# 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 ${\displaystyle q_{j}}$, but my intuition tells me one of these should be ${\displaystyle q_{k}}$, right? —Preceding unsigned comment added by Wildekoen (talkcontribs) 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 ${\displaystyle \gamma }$ 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
${\displaystyle m_{j}{\ddot {q}}_{j}=\gamma \sum \limits _{k\neq j}^{n}{\frac {m_{j}m_{k}(q_{j}-q_{k})}{|q_{j}-q_{k}|^{3}}},j=1,\ldots ,n\qquad \qquad \qquad (1)}$
not
${\displaystyle m_{j}{\ddot {q}}_{j}=\gamma \sum \limits _{k\neq j}^{n}{\frac {m_{j}m_{k}}{|q_{j}-q_{k}|^{3}}},j=1,\ldots ,n\qquad \qquad \qquad (1)}$
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
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 (talkcontribs) 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 ${\displaystyle n}$ 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.

To understand the motion of celestial bodies, the sun, planets and the visible stars has been the main motivation for the ${\displaystyle n}$-body problem. The first complete mathematical formulation of this problem appeared in Newton's Principia (the ${\displaystyle n}$-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 ${\displaystyle n}$ point masses ${\displaystyle m_{1}}$, ${\displaystyle m_{2}}$ 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 ${\displaystyle t_{0}}$, 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 ${\displaystyle n}$-body problem.

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

I'm not an expert in the field, but this paper casts a very serious doubt on the originality of the paper by Quidong Wang: [1]. I was not able to find any couter-argument browsing the web. --Fioravante Patrone en (talk) 17:48, 17 October 2009 (UTC)

Re: Fioravante This looks serious. The article is published in the same journal as Wangs. The issue is die Babadzanjanz prove really the global existence theorem in his 78/79 papers. I can't access the 79 paper of Celestial Mechanics. May be best would be to contact an expert in the field. Oub (talk) 11:25, 20 October 2009 (UTC):
[From Terry0051] The Babadzanjanz 1979 paper is accessible here. The trouble in locating it was probably due to an issue of name-transliteration (from the Russian or other original) in the English indexing. Terry0051 (talk) 12:28, 20 October 2009 (UTC)
Re: Terry0051 Excellent, thanks I will have a look. Oub (talk) 12:58, 20 October 2009 (UTC):
I read the paper although not in full detail and I "talked" with an expert in the field, who confirmed what I suspected: Babadzanjanz did not prove global existence in his paper from 1979, he does it in the paper from 1992 making reference to the 1979 article. So it seems to me that he could have proven that result but he did not. Oub (talk) 14:38, 14 December 2009 (UTC):
Ok, thanks for the "investigation" ;-) --Fioravante Patrone en (talk) 16:24, 14 December 2009 (UTC)
It is not true: all the principal results described in "On the Global Solution of the n-body problem" one can find

(using the instructions given there) in papers 1978, 1979 by Babadzanjanz. All three papers by Babadzanjanz and his doctorate thesis (in russian, 1985) one can read here http://www.apmath.spbu.ru/ru/staff/babadzhanyants/ . Poupycheva —Preceding unsigned comment added by 195.19.226.194 (talk) 16:47, 17 December 2009 (UTC)

## Recent large change

Wolfkeeper recently made a large edit to the section entitled "General Considerations: solving the n-body problem. While I am not sure whether or not this is a valid change, I do not believe that this has been discussed, as any major edit warrants. I will not revert this, but I encourage some comment on this action.Jame§ugrono 22:43, 4 November 2009 (UTC)

[From Terry0051] I agree. Other editors look for some good WP reason given for jettisoning subject matter. And I doubt if it's encyclopedic style to say 'of course' about anything, this begs the question whether there are relevant reliable sources. If an editor wants to shorten the material, it perhaps wouldn't be too much trouble to find and cite suitable sources, e.g. on computer implementation of approximate solutions. I think there's likely to be plenty around, but the readers need the facts, not just an 'of course'! Terry0051 (talk) 23:20, 4 November 2009 (UTC)
The nature and phraseology of the material didn't seem to be in keeping with the wikipedia's general structure. To be honest this is a large subject.- (User) Wolfkeeper (Talk) 23:53, 4 November 2009 (UTC)

## Solved or open?

Reading this article has left me confused: is this problem solved or not? On one hand the article states: " or n ≥ 3 very little is known about the n-body problem. The case n = 3 was most studied and for many results can be generalized to larger n. " On the other hand it says:

"In 1912, the Finnish mathematician Karl Fritiof Sundman proved that there exists a series solution in powers of t for the 3-body problem" and "Finally Sundman's result was generalized to the case of n>3 bodies by Q. Wang in the 1990s".

So what did they prove? Just that the solutions exist, but they are not possible to find?

--78.157.76.143 (talk) 20:51, 23 January 2010 (UTC)

RE: could you please sign your comments? Sorry for the confusion (that is most likely be caused by the fact that there are many authors contributing). First of all the solution found by Sundman (and Wang) is a constructive one. That is they don't just prove the existence of the solution, no they even give a constructive method, moreover they do it precisely in the spirit of the King Oscar prize, that is the solution is represented as a convergent power series. What came out as a surprise was the fact that the convergence is so slow that it is practically useless for obtaining quantitative or qualitative results and one again is stuck with numerics. Hope this helps. Oub (talk) 09:50, 25 January 2010 (UTC):

## Has anybody pursuited the orbital resonance?

The article Orbital resonance is talking about this phenomenon as an experimental fact. I wonder whether it was studied theoretically? I have a prejudice that there may be an analytical expression for the resonance. --Wladik Derevianko (talk) 20:47, 13 June 2011 (UTC)

Why bring this up here rather than in Talk:Orbital resonance? —Tamfang (talk) 18:58, 14 June 2011 (UTC)
Because resonance is a particular solution of N-body problem. In my opinion the two articles should be cross-referenced.--Wladik Derevianko (talk) 06:44, 15 June 2011 (UTC)

## Lagrange

The Article currently includes "The restricted problem (both circular and elliptical) was worked on extensively by many famous mathematicians and physicists, notably Lagrange in the 18th century ... ".

Minor point - "Euler and Lagrange" ?

Major point - Lagrange's Essai sur le Problème des Trois Corps dealt with the full three-body problem (translation: http://www.merlyn.demon.co.uk/essai-3c.htm) - no restriction to circularity or to one body being a mere particle. Euler dealt with three masses in a line, I believe. If that is Lagrange's only relevant work, the statement needs correction. Otherwise, a URL is needed.

94.30.84.71 (talk) 20:16, 16 August 2011 (UTC)

## Descriptions of 1767 & 1772 papers; general on dates

Article has : "In 1767 Euler found the collinear periodic orbits, in which three bodies of any masses move such that they oscillate along a rotation line." Euler specifically stated that the line is a fixed line = the title of the 1767 work is "De motu rectilineo ...". I recall no mention of oscillation, and do not see how it can occur in rectilinear motion. There is a very rough translation of that work at http://www.merlyn.demon.co.uk/euler327.htm. 94.30.84.71 (talk) 11:00, 28 August 2011 (UTC)

Article has : "In 1772 Lagrange discovered some periodic solutions which lie at the vertices of a rotating equilateral triangle that shrinks and expands periodically." Lagrange did not write, and the solution does not require, either "rotating" or "periodically". He wrote "conic section", which includes hyperbolic and parabolic solutions, which are not periodic, and includes non-rotating solutions, in which the bodies fall to the barycentre. There is a reasonable translation of that paper at http://www.merlyn.demon.co.uk/essai-3c.htm. 94.30.84.71 (talk) 11:00, 28 August 2011 (UTC)

General : "in date ... discovered" is commpnly unsustainable, at least for early work. The facts may have been discovered in one year, read to a learned society in a later year, and printed on an even later year. Normally, only "published" or "announced" can be accurate. 94.30.84.71 (talk) 11:00, 28 August 2011 (UTC)

## Suggestion for lede

On this talk page several people have asked "Is this problem solved or not?" At one point a partial answer is given as "So to summarize, the problem is solved but the solution is not very useful, the issue is to find a better one." Perhaps this question keeps arising because it is not answered in the lede. Judging by the rest of the article, the answer is complicated because it depends on n=3 versus n>3, whether collisions are assumed away, whether special cases have been solved, and (if I understand correctly) precisely what is meant by "solved" (and in particular depends on the distinction between being solved in principle and being solved in a way that is useful for computer implementation).

Since the non-expert reader will want most of all to know (1) specifically what is the n-body problem, and (2) has it been solved, and since the lede only covers the first of these, I suggest that someone write a paragraph in the lede addressing whether and in what sense it has been solved. Duoduoduo (talk) 17:45, 13 December 2011 (UTC)

## What do we know about when it converges to an attractor?

For special cases or the general case, what if anything do we know about whether there exists a positive-measure range of initial conditions that converge to an attractor? I.e., in the absence of collisions, is it known that from some initial conditions none of the objects ever gets flung away to infinity? I think this should be dealt with in the article. Duoduoduo (talk) 17:45, 13 December 2011 (UTC)

## Numerical Integration

"Numerical integration is O(N2), but tree structured algorithms can improve this to O(n log(n))."

Fast Multipole Alorithms have a complexity O(N), or did I miss something? -- Crypticalcode 15:39, 12 January 2012 (UTC)

## Theology aspect

The article lacks a theology aspect. Apparently K.F. Sundman's result suggests that an infinitely powerful being, able to calculate anything by just saying "let"s see", would see the Universe as a clock-work machanism, while humans and their finitely powerful computers would see it as chaotic. I would guess that dichotomy was not lost to science-aware theologicians of the 20th century? 91.82.37.38 (talk) 21:16, 10 March 2013 (UTC)

## status of the theoretical solution?

On this page, there is a section with the headline "theoretical solution". The solution method which is described therein seems quite straightforward (at least conceptually). After reading through, however, I'm confused: does the series solution one obtains actually converge for a certain time interval (given some valid initial conditions)? If so, I don't understand what Sundman's series in powers of t^(1/3) is about: we already have an easier series consisting of powers of t that does the job. As soon as the series diverges we either have a collision or else you can start a new series expansion at this point. 193.190.253.144 (talk) 10:38, 2 September 2013 (UTC)

## Re-write

I have taken the liberty to edit the whole page, starting from the opening paragraph. I think there are several paragraph sections which desperately need clarification to say the least; or maybe they should be just removed (?), beings there is now a closed-form solution for the reactive forces. Real editing doesn't begin until the second part of the Theoretical Solution (HOOKEAN Switches...). Most being developed, although simple, is new, so any help or suggestions would be greatly appreciated. And there is still much work to do: add a numerical example; add a short FORTRAN source code to calculate inertia and its inverse, and the reactive forces; discuss the Solar System model; and maybe more. Talk to me. (rudrene)

## Source code

I don't like the way the source code looks and I'm not sure it's appropriate to the article. Using images makes it look bad and readers can't copy it electronically. I'd suggest we remove the code, have an paragraph of prose describing it and, if we can find one, a link to source code if we're going to have it at all. RJFJR (talk) 02:08, 20 September 2013 (UTC)

I went bold and removed the section (which I've placed below). Let's discuss the formatting before we consider put it back.

### Compute Program

The FORTRAN source code (4 pages) is a computerization of the above general n-body Problem given above.

File:The Force FOR 01.jpg
The Force Code, 1
File:The Force FOR 02.jpg
The Force Code, 2

File:The Force FOR 03.jpg
The Force Code, 3
File:The Force FOR 04.jpg
The Force Code, 4

I'm not sure this much code, in a particular language, belongs in an encyclopedia. (Is this more wikisource material?) Using images don't seem to me to be the best way to do this. Having it run down both sides of the text doesn't look good and is distracting and confusing. Is Fortran the best language (it isn't as much a lingua-franca as it used to be)? RJFJR (talk) 16:27, 20 September 2013 (UTC)

(oh. And I just saw that someone has nominated the images for deletion.) RJFJR (talk) 16:30, 20 September 2013 (UTC)

## Where did Newton say it was unsolvable?

Where did Newton say this: "Newton actually stated the solution to the n-body problem was unsolvable"? Was it in the Principia or was it elsewhere? RJFJR (talk) 14:22, 20 September 2013 (UTC)

## Warning?

What is this?

== Warning == Do not use or employ a rigid-body solution (A's = 1.0) to an elastic structure (σ = εE) -- the results will be wrong.

And why does it need to be a whole section? RJFJR (talk) 14:09, 23 September 2013 (UTC)

## Wrong?

I removed this from the article:

"This is not only informal, it is wrong. Wrong because Newton, by employing three orbital points (positions) for any planet -- obtained from John Flamsteed [1] -- could produce an equation describing the motion of its orbit. This was true all except for the two outermost planets (known then), whose orbits even today have to be constantly updated."

I don't see how this makes the statement that preceded it wrong. RJFJR (talk) 14:47, 25 September 2013 (UTC)

## Needs rewriting

Thanks to RJFJR for pointing this article out here Wikipedia talk:WikiProject Physics#n-body problem being heavily revised, additional opinion appreciated.

The new revisions may be in good faith, it needs a lot of work in rewriting and trimming... Especially for topics which already have main articles like the Two-body problem and Newton's law of gravity even more so. I don't have time now but will have a look in more detail tomorrow. M∧Ŝc2ħεИτlk 19:55, 25 September 2013 (UTC)

I agree with your assessment. There are a number of articles that overlap with this one and it would probably be good to reduce duplication. Some extra articles worth considering in a future cleanup are
and the somewhat related
--Mark viking (talk) 20:24, 25 September 2013 (UTC)
Thanks for providing these links, they will help. M∧Ŝc2ħεИτlk 08:41, 26 September 2013 (UTC)

## Misconceptions of Newton's third law

I was surprised to find in places that Newton's third law is apparently "Fij = Fji" (where Fij is the force on particle i by particle j). By this logic, the forces each particle exerts on each other are the same, not oppositely directed!

Even more strange, it was written that this leads to "two independent equations for each particle", despite the fact that the forces are apparently the same.

There are not "2n" independent equations from Newton's third law, since the forces are only negated.

I hopefully cleared this up. M∧Ŝc2ħεИτlk 08:40, 26 September 2013 (UTC)

## Hookean switches??

I have no clue what this section N-body problem#Hookean switches and springs, point-masses and the inertia tensor is on about. Some of it even looks like OR. What exactly does:

"Hookean Switches: Let there be a vector function, 0 ≤ Aζi ≤ 1"

even mean if A is a vector or tensor? What are the "Hookean switches"?

I tried to clarify the section, but failed because it is so incoherent and hard to read. I'll try again to interpret the entire meaning of the section. In the meantime, if someone wants to simplify the section, please don't hold back! M∧Ŝc2ħεИτlk 10:37, 26 September 2013 (UTC)

## Newton's Third Law and Hookean switches

• I left out the sign in the third law (obviously, I'm surprised that wasn't realized). I was still in the process of input and cleaning the page up when you came along. The largest task yet to complete is the Numerical Example: the coordinates were rotated just after calculating the inertia tensor and just prior to applying the loads: and the last two equation reflect that. Take a look at the numerical example and see how the A-operators work. I put in an numerical example for all those who couldn't or can't follow the math.
• In the general Law of Universal Gravitation section I was trying to show how it broke down for the n-body problem and was the wrong approach with n > 3 or 4. Too bad you removed it. Some of what you added is good.
• It is too bad you removed the concept of mappings of space (geometry) phenomenon to force and mass phenomena.
• Hookean switches are similar to grid point release in a FEM: There the DOF are restricted (from spinning) by freezing them to the "wall"; here Hookean switches remove DOF (new concept), allowing only feasible DOF. For example, if a problem is planer the all the out-of-plane DOF are removed (Az'k''s = 0). Hookean switches for rigid bodies problems are logical on-off switches. "A" is a vector, so "s" is a vector. Think of it as a 3D rivet analysis, where some fasteners resist load only in one direction -- the A's remove all the other directions from the equations. Mathematically the "A's" components are mathematical tools that greatly simplify the equations of motion -- that's all. It should be obvious what these operators do.
• I'm surprise you find it incoherent. You should have left the section alone until you understood what it was saying...
• We are not on the same page and I don't think we are going to get there with the current reviewer, so I think I should just bow out. It's not worth the effort. It's too bad because this solution really works.24.184.180.233 (talk) 15:16, 26 September 2013 (UTC)

Second I was simply trying to clarify the text and equations in good faith. The whole article is cluttered and needs to be written in clearer, plain english, to the point, not dwell on irrelevancies which can and should be in the main articles, not littered with different notations for the same thing (for example: r, x, P, and q were all used for position, there were some awkward notations for derivatives and labels for particles, why use zeta ζ for a label when you could just use k or something else?).

Still, I apologize for any offense, by all means clean the article up if you are going to. I'll stay off the page from now on. M∧Ŝc2ħεИτlk 15:33, 26 September 2013 (UTC)

In response to some of the points above, here is an earlier version of the section: n-body problem#Classical Mathematical formulation before I started to edit:

Classical Mathematical formulation

Since gravity is responsible Newton reasoned for the motions of planets and stars, he expressed their gravitational interactions in terms of his Law of Universal Gravitation, but soon found, and like everyone else since, it was not a general solution to the n-body problem (see below). However, his Law was the catalysis which started the modern quest for the solution to the n-body problem.

The Law of Universal Gravitation[2] is expressed in scalar form as:

${\displaystyle {\vec {F}}={\frac {Gm_{1}m_{2}}{r^{3}}}{\vec {r}}\qquad (1)}$

where F, a scalar, is, by Newton's 3rd Law, the force between the two masses; m1 and m2 are two mass bodies attracted to each other, assumed reduced to point-masses; and r, a scalar too, is the distance between the masses. Newton proved in his Principia a spherically-symmetric body can be modeled as a point-mass.

Notice the difference between the scalar form and the vector form:

${\displaystyle {\vec {F}}={\frac {Gm_{1}m_{2}}{r^{3}}}{\vec {r}}\qquad (1)}$

where both F and r are vectors giving both magnitude and direction in some pre-defined Euclidean coordinate system.

The constant G is special, it is the gravitational constant. G, the constant of proportionality, is the parameter making Newton's Law emperialistic (i.e., capable of being verified or disproved by observation or experiment). To this purpose see Lord Cavendish's experiment measuring of G. Then what is G physically? G allows mass (matter phenomenon) when combined with distance (spatial phenomenon) to be linked to force ("force" phenomenon). All three phenomena have different units of measurement. Newton's equation is empirical because G is empirical, but (m1m2/r3)r is not a force and so G(m1m2/r3)r is not equal to Fg (gravitational force) -- but only equivalent to it (and that only by assuming we know what "force" is: in this case it is a field force). His equation establishes only an equivalence relationship via G: the equation maps the mass and spatial phenomena -to- force. As will be explain below, force is a functional.

G allows Newton's 2nd Law and all others too, to not require incorporating any constants of proportionality.

Either the scalar or vector forms can be set equal to Newton's 2nd Law, F = m a , where a is the directed acceleration. Notice the absents of a constant of proportional. Continuing with this purpose of equating and using the vector form, let F ij = f(mi,mj,rij), where i and j are integers, 1, 2, 3,... Otherwise, F is a function of mass and geometry. Then to employ Newton's Law Universal, let F12 = (Gm1m2/r3) r be the force on m1 owning to mass m2; likewise, let F 21 =(Gm1m2/r3)r be the force on m2 owning to m1. This way one arrives at two separate equations, and by Newton's 3rd Law, F 12 = F 21 (see the Two-body Problem below). Imagine if there were three masses involved: there would be six separate equations, F12, F21, F13, F31, F23, F 32. If four masses (or more) were involved there would be a mathematical mess.

Newton's Law of Universal Gravitation is generally considered applying only to celestial objects, but is equally useful for Earth science (calculating Ocean tides, calculating the 13-mile high ellipsoidal budge at the equator, calculating the fuel requirements verses weight for rockets, etc.). For those applications there is the "Laws of Weights".[3] Bodies weigh most at the surface of the Earth of course. Below the surface the weight decreases as the distance to the center decreases linearly; above the surface the weight decrease as to the square of the distance increases (Newton's Law).

Newton thought in terms of a change in momenta ∆m''v; where as Euler, for changing parameters as a function of other variables, thought in calculus terms, and momenta became a rate of change between variables, like F = m''a = m d2r/dt2. This was not just a change in notation started by G. W. Leibniz (1646-1716) but a change in concept. In calculus terms Newton's notation indicated a rate of change in variables—with variables wearing hat-like dots (for, say, a change in velocity with one dot), double dots for acceleration a. Newton's momentum became Euler's Eulerian acceleration with a = d2r/dt2. In what follows Newton's notation is used.

Most textbooks present the following general solution approach for the n-body problem: it means in mathematical terms finding a global solution describing the ${\displaystyle n}$-body problem of celestial mechanics, which is an initial-value problem for ordinary, second order, differential equations. Given initial values for the positions ${\displaystyle \mathbf {q} _{j}(0)}$ and velocities ${\displaystyle {\dot {\mathbf {q} }}_{j}(0)}$ of n particles (j = 1,...,n) with ${\displaystyle \mathbf {q} _{j}(0)\neq \mathbf {q} _{k}(0)}$ for all mutually distinct j and k , find the solution of the second order system:

${\displaystyle m_{j}{\ddot {\mathbf {q} }}_{j}=G\sum \limits _{k\neq j}{\frac {m_{j}m_{k}(\mathbf {q} _{k}-\mathbf {q} _{j})}{|\mathbf {q} _{k}-\mathbf {q} _{j}|^{3}}},j=1,\ldots ,n}$

where ${\displaystyle m_{1},m_{2},\ldots m_{n}}$ are constants representing the masses of n point-masses, ${\displaystyle \mathbf {q} _{1},\mathbf {q} _{2},\ldots ,\mathbf {q} _{n}}$, are 3-dimensional vector functions of the time variable t, describing the positions of the point masses, and G is the constants of proportionality; i.e., gravitational constant.

This equation is Newton's second law of motion on the (left side) combined with his Law of Universal Gravitation (right side). The left-hand side is the mass times acceleration for a particular jth particle (F = ma in vector form); whereas the right-hand side is the sum of the forces on that particle relative to another particle. The forces are assumed here to be gravitational as given by Newton's law of universal gravitation; thus, they are proportional to the masses involved, and vary as the inverse square of the distance between the masses. The power in the denominator is three instead of two to balance the vector difference in the numerator, which is used to specify the direction of the force. An aside, Euler actually transformed Newton's 2nd Law equation — of changing momentum, assumed equivalent to net force - into an ordinary, second order differential equation.

A general, closed-form, n-body problem solution (i.,e., n-formulas) can not be derived employing Newton's Law of Universal Gravitation equation as a solution approach.

see General considerations for the Classical Solution below for further discussion.

In this extract written by you:

• There are paragraphs which dwell too much about the background and applications on Newton's law of gravity, about the nature of the gravitational constant, and about Newton's second law in different forms. All descriptions in this article should be streamlined and to the point about the n-body problem. We have other articles on these topics.
• there is no difference between F = dp/dt = ma since the mass of a particle is constant in classical mechanics. Why take up so much space making a big difference between them as "Eulerian" and "Newtonian", and Leibniz's notation? Why such a long discussion? Why not just introduce Newton's 2nd law when needed? There is also so much repetition on what "Newton's Law of Universal Gravitation" states, over and over and over again.
• why do we have a mixture of arrow notation and boldface and bold-italic for vectors? Why do we have r and q for the positions?
• What is the point of writing " F ij = f(mi,mj,rij)"? Newton's law of gravity already states the relation "to mass and geometry" and in the space it takes to write the function notation, you could just write the law itself and it would be the precise formula. Also you write "As will be explain below, force is a functional", which has various meanings in mathematics. Is force not just a vector (or vector field in this case)? Why not just say it?
• where did I remove content about "In the general Law of Universal Gravitation section I was trying to show how it broke down for the n-body problem and was the wrong approach with n > 3 or 4. Too bad you removed it."?

Let's at least clear up why my edits are bad or wrong to this section, before the other sections. M∧Ŝc2ħεИτlk 15:58, 26 September 2013 (UTC)

## Reflist

1. See David H. and Stephen P. H. Clark: Newton's Tyranny, The Suppressed Scientific Discoveries of Stephen Gray and John Flamsteed, W. H. Freeman and Co., 2001.
2. Newton's Law of Universal Gravitation can be derived via the potential function V(r)  =  m/r, where m is mass and r is the spatial vector. See Brouwer and Clemence book: their Chapter III (the heart of the book), Gravitational Attraction Between Bodies of Finite Dimensions, assumes Newton's Law of Universal Gravitation in order to prove it yields the potential function V(r) = m/r. Going the other way (derivation not in their book) and deriving Newton's Law raises big questions as to which is more fundamental, Newton's Law; or the potential function and LaPlace's equation?
3. The Elements of Mechanical And Electrical Engineering, Volume I, Chapter Elementary Mechanics, pp. 318-319, The Colliery Engineer Co., 1898.

{{reflist}} is placed here because there are inline refs on this talk page, causing:

"Thereare<ref>tagsonthispage,butthereferenceswillnotshowwithoutareflisttemplate(seethehelppage)."

to continually appear below editor's posts near the bottom of the page. It should stop with this section to absorb the references, so posts can continue below. Please do not delete this section.

M∧Ŝc2ħεИτlk 16:10, 26 September 2013 (UTC)

## Revert to yesterday

Sorry about so many new sections in a single day: this is the last new section I'll introduce.

Owing to this discussion - the recent main editor of this article declared to clean up the article if I revert to the version before. So it shall be done.

The problem is that other editors/bots have contributed to the article as well.

M∧Ŝc2ħεИτlk 18:58, 26 September 2013 (UTC)

Well, I object to this reversion of all of our work. As far as I can tell your edits were reasonable and I believe that all of my edits were reasonable. Bold edits, reversion and discussion are a normal part of editing. If there are particular disputed edits, they can be discussed on this talk page. But a rollback of everything throws the baby out with the bathwater. --Mark viking (talk) 20:08, 26 September 2013 (UTC)
Apologies for not responding to this earlier, but I think the main issue was with my edits. I reinstated your edits to the lead. The other edits were bots, which perform cleanup tasks routinely anyway. M∧Ŝc2ħεИτlk 16:17, 27 September 2013 (UTC)
Thanks for reinstating my edits and I'm sorry for putting you in the middle of this. --Mark viking (talk) 16:36, 27 September 2013 (UTC)
?? How? You haven't, and there is nothing to apologize for. M∧Ŝc2ħεИτlk 20:39, 10 October 2013 (UTC)

## References needed for the section Practical n-body Problem Theoretical Solution

Looking through the section N-body problem#Practical n-body Problem Theoretical Solution, it appears to me that the bulk of what appears in this section may be synthesis and original research. Synthesis of new results and original research are generally disallowed at Wikipedia; see WP:SYN and WP:OR for details. Right now there are only references to D'Alembert's principles and basic texts on matrix and tensor analysis. But there are no references to use of the FEM in the context of the n-body problem. The only paper I know of that uses FEM in this context is that of Betsch, P., and P. Steinmann. "Conservation properties of a time FE method. Part I: time‐stepping schemes for N‐body problems." International Journal for Numerical Methods in Engineering 49, no. 5 (2000): 599-638., but their method doesn't seem to be same approach as is described here. We need citations in reliable sources, per WP:RS, to support the content of this section. Thanks, --Mark viking (talk) 20:29, 26 September 2013 (UTC)

## Re Roll-back

Thank you for the roll-back. I totally believe you when you say you did all the revisions in good faith -- some were very good. My main objection is deleting without indicating what went before. Stating a kind of "is," -- "was" format may be helpful. Also, I do appreciate the criticism. RJFjR has been good at this too and has been a wise council -- we got you.

• Ok. Go back to the message part wherein I said I wanted to delete the old parts, relegating them to notes, and if it is Ok, I will proceed. Improve readability.
• This is not a research project: it gives a method or rather an algorithm for calculating reactive loads owning to an applied load. The validity is by calculating equilibrium (the acid test). It is true it is a new algorithm and there are no direct references per se.
• Interesting in how you made me write all that about the mechanism of gravity, which is new too, and its relationships to the n-body problem, etc. I think that was constructive and helpful.

I will proceed with the clean-up and also incorporate some of your suggested remarks and maybe we can get on the same page. Don't worry about style yet, worry about content. Thank you for the help.Rudrene (talk) 21:36, 26 September 2013 (UTC)

Gratitude accepted.
Also, as I said above, the content of this article is the very issue I was trying to get at. There is too much duplication/repetition of content with other articles that needs streamlining. Looks/appearances of notation are the least of the problems.
Regards, M∧Ŝc2ħεИτlk 11:03, 27 September 2013 (UTC)

## Hookean Switches

I need to know if you understand what Hookean switches are and if not what don't you understand. How do I write it clearer? Look at the tables in the Numerical Example.Rudrene (talk) 21:43, 26 September 2013 (UTC)

Many people (myself included) would have come across springs which obey Hooke's law even at high school level, that's it. As said above, I can't follow the entire section. It needs to be written clearer:
• without rambling and distracting repetition of what things mean (if you have explained the meaning of an equation once already, don't keep repeating it, in this case the positions of masses Pζ, (xζ, yζ, zζ), and r).
• when writing definitions, keep in all the necessary detail, but use the minimum number of words/sentences in the definitions, currently there is one paragraph after another with comments like "(Definition continues.)" which can be condensed and still provide the same information.
About Hookean switches, I'm geussing A is something to select components of the radial vector? (Radial vector of what? any particular mass or for all of them? This should be stated at the outset). For starters,
• simply state what Hookean switches and Hookean springs are in this context,
• followed by why they are important,
• then you can go on to use them in the analysis.
The numerical calculations aren't too helpful. Most readers will just see lots of numbers and formulae, and will know this is obviously a calculation, but will have little clue what it's about, will have to keep scrolling up referring to the already confusing explanations, and even then reading the calculation requires full attention to detail to follow. Is this going to interest and stimulate readers to learn more on the topic, or would they just become board? Are calculations like this encyclopedic? WP is not a textbook, journal, or manual. It's actually better to describe the theory/formalisms in an article, then towards the end state "this model/theory/method/etc predicts that these quantities ... will have these values..." with some interpretation of the results.
Hopefully this helps. M∧Ŝc2ħεИτlk 11:03, 27 September 2013 (UTC)

## Changes

If you change something in the Article please indicate what you have changed on the Talk Page. Thank you.Rudrene (talk) 16:34, 29 September 2013 (UTC)

## Ph.D. Review

Waiting for review by two Ph.D physicist.Rudrene (talk) 16:57, 29 September 2013 (UTC)

Credentials don't count for much at Wikipedia; what does count is the ability of an editor to evaluate the content of an article and assess its accuracy and neutrality based on the reliable sources from which the article was created. If it helps, I have a PhD in theoretical physics and have been a practicing physicist for years. I've worked before on gravitational simulations using techniques like symplectic integration and multipole approximations.
I'll concentrate on the section Practical n-body Problem Theoretical Solution since that seems the core of the new material. As mentioned in a previous section of this talk page, this section looks like original research to me. I have not been able to find any peer-reviewed papers or books that apply the finite element method in this way to the n-body problem. For any such approach to remain in the article, it must be based on a reliable source such as a peer-reviewed paper or a book. Providing references for aspects of the approach, such as vectors, tensors and the FEM is good, but what is really needed is a reference for the application of FEM to the n-body problem in the particular way stated in the section. Ideally this section is merely a summary of one or more reliable sources. But right now, idiosyncratic terminology, such as Hookean switch (which gets zero hits in Google scholar) and mentions of human body loads, luggage, and helicopter airframes, suggest to me that this is a synthesis, per WP:SYNTH, of an FEM in an aeronautical context with the gravitational n-body problem. Both synthesis and original research are not allowed in Wikipedia articles.
As for the physics itself, here is my personal assessment: FEM simulations with materials like rigid bodies in the Hookean regime are based on the assumption of a system at equilibrium with any deformation from equilibrium resisted by a Hooke's law restoring force. But in the n-body problem with three or more objects, there is no general concept of equilibrium. It is well-known that chaos is a generic feature of n-body gravitational systems; this means that even small perturbations can grow large over time. A Hamiltonian system with chaos doesn't have a simple equilibrium point or limit cycle along with a basin of attraction, it is in general (depending upon parameters and initial conditions) a complex interleaving of chaotic regions, separatrices, KAM tori, and stable regions. So from general principles, a Hookean FEM approach cannot always apply because the assumption of equilibrium is general violated. There are restricted situations where resonance and entrainment can provide a kind of restoring force (e.g., the Lagrangian points in the perturbative regime, the bands of Saturn's rings), but these are special exceptions to the general rule. --Mark viking (talk) 21:36, 29 September 2013 (UTC)

Pacificscottsman (talk) 03:59, 13 October 2013 (UTC) FEM's are based on conservation of energy, not equilibrium as such, and use several mathematical techniques, Hooke's Law, Stress-Strain, free body diagramming, etc, to create a simulation of physical events. (If FEM's were not complete mathematical systems, then a whole lot of physical and biological systems created by man would not work, or rather, would fail unpredictably. It is as simple as that.) Your representations above are a subset of the total methods available to the user of FEM's for modeling purposes, and yes, would be extremely sensitive to variable parameters depending on how the parameters would be chosen, and therefore, would be inappropriate to use as you have described. The technique presented for the N-Body problem by the authors is not what you have indicated. There is such a thing as scale, where variable sensitivity is not significant with respect to the outcome of the model (conservation of angular momentum and energy) and its representation of the physical event, e.g., N-Body's gravitational interactions. Moreover, the critical thinking illustrated in the paper is quite worthy of more review and consideration, especially if more detail can be added by the authors with respect to the complete demonstration of the mathematics and computation of the simulation results as applied to actual events. It looks to me like the analytical method is applicable to the conditions of the N-Body problem as stated. I wonder if Einstein's original ideas would have survived the Wikipedia filters you are suggesting above with your statement "But right now, idiosyncratic terminology, such as Hookean switch (which gets zero hits in Google scholar) and mentions of human body loads, luggage, and helicopter airframes, suggest to me that this is a synthesis, per WP:SYNTH, of an FEM in an aeronautical context with the gravitational n-body problem. Both synthesis and original research are not allowed in Wikipedia articles.", as "Hookean Switch" is a well defined modeling term, so perhaps, Dr. Viking, you should do a little more digging, or ask for an in-depth explanation of the terms and mathematics you mentioned before suggesting that an idea as lateral as what has been presented is dismissed out right just because your unique experience does not include the information and techniques presented. And, as far as Symplectic integration and multipole approximations, they are just that, approximations and simulations of events that are not supported by mathematical rigor, whereas, the new method proposed appears to be based in mathematically proven techniques. I suggest more review is definitely in order for the ideas presented by the authors, and the article should not be "rolled back" to a previous meaningless version. And, so you don't think I am simply speaking out of turn, I am a math modeler with many year's experience creating chaotic and stable models in order to study dynamical systems of difference equations, linear and non-linear. --User:Pacificscottsman

I said: "This Section presents a closed-form, algebraic algorithm (just algebra and simple statics) for calculating the response of an arbitrary n-number (n > 3) of spatial reactions owning to a single arbitrary concentrated applied load (a force and moment), when distributed via their centroids and the massless geometric inertial properties of the reaction pattern. Simply put, this solution to the n-body problem takes an external load and moves it over to the centroids of the reactions or supporting pattern; and by incorporating the massless geometric inertial properties of that pattern, beams (maps) the applied load to those reaction points, resulting in equilibrium between the applied load and the reactions. Although somewhat messy algebraically, it's that simple. The solution can be somewhat analogous to a 3D rigid-body, rivet analysis, but it is more, as the “load” can be any physical quantity. This algorithm when only a function of load is only a function of geometry."

I have used the rigid-body form of this solution for over forty years and it always yielded equilibrium: internal reactions = applied load.

I have no argument with what you said and I can see you put a lot of thought into what you said...but, we are still not on the same page.

Please explain to me how this solution fundamentally is different than just a plain application of simple statics. Since it is a new application of statics, there are no direct references except Gallian, Dave A. and Wilson, Henry E.: "The Integration of NASTRAN Into Helicopter Airframe Design/Analysis," paper. Please help.Rudrene (talk) 00:23, 30 September 2013 (UTC)

You said: "But in the n-body problem with three or more objects, there is no general concept of equilibrium." In a dynamic situation that is true. But this is not a dynamics situation. It is static situation. In the first instant, the Practical n-body Problem Theoretical Solution, only addresses the static reactions and a load. If you will, quasi-steady loads, velocities and accelerations are employed in dynamic cases. In the Solar System model following, quasi-steady loads, etc., are applied. When analyzing a moving structure quasi-steady loads, etc. are always used.Rudrene (talk) 01:01, 30 September 2013 (UTC)

## Generalizations

It seems to me that we may be being to restrictive in repeatedly stating that the force in Newtonian and that there are many possible generalizations where we would still be discussing the n-body problem. We could limit it to 2-dimensional space. We could even expand it to 4-D space. We could change the force equation to be inversely proportional to the distance. Instead of x-2 we could make the force proportional to eaxx-2 (with a=0 it is Newtonian). We'd still be talking about the n-body problem. To accommodate this we could add a section on generalizations and reduce repetition of the Newtonian force model.

## Informal version

The material in the section 'Informal version of the Newton n-body problem' still doesn't make sense to me. The objection that it is misleading doesn't make sense to me (rather it seems to say the model wasn't complete.) Further, it seems to be an attempt to make a formal rather than informal statement by making simplifying assumptions explicit. When it describes the bodies as points it is specify that we assume the bodies don't collide. And it specifies the boundary conditions for the problem as position and velocity at time 0 rather than, say, position at times 0 and t. RJFJR (talk) 02:19, 3 October 2013 (UTC)

## Generalizations and Informal Version

The first paragraph "Generalizations" I assume was written by RJFJR, is hard to understand: "repeatedly stating that the force (s?) in Newtonian"...is what?...The goal is to have a math model echoing real-world physics; not mathematics for mathematics sake.

As to the second paragraph, I tried to say this initially: the first sentence is wrong. Whoever wrote it didn't understand the problem correctly; and in that Section I tried to explain why it is wrong. Drop the first sentence. I will re-write that section and try to be more clear why the first sentence is wrong. Essentially Newton already had equations (math models) for the planet's orbits, but they didn't fit the observations very well. Question was why? Newton realized it was the effects of gravity (Books 1 and 3), but he didn't know how to calculate those forces of gravity: is the n-body problem. Thank you for your questions. Questions help clarify the subject.Rudrene (talk) 14:16, 3 October 2013 (UTC)

## Re Generalizations

You said: "force in Newtonian" should be "force is Newtonian," and where upon first reading I misunderstood it. Anyway...

I wrote: "This angular rotations -to- moments assumption will be incorporated below. Interestingly, this same assumption is employed in the displacement method in finite element analysis, but there in a different context. (Other assumption relationships may be employed too.)"

Key is the last sentence: all new concepts could be tried. I didn't expand this idea because the scope was to be limited only to the classical solution. The solution presented is already nonlinear via the inertial tensor. Time factors could be incorporated into the transfer matrices. And on and on. Your gut feel about the flexibility of the nonlinear spatial parameters assumptions is correct: go for it.Rudrene (talk) 17:40, 5 October 2013 (UTC)

## Semi-Protected

Please provide semi-protection until I finish cleaning it up: Template:Edit semi-protected. Thank you.Rudrene (talk) 16:32, 8 October 2013 (UTC)

Template:Replyto. The place to request semi-protection is Wikipedia:Requests for page protection (shortcut: WP:RFPP), not on the talk page of the article you want to edit. However, semi-protection is unlikely to be applied just for this purpose. The normal way to reserve an article for serious editing while avoiding edit conflicts is to place {{In use|time=~~~~~}} at the top of the article. This is for intensive editing over a short period, and you should remove it again promtly when you're done. If I can offer further help, you're welcome to ask on my talk page (and watchlist it, as I will reply there). Cheers, --Stfg (talk) 17:04, 8 October 2013 (UTC)

## Progress?

The many recent edits do not seem like it.

Please see this section. If this article does not improve soon, it may be reverted to a much earlier version. M∧Ŝc2ħεИτlk 20:39, 10 October 2013 (UTC)

It appears to me that you simply don't understand the author's mathematics and use of physical ideas to describe the behavior of N bodies interacting with each other. Perhaps you should try and learn the material in question before threatening to roll back an article that looks to me to be correctly stated with respect to conservation of angular momentum and energy. I admire the clever use of widely proven mathematical techniques and proven fundamental ideas of physics. --User:Pacificscottsman

## Math Mess

I said at the beginning of the Section: "Index notation is used in this section.[30][31] Indexes and scalars are plain italic text; vectors are in bold, italic text; Cartesian tensors[32] are in plain italic text." Implying index notation and matrices were to be the mathematical vehicle for this Section (need I say more?). It's not my fault you're not up to speed and don't know these combined notations (that's why I suggested your teachers review this Article). The fact that I'm mixing tensors (i.e., index) with matrices, if unknown or not understood by the reader, would indeed certainly present a problem. Since the problem model is 3D, it is hard and very mess to use any other notation (like say vector analysis) for deriving the inertia tensor. So I can understand where you are coming from. The math is not a "mess." It's true I didn't develop the two Hermitian, but I felt it was not necessary because the reader, knowing and understanding this mathematical vehicle, would know how it is developed. Maybe I'm wrong. (In graduate school one of the first textbooks for example you will have to read will be Morse and Feshbach's classic Methods of Theoretical Physics, which uses this mixed format; and without the proper mathematical tools you won't be able to read it either.)

You said on Oct. 10th: "If this article does not improve soon, it may be reverted to a much earlier version." What am I missing here?

I became very possessive of the Article because people, who didn't understand a word said, were changing the Article without any rational basic for what they were doing. 1.) The algorithm's proof is via calculating equilibrium, and therefore needs no references (Q.E.D.). 2.) If you limit your articles to what has transpired in the pass, and some of that has been wrong (is why we keep testing and exploring, etc.), than by default your going to end up with outdated Articles.

I would suggest you are not reading what I've been saying. Thank you for your answer back.Rudrene (talk) 18:13, 13 October 2013 (UTC)

Rudrene I know that the components of a 2nd order tensor can be displayed as a matrix. The section itself is hard to read because it rambles on and on and on.
As for:
"Please see this section. If this article does not improve soon, it may be reverted to a much earlier version."
this does not imply that I personally will do the rollback, perhaps if you read the link there are also two other editors who agreed to rollback (Mark Viking and Jayanta Sen). The statement never said which editor, only that it COULD be reverted. For the nth time now, I'm not going to revert. If others do so, that's up to them.
I have read what you have written. Try doing the same yourself.
It would be good to see user:Pacificscottsman rewrite the article in a professional prose style. M∧Ŝc2ħεИτlk 19:07, 13 October 2013 (UTC)

## Improvements to the Aug 26, 2013 version?

Hi Rudrene,

I looked at an earlier version of the article, specifically the section that gave the mathematical formulation of the "problem".

I found the above section quite precise. It simply expresses in vector notation that every body in the "n-body system" exerts a force on every other body according to Newton's law of universal gravitation, and the accelerations resulting from these forces are given by Newton's Second Law.

The n-body problem isn't really any more esoteric than the above.

Quite simply if I were to use English the equation says that if you have n bodies, then they exert a (gravitational) force on every other body, that these forces are proportional to the masses of the two-bodies under consideration, and these forces cause the bodies to accelerate in the direction of the force with the acceleration proportional to the force and inversely proportional to the mass of the body being accelerated.

That's it! That's all the n-body problem is as far as I can see, unless you go beyond classical mechanics into relativity, quantum mechanics etc. And as far as I understand, the n-body problem as traditionally formulated is restricted to classical mechanics.

Quite honestly, the "n-body problem" doesn't seem like much of a problem in classical mechanics, beyond the fact that an analytical solution (or whatever other name that you may call it by like "closed-form solution") is not available.

Analytical solutions are not available for many things, that doesn't mean a solution doesn't exist, just that we may have to grind through to get a numerical solution whose accuracy of course depends upon computing resources available.

To modern physicists the "n-body problem" is trivial, no more than high-school physics. However for the lay Wiki reader a clear explanation of the "n-body problem" is beneficial.

You wrote "This Section relates why the classical approach won't work." Can you explain why you think gravity and the second-law are not sufficient to explain the movement of the n-bodies?

Maybe you can take a look again at the section, the link again is:

Possibly you can explain to Mark, Maschen, me and others what precisely is wrong with the above section and why there need to be changes and additions?

Thanks,

JS (talk) 04:52, 14 October 2013 (UTC)

## Stupidity

Thank you for responding and asking questions on the meaning and content of the Classical Mathematical Formulation section, because it clearly demonstrates your lack of mathematical and physical science comprehension and understanding of what is being presented there. For example my statement "Imagine if there were three masses involved: there would be six separate equations, F12, F21, F13, F31, F23, F32. If four masses (or more) were involved there would be a mathematical mess. This is the reason this approach fails [16].", went completely over your head. You not only didn't understand that sentence, and you didn't understand Note 16 either. The classical approach as demonstrated by the above sentence leads to an overwhelming set of intractable equations -- a point you completely missed. But as a lay reader you bring up a good point: how deep does the mathematics and physics have to be explained for lay people like yourselves? Because of your lack of basic knowledge of mathematics and physics (it clearly comes out) you missed so much of what was being said. Your first edit was to delete the first two critical paragraphs in that Section, which are the essence of what gravity is and what Newton in part based his equation on. Why don't you try to understand what those two paragraphs are saying before you delete them?

In the old version they just presented the main equation (which was mathematically wanting) without much explanation as to where it came from or how it worked functionally. In the new version I very carefully developed the 2nd Law -to- Universal Law connection (a point you completely missed too); and the roll G plays in the scheme of things. In the old revision the Two-body Problem following the above Section didn't demonstrate how the above main equation worked; but I did by giving its classical Two-body Problem solution. There I also pointed out how Newton created a mathematical toy -- which you as a lay reader (and everybody else) really don't realize how misleading this toy has been.

You said: "And as far as I understand, the n-body problem as traditionally formulated is restricted to classical mechanics." Nothing could be further from the truth -- again demonstrating your ignorance. The solution to the n-body problem is very important to molecular physics, astrophysics, and FEA; and the list goes on. You don't know what a closed-form solution is; but that said, the solution to the n-body problem is extremely important to science (and military science). Putting it on Wikipedia, everyone can see and use it. No one in the pass has ever been able to solve the n-body problem in closed-form!

You clearly don't understand how awful and wrong the old Article version is and quite frankly, you're not capable, nor are some other lay Editors capable (Maschen can't read the math; according to Pacificscottsman, Dr. Mark interpretation of the new Article was completely wrong...off the wall) of making a sound judgment call. The old Article awfulness was what started me re-writing it in the first place. You need to go outside of Wikipedia and fine people qualified to review the new Article's version. I suggested Maschen get his professors to review what I have written.

Enough!

You Editors have so harassed me ― the whole effort has become ugly ― but maybe that's your purpose.Rudrene (talk) 15:46, 14 October 2013 (UTC)

The Classical Mathematical Formulation section is rambling and pedantic and does not elucidate any concepts relating to the n-body problem. Your main issue seems to be that the classical formulation "does not work". Of course it does not. The universe is not classical. The n-body problem is more a problem in applied mathematics than physics. There is already a quantum equivalent at many body problem and there could conceivably be a general relativistic many body problem. The presence of a dimensionful constant isn't a problem either; a fully general relativistic treatment of the movement of two bodies under mutual gravitation would still include G. Griping because a classical problem is treated in a classical is like attempting to factor quantum fluctuations in a highschool mechanics problem. -Anagogist (talk) 16:54, 14 October 2013 (UTC)
I think you are confusing the terms N-body and Multibody. Multibody is a term typically used by engineers for systems with many interacting elements. As seen in the articles Multibody system and Multibody simulation, these systems typically have rigidity constraints, elastic forces, and so on, that you have been discussing in this article. Typical simulation techniques for multibody systems include finite element analysis. As mentioned in Multibody system, applications include aerospace engineering and military science, which you have also mentioned.
By contrast, N-body is a term used by physicists and mathematicians to describe systems of multiple bodies/particles independent except for interacting through a physical force, commonly gravity. In these sorts of problems, there are no rigidity constraints between objects like planets and there is no concept of elasticity between objects. If I take the moon in an orbit around the earth, for instance, and give the moon a kick, the moon's orbit will change. There is no restoring force to bring the moon back into the original orbit. The assumptions that are usually true in a multibody system--rigidity constraints, elastic interactions among the multiple bodies--do not in hold in n-body systems that physicist and mathematicins look at.
So I think you have been editing the wrong article. I stand my by judgement that with respect to gravitational n-body systems, your work (by your own admission) is original research, and original research does not belong on Wikipedia. --Mark viking (talk) 18:20, 14 October 2013 (UTC)
Hi Rudrene, Wikipedia is a collaborative effort. So if Mark, Maschen, Anagogist, or I do not understand your edits we can in good faith ask you to explain.
1) I wrote "And as far as I understand, the n-body problem as traditionally formulated is restricted to classical mechanics". You replied "Nothing could be further from the truth". Anagogist writes "Griping because a classical problem is treated in a classical is like attempting to factor quantum fluctuations in a highschool mechanics problem" by which, I assume, Anagogist means it is okay for the article to be restricted to classical mechanics. Of course if you wish you can can introduce quantum mechanics and relativity, preferably as a separate section. Your current version does not seem to have any of these.
2) You wrote "Imagine if there were three masses involved: there would be six separate equations, F12, F21, F13, F31, F23, F32." And for n bodies there will be n x(n-1)/2 pairs of forces. Unless you can overturn gravity, that is reality.
3) You wrote "In the old version they just presented the main equation (which was mathematically wanting) without much explanation as to where it came from or how it worked functionally." Actually the old version did explain concisely and accurately where the equation came from and how it worked. It says that the forces are gravitational, the acceleration given by Newton's 2nd-law. I quote "This equation is Newton's second law of motion; the left-hand side is the mass times acceleration for the jth particle, whereas the right-hand side is the sum of the forces on that particle. The forces are assumed here to be gravitational and given by Newton's law of universal gravitation; thus, they are proportional to the masses involved, and vary as the inverse square of the distance between the masses."
4) You wrote "No one in the pass has ever been able to solve the n-body problem in closed-form!" If it is true that you have indeed found a "solution" to the "problem" that has not been solved before, then as Mark mentions, Wikipedia is not the proper place. There is a strong prohibition on original research. You should send it to a journal, probably one whose subject is applied mathematics.
Best Wishes, JS (talk) 19:41, 14 October 2013 (UTC)

## Edits by Rudrene

Since Sept 3 2013 Rudrene has made hundreds of edits to this article, including the content of most sections and references that are questionable. The article has become disorganized and incoherent. I am not sure what Rudrene's motives are. For example, a couple sections I have issues with:

"Rigid-body applications (i.e., where A 's = 1, see below) include distribution of useful loads in a finite element model (FEM); 3D rigid-body rivet analyses; and astronomy problems. Astrodynamics problems and the like are particular solutions that use this new, general n-body problem solution as an algorithm; and once the forces of the bodies, {mζ}, ζ = 1, 2, ... N, are known, velocities and accelerations, i.e. their motions, may be determined. Determining equations and numerical values for astronomy problems is beyond this page's scope. Structural analyses or soil analyses elastic solutions (A 's < 1) are possible too. Again, the latter applications are beyond the scope of this page. Note especially: rigid-body solutions for elastic structures are not valid.

An example problem is given." - This is barely relevant information to this article and should not be in the introduction, if included at all.

"It further needs to be pointed out the total mass orbiting the Sun is probably equal to the Sun's own mass." - False, and contradicts an earlier statement.

"Both Robert Hooke and Newton were well aware Newton's Law of Universal Gravitation did not hold for the forces associated with elliptical orbits.[16] In fact, Newton's Universal Law doesn't account for the orbit of Mercury, the Asteroid Belt's gravitational behavior, or Saturn's Rings." - Newton's Law of Universal Gravitation holds just fine for elliptical orbits, partially accounting for precession, relativistic effects notwithstanding.

Turidoth (talk) 20:57, 16 October 2013 (UTC)

I propose that we revert the article to the August 23, 2013 version[[2]]. If I do not hear any dissenting views in a couple of days I will do so. Thanks, JS (talk) 19:27, 19 October 2013 (UTC)
With due apologies to anyone who may have made any changes to the article over the past month, I am reverting it to the August 23, 2013 version. I believe it is much easier to follow. Thanks, JS (talk) 19:24, 23 November 2013 (UTC)

## intro

I've modified the intro sentence by removing the phrase struck through below on the grounds that given the position I know the forces so the motion is hard but the forces easy.

In physics, the n-body problem is an ancient, classical problem of predicting the individual motions, and forces on same, of a group of celestial objects interacting with each other gravitationally.

If there is disagreement about the need for the phrase, which seems to make the sentence much more complicated to me, we can put it back. RJFJR (talk) 17:35, 25 February 2014 (UTC)

I agree with removing that term. FWIW, the term was added in this edit by User:Rudrene. Yaris678 (talk) 10:45, 26 February 2014 (UTC)

## Areas for improvement

• The sequence of sections is disorganized, with, e.g., history spread here and there
• The is no clear mathematical statement of the n-body problem in general
• Methods for solving the problem are given little attention, e.g., mesh methods, tree methods, multipole methods, few body vs many body approaches, etc.
• For a general article on the n-body problem, some aspects, such as N=3 Sundman's theorem, seem to have undue weight
• It is not clear from the article that there are two (overlapping) cultures that study this problem
• The mathematical analysts who may be interested in pure questions like the topological structure of the problem, exact solutions and formal techniques
• The physicists and engineers, who may be more interested in simulation and predicting orbits

Fixing all of these is going to be a good bit of work. Because this article has been subject to a good bit of controversy, getting consensus will be important.

As a start, I plan to rearrange and rename some sections while reserving all the material--hopefully this won't be too controversial. The top level sections would be something like

1. History
2. Formulation - general statement of the problem
3. Special cases - two, three and few body
4. Mathematical properties - stability, small denominators, resonance, singularities, choreography, ...
5. Solution and simulation - perturbative techniques, particle-particle, mesh, tree, multipole, ...

Second, I plan to add a section giving the general statement of the problem and the overall symmetries that any self-contained n-body system will generally satisfy.

Third, I'd like to move the Sundman stuff over the the Three-body problem, where it seems more appropriate.

Comments and suggestions? --Mark viking (talk) 21:26, 26 February 2014 (UTC)