Talk:N-body problem

message: 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)
thread_title: Name of the page 

message: 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)
replies: text: 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)

text: 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)
thread_title:  Cannot Be Solved vs.  Impossible to Solve 

message: 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)
replies: text: 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)

text: 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)
replies: text: 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)
thread_title: Animated GIF

message: 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)
thread_title: Euler

message: 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)
replies: text: Re User talk:118.208.184.222
#Please create an account and log in.
#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:
#The computer can only calculate for a certain time interval not for all times .
#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)

text: 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)

text: [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)
thread_title:  Suggestion 

message: 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)
replies: text: [From Terry0051] See recent edit which attempts to clarify the opening statement of the problem. Terry0051 (talk) 12:41, 21 July 2009 (UTC)
replies: text: 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)
replies: text: 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)
thread_title: But what is the n-body problem? 

message: 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)
thread_title:  A new formulation of the introduction. 

message: 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)
thread_title: Informal version of the Newton n-body problem

message: 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: . I was not able to find any couter-argument browsing the web. --Fioravante Patrone en (talk) 17:48, 17 October 2009 (UTC)
replies: text: 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)

text: [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)
replies: text: Re: Terry0051 Excellent, thanks I will have a look. Oub (talk) 12:58, 20 October 2009 (UTC)

text:  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)
replies: text: Ok, thanks for the "investigation" ;-) --Fioravante Patrone en (talk) 16:24, 14 December 2009 (UTC)

text: 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)
thread_title:  Babadzanjanz 

message: 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)
replies: text: [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)

text: 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)
thread_title:  Recent large change 

message: 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)
replies: text: 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)
thread_title:  Solved or open? 

message: 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)
replies: text: Why bring this up here rather than in Talk:Orbital resonance? —Tamfang (talk) 18:58, 14 June 2011 (UTC)

text: 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)
thread_title:  Has anybody pursuited the orbital resonance? 

message: 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)
thread_title:  Lagrange 

message: 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)
thread_title:  Descriptions of 1767 & 1772 papers; general on dates 

message: 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)
thread_title: Suggestion for lede

message: 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)
thread_title: What do we know about when it converges to an attractor?

message: "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)
thread_title:  Numerical Integration 

message: 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)
replies: text: I doubt theology will find a place in this article, but if you have references, and wish to add to this article, please do so. JS (talk) 00:31, 1 November 2013 (UTC)
thread_title:  Theology aspect 

message: 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)
thread_title:  status of the theoretical solution? 

message: 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)
thread_title:  Source code 

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

message: 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)
thread_title:  Where did Newton say it was unsolvable? 

message: 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)
thread_title: Warning?

message: 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 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.  -- 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)
thread_title:  Wrong? 

message: 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)
thread_title:  Misconceptions of Newton's third law 

message: 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)
thread_title:  Hookean switches?? 

message: Dear Administrator:
*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'ks = 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)
thread_title:  Newton's Third Law and Hookean switches 

message: 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)
thread_title: Reflist

message: 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)
replies: text: 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)
replies: text: 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)
replies: text:  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)
replies: text: ?? How? You haven't, and there is nothing to apologize for. M∧Ŝc2ħεИτlk 20:39, 10 October 2013 (UTC)
thread_title:  Revert to yesterday 

message: 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)
thread_title:  References needed for the section ''Practical n-body Problem Theoretical Solution'' 

message: 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)
thread_title:  Changes 

message: Thank you for your review.
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)
thread_title:  Reading 

message: 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)
thread_title:  Quasi-Steady Loads, Velocities and Acceleration 

message: 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)
thread_title:  Informal version 

message: 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)
thread_title:  Generalizations and Informal Version 

message: 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)
thread_title:  Re Generalizations 

message: Please provide semi-protection until I finish cleaning it up: .  Thank you.Rudrene (talk) 16:32, 8 October 2013 (UTC)
replies: text: . 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  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)
thread_title:  Semi-Protected 

message: 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)
thread_title: Progress?

message: Hi Rudrene,
I looked at an earlier version of the article, specifically the section that gave the mathematical formulation of the "problem".
https://en.wikipedia.org/w/index.php?title=N-body_problem&oldid=570194946#Mathematical_formulation_of_the_n-body_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:
https://en.wikipedia.org/w/index.php?title=N-body_problem&oldid=570194946#Mathematical_formulation_of_the_n-body_problem
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)
thread_title: Improvements to the Aug 26, 2013 version?

message: 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)
replies: text: 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)
thread_title:  intro 

message: This article is still a bit of a mess:
* 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
# History
# Formulation - general statement of the problem
# Special cases - two, three and few body
# Mathematical properties - stability, small denominators, resonance, singularities, choreography, ...
# 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)
replies: text: Considering the recent turmoil, this article could do with the above changes. You've made nice progress so far - keep up the good work! M∧Ŝc2ħεИτlk 21:43, 25 March 2014 (UTC)

text: Sorry if I edit conflicted, I'll stay out of the way, there is not much more I can do for now. M∧Ŝc2ħεИτlk 21:56, 25 March 2014 (UTC)
replies: text:  Thanks for taking a look at this and your encouragement. I've mostly accomplished the first two steps. Moving the N=3 Sundman material remains to be done. No worries about edit conflicts, please feel free to jump in. --Mark viking (talk) 22:04, 25 March 2014 (UTC)
replies: text: Cool. Moving the Sundman section over to the Three-body problem article can be done easily enough. M∧Ŝc2ħεИτlk 22:11, 25 March 2014 (UTC)
thread_title:  Areas for improvement 

message: With Maschen's help, the suggested improvements in the previous section have largely been implemented. Parts of the article are now in reasonable shape, but other parts are problematic and need reworking. As I see it, the next steps are to improve some of the particular sections and subsections:
* The two-body and three-body sections are filled with confusing turgid prose that do not explain the basics well nor summarize their respective main articles. These need a rewrite.
* The history section is incomplete, e.g., nothing about the "ancient problem", nor anything about later 20th century developments. The prose is also has some undue details. 
* The analytic approaches section has some miscellaneous subsections not really tied together. I gather that the power series section was meant to be the start of a section on perturbative expansion in for instance the planetary problem, but it is confusing and doesn't get anywhere.
* In time, when the article has converged to a new state, the lead needs to be rewritten to reflect the article. 
Because these involve changing sourced material, I expect they might be more controversial. Comments or suggestions?
--Mark viking (talk) 18:38, 28 March 2014 (UTC)
replies: text: 11 Months later...The article remains unsatisfactory. A real mess. It is claimed that the "n-body problem" is "ancient" in the lede while in the history section it claims a 17th Century "awareness and rise" ["rise" ??] of the problem. MAKE UP YOUR MIND.  Since gravity wasn't (afaik) appreciated by the ancient Greeks, I really doubt that they were concerned with anything more than predicting the paths of the planets (or rather predicting location of the stars), a totally different subject (unless underlying Laws are applied, laws which certainly didn't exist prior to the 16th Century.) I also think the lede continues to be turgid and far too awkward. Let me give some examples: 1. An "ancient, classical problem..." what does this mean? Compared to modern classical problems? or compared to ancient, non-classical problems?? Noise without significance, imho. "individual motions of a group" ...is there some GOOD reason that this isn't "the motions of a group"? Clearly, the claim that Greeks were involved REQUIRES a very good source of reference, since I doubt it is true. You claim that the Solar system problem has been "the" motivation but then admit that study of globular clusters is also a problem (and you also confuse what the subject IS: is it solving a problem or understanding a system??) Your informal problem statement uses technical jargon which I've never run across before, meaning 99.99% of the readers here will also find useless. quasi-steady?? orbital properties? interactive forces? Terrible. If the n-body problem assumes n objects, each possessing ONLY mass, velocity (3-vector) and position (3-vector) then say so. If other properties (for instance diameter, density, spin) are (sometimes) included then say so. A SIMPLE problem statement is: Given n objects interacting with a known force law with positions and velocities known at one instant of time, predict their positions at any other time. Predicting their "interactive force" is NOT the object, is it? (Although we need that intermediate step, it is NOT the goal and unnecessarily complicates the sentence, imho.) "as is the famous 3-body problem" ?? WTF does that mean, that its discussed or that it has been "completely" solved???? Are you TRYING to obfuscate?! The whole lede vascilates between "the" n-body problem and a variety of n-body problems. MAKE UP YOUR MIND. To my mind (but I'm only vaguely familiar with the subject), the n-body problem arose out of celestial mechanics but NOW encompasses more than simply gravitational problems. The lede also claims that the General Relativistic problem is more difficult to solve...am I wrong in thinking that the Classical problem is (almost certainly) impossible to solve? So, how does it get "more difficult" than that?? Or is it referring to (practical) numerical approximations of 'the solution'? Next Section:"Knowing three orbital positions of a planet's orbit..." OMG! Do you mean "knowing the position of a planet at three different times during a single orbit"?? (I am hoping I don't have to explain how horrible the first phrase's wording is...). "did not predict...very well or even correctly." WTF AGAIN! "or even correctly" as an afterthought??! Isn't being incorrect a logical subcategory of not doing "very well"? I'm not being pedantic (I hope) here, I'm trying to indicate how such lousy prose reads - redundant and excessive. You can't simply say they found those equations incorrectly predicted the orbits? Why not? Finally, and I've only read the first several paragraphs of the article, I want to point out that I'VE NEVER HEARD OF "(gravitational) interactive force" as a technical term. WHY do you include "interactive"? What forces are NOT "interactive" ? Depending on your choice of Hamiltonian, LaGrangian, or Newtonian Mechanics, the importance (meaning?) of "force" in the various solutions is NOT necessarily a fundamental part of the problem, is it? Arguably, it is position that is the fundamental observable and also must be the fundamental dependent variable (that which is predicted). At least, so it seems to me.Abitslow (talk) 15:26, 1 February 2015 (UTC)
thread_title:  Further areas for improvement 

message: "Second, in general for N > 2, the N-body problem is chaotic". The link under "chaotic" links to "Chaotic (TCG)" I don't know where it *should* link to, but I do know where it *shouldn't* link to.  — Preceding unsigned comment added by 77.173.20.75 (talk) 13:59, 3 July 2014 (UTC)
replies: text: Thanks for catching that problem. I have redirected the link to chaos theory. --Mark viking (talk) 15:35, 3 July 2014 (UTC)
thread_title:  The link 'Chaotic' 

message: Yes, a mess.  I wish I had time to learn more so I could help fix this.  Much is known about computational orbital mechanics, and there are probably many open-source simulation tools available.  Helping readers understand the real opportunities and challenges (not sterile historical arguments about what we can or cannot accomplish with closed form algebra) would help them understand the state of knowledge.  Directing them to code they can run on their own desktops would help them play with the concepts.
The Clemishaw paragraph starting with "The Sun contains 98 per cent of the mass ... " is vaguely cited and does not appear to be a direct quote.  For example, the Sun is closer to 99.8 per cent of the mass, the exact value impossible to determine without better measurements of the Oort cloud, but no less than 99.7%.  A "quote" that may be a paraphrase from a 64 page 1966 children's book is a waste of space.  
This matters to me because I am in a heated disagreement with an AGW skeptic, who exploits confusions in this particular article to make arguments about Earth-Sun radius and solar insolation that aren't true, either empirically or theoretically. "Wikipedia says ..." can lead to awful consequences.  KeithLofstrom (talk) 22:07, 5 June 2016 (UTC)
thread_title:  Agree that rewrite is needed, Clemishaw quote 

message: I was checking out the further reading section and I noticed this one:
Havel, Karel. N-Body Gravitational Problem: Unrestricted Solution (). Brampton: Grevyt Press, 2008. http://www.grevytpress.com
I clicked the link and went to have a look, thinking that it was a book about Sundman/Wang's solution. Unfortunately it seems to be some crank guy? If you click through his website to read "selected papers" you find this gem: http://www.grevytpress.com/nprob1.pdf
Which includes the line: "The general unrestricted analytical solution of the n-body gravitational problem is
disclosed in [1]. Even though it sells in bookstores, the book has been ignored so far."
Another quote: "To summarize:
- the n-body gravitational problem is impossible to solve;
- the n-body gravitational problem is trivial;
- the n-body gravitational problem does not exist;
- the n-body gravitational problem has been already solved;
- the solution of the n-body gravitational problem has been ignored.
Instead of trying to reconcile these incompatible views, I shall briefly explain how I
solved the problem. I did it in two steps. The first step was to examine in detail how the
rocks fly in space under the influence of gravitation. The second step was to convert the
physical operation of gravitation to an analytical mathematical model."
The guy is selling another book titled "Gravitation: master key to the universe". I'm not about to buy his book to check if he is selling snake oil but it sure smells like it to me. I'm gonna go ahead and remove his book from the suggested reading list. Please feel free to revert if I have done so in error.
CanisStultus (talk) 14:18, 5 February 2017 (UTC)
thread_title:  Further reading? 

message: I see it mentioned that taylor series formulations can be found in the Taylor section of the article.
Please see external link #2 in the Wikipedia entry under "Parker-sochacki method".  
https://en.wikipedia.org/wiki/Parker%E2%80%93Sochacki_method
The reference is Joseph W. Rudmin (1998), "Application of the Parker–Sochacki Method to Celestial Mechanics", Journal of Computational Neuroscience, 27: 115–133, arXiv:1007.1677, doi:10.1007/s10827-008-0131-5. A demonstration of the theory and usage of the Parker–Sochacki method, including a solution for the classical Newtonian N-body problem with mutual gravitational attraction.  
Link ( https://arxiv.org/abs/1007.1677 )
That source, in ArXiv, gives the exact system of equations for Taylor series of the N-body problem.  Moreover, since it is Initial-Value dependent, it provides a piecewise continuous way to handle collisions.
Note that the equations as given there are only for Newtonian Mechanics -- they won't handle relativistic motion, nor extreme proximity to large gravitational sources (such as Mercury) which require general relativity -- but the method can handle general relativity as well.  12.160.67.170 (talk) 12:20, 9 September 2019 (UTC)
thread_title:  Taylor Series