <formula> ...." It says in the beginning both the position and the velocity are represented by $q_{j}$, but my intuition tells me one of these should be $q_{k}$, right? —Preceding unsigned comment added by Wildekoen (talk • contribs) 18:58, 29 July 2009 (UTC)

I have the felling that the section General considerations: solving the n-body problem is erroneous but may be I am wrong. 1. The dimension of the Gallilean group is 6 so the number of independent first integral can hardly be equal to 9. The integrals you consider have relations? Each first integral decreases the dimension by 2 and not by one (see any book on classical mechanics starting from Jacobi or Poincaré Lectures). 30 june 2008, Mo (please apologise if i did not write at the correct spot). —Preceding unsigned comment added by 193.55.10.104 (talk) 13:55, 30 June 2008 (UTC)

The parameter $\gamma$ 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).

The section which describes numerical solutions to the n-body problem needs to be edited for style, there are numerous "we"s and "you"s Wikipedia:Manual_of_Style#First-person_pronouns. The errors described associated with approximate numerical solutions in this section are common for many numerical integration schemes and this discussion should probably be replaced by a link to finite differencing schemes or other relevant techniques. Is it necessary to mention "Vpython" by name? What is it and why include it?

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):

I'm having a hard time finding infomation on Sundman's Theorem. If you have any suggesions on resources I should look at, please list them here. Thanks. —Preceding unsigned comment added by 99.6.253.103 (talk) 19:39, 11 November 2008 (UTC)

Re: talk

Well have a look at the reference I'd say. In any case

• Diacu, F.: The solution of the n-body Problem, The Mathematical Intelligencer,1996,18,p.66–70
• Saari, D.: A visit to the Newtonian n-body Problem via Elementary Complex Variables, American Mathematical Monthly, 1990, 89, 105–119

Saari is more technical Diacu more informative. If you really want all the mathematical details, you should read Sigel/Moser Celestial Mechanics. Oub (talk) 12:28, 12 November 2008 (UTC):

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):

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.

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)

Suggestion

Can we say in the article whether the problem has been solved or not. Also can someone please explain exactly what the problem is. Seems pretty easy if you are trying to find the movements of n bodies. A computer can work it out. Obviously I'm missing something (as well as the article). 118.208.184.222 (talk) 08:21, 21 September 2008 (UTC)

Re User talk:118.208.184.222

1. Please create an account and log in.
2. Has the problem been solved? It depends what the problem is and what one is willing to consider as a solution.

Take the King Oscars price (section 4). The announcement states clearly

Given a system of arbitrarily many mass points that attract each according to Newton's law, under the assumption that no two points ever collide, try to find a representation of the coordinates of each point as a series in a variable that is some known function of time and for all of whose values the series converges uniformly.

So given that announcement, this problem was clearly solved by Sundman (n<=3) and later by Wang (n.3) (Singularities apart: Collisions singularities have been proved to be non generic and it has been conjectured by Saari, although not proven, that non-collisions singularities are non generic as well). So all that is pretty satisfactory, although may be from an esthetic point of view one would like to have a real restriction on the initial data: say in form of an inequality, so that all solutions corresponding to these data do not develop singularities. However such a theorem does not exist. So far so god. Now if you ask whether one could calculate , using these results, the positions say of certain meteors for the next 20 years, in order to be sure that the don't collide with earth then the answer is NO . Now with respect to numeric calculations. Keep in mind:

1. The computer can only calculate for a certain time interval not for all times .
2. the computer does not solve the correct equations but solves approximations. Usually for ordinary differential equations it is known that as finer the grid gets the better the approximation becomes. However in the case of possible singularities and instabilities this is far from being clear. So in short the computer is useful but does not solve the problem.

So to summarise, the problem is solved but the solution is not very useful, the issue is to find a better one.Oub (talk) 13:53, 21 September 2008 (UTC):

The problem is hard because the if the system must be solve as a function of t, with respect to what t=0 will that t be solved? There seems to be be an infinity of possible arrangements that a system may evolve (or have evolved from), but to find an equation that corresponds to a continuity of all of them must involve solution data of greater hierarchy than the variables to be predicted. Because of this, I do not believe that we cannot solve the n-body problem by solving for t. Are well known physical dimensions enough to characterize this greater hierarchy? Do conservation laws of momentum, angular momentum, and energy have to be merged into a singular concept of conservation (of different type of physical dimension) in order to solve this problem? This to me seems very likely.Kmarinas86 ^(6sin8karma) 16:52, 20 July 2009 (UTC)

[From Terry0051] The problem does not have to be solved as a function of t, in principle another independent variable can be defined and used, and historically this has been done. Also, a useful distinction to make when talking about solutions is the distinction between an exact solution and an approximate solution, e.g. a solution by series approximations, which can be taken to any desired level of accuracy, if the series is continued long enough before truncation. Terry0051 (talk) 12:41, 21 July 2009 (UTC)

But what is the n-body problem?

Great page, if you are a physicist, but the page means nothing to the average reader. Please may I suggest that someone adds an initial paragraph stating in plain English exactly what the n-body problem is. Is it that if there are more than three physical objects all moving independently in different trajectories, then it is very difficult to work out how to get from one to the other? If this is the problem, let's say so. If it isn't, then please say what it is. Thanks. —Preceding unsigned comment added by NOKESS (talk • contribs) 15:39, 20 July 2009 (UTC)

[From Terry0051] See recent edit which attempts to clarify the opening statement of the problem. Terry0051 (talk) 12:41, 21 July 2009 (UTC)

Please follow the standard convention: the most recent comments are at the end of the page not at the beginning. This is the way books are read. As for the problem, Terry already tried to clarify. Let me add some remarks. You can think of n bodies idealised as point particles that is they don't have a spatial extension (say the stars at the sky) these body have to satisfy the laws of motion, namely Newton's law which I am not going to recall here. Moreover these bodies attract each other by the law of gravitation. So the question is for a given initial configuration (initial data) say the bodies are all in a line or in a circle or whatsoever, and have a certain velocity, how will this configuration evolve, will it collapse, will the bodies rotate around each other, will they move chaotically etc etc. You can even express it in a simpler form: try to predict the movement of all stars in the universe. Think of the 2 body problem. There you get an idea what one is looking for: an explicit expression which allow to predict the movement of the particles. Did this clarify your question? Oub (talk) 16:41, 21 July 2009 (UTC):

It's even harder to understand what the n-body problem is when you ignore the conventions of English grammar. (I did understand it after a few re-readings, however. Thanks.) --V2Blast (talk) 06:56, 22 July 2009 (UTC)

A new formulation of the introduction.

Hello

I am not very fond of the new introduction of the $n$ problem. First of all it is not a class of problems. There are several parameters if you want such as number of bodies, their masses and there initial data, but still it is not a class of problems it is one problem and this is the notation used in the mathematical and physical literature. Also this paragraphs fails to explain why it is a class of problems.

I am also not so sure that the new formulation is clearer than the old one

The n-body problem is the problem of finding, given the initial positions, masses, and velocities of n bodies, their subsequent motions as determined by classical mechanics, i.e., Newton's laws of motion and Newton's law of gravity.

But maybe some remarks would be good.

So what about the following:

To understand the motion of celestial bodies, the sun, planets and the visible stars has been the main motivation for the $n$-body problem. The first complete mathematical formulation of this problem appeared in Newton's Principia (the $n$-body problem in General Relativity is considerably more difficult (citation needed)). Since gravity was responsible for the motion of planets and stars, Newton had to express gravitational interactions in terms of differential equations. An important fact, which Newton proved in the Principia, is that celestial bodies can be modeled as point masses.

Informal version of the Newton n-body problem

The physical problem can be informally stated as:

Given only the present positions and velocities of a group of celestial bodies, predict their motions for all future time and deduce them for all past time. More precisely, consider $n$ point masses $m_{1}$, $m_{2}$ in three--dimensional (physical) space. Suppose that the force of attraction experienced between each pair of particles is Newtonian. Then, if the initial positions in space and initial velocities are specified for every particle at some present instant $t_{0}$, determine the position of each particle at every future (or past) moment of time. In mathematical terms, this means to find a global solution of the initial value problem for the differential equations describing the $n$-body problem.

Oub (talk) 15:10, 13 October 2009 (UTC):