Talk:Three-body problem

Relation to Laymen

Like all the other science and technical articles are going in Wikipedia, there is no synopsis in the intro to try to explain at a high level what it means to the non-technical, nor why it matters. Can someone climb out of their specialist hole and please add something like this. Maybe get a less technical friend to help you word it. When discussing with them, use the phrases, "did you understand that?" and "can you see why this might be of interest or useful in some way?" — Preceding unsigned comment added by 216.19.181.213 (talk) 19:12, 21 January 2023 (UTC)

I would think a major contribution to unpredictability in chaotic systems is simply rounding errors. Any initial error in measurement accumulates, given the imprecision of instruments that cannot measure far away objects to Planck length. I could be wrong. 47.157.95.50 (talk) 20:58, 24 November 2024 (UTC)

The section called "Classical versus quantum mechanics

This section was just fraught with problems. I contemplated deleting the whole section on the grounds that it had more disinformation than information. Basically, the "quantum 3-body problem" is stated here as finding the ground state and first excited state. But this isn't the same thing as finding the dynamics of an arbitrary state, which is quite complicated still in the quantum case. This is what we are asking for in the classical problem: the complete dynamics. So it isn't a fair comparison. The second sentence of the second paragraph still doesn't make sense, but I left it there in the hope that someone might be able to understand what the author intended and fix it. — Preceding unsigned comment added by 203.116.114.162 (talk) 08:06, 27 November 2011 (UTC)

I added a piece to this section since originally it was literally MISINFORMATION.

In quantum mechanics, analytical solutions of even 2-body problems (let alone 3 or more!) are impossible (except some special cases perhaps) since the Schrodinger differential equation cannot be solved analytically for 2 or more bodies interacting. That is why most "analytical solutions" involve approximations that reduce the problem to a single-body problem or an approximation that allows for an analytical solution. — Preceding unsigned comment added by 131.175.161.14 (talk) 14:27, 4 February 2015 (UTC)

Relation and possible merger between articles on 3-body problem and n-body problem

"It has been suggested that this article or section be merged with n-body problem."

Most of the material currently in Wikipedia on the 3-body problem already seems to be in the "n-body problem" article. There is very little of anything in the "3-body" article right now. That might seem to support a merger. On the other hand, much of the most intense and historically early work was focussed on the 3-body problem; and those are also the results that figure in much of the foundational description that readers would want to find in an encyclopedia. If this subject is developed in an appropriately encyclopedic way, I suggest that the 3-body problem would make a substantial and potentially very good article, probably too big to be a section in the "n-body problem" article. Terry0051 (talk) 20:51, 4 June 2009 (UTC)

• Oppose merge. I agree with Terry0051. Although the math for n-body includes the narrow case of 3-body, the separate article for the special case is useful for historical and foundational reasons. --Jack-A-Roe (talk) 05:58, 11 June 2009 (UTC)
• The articles should not be merged. A wide variety of research was done on topics specifically related to the three body system. i.e. the Pythagorean three body system. There is no space to include all this information in the n-body problem article. 217.93.180.18 (talk) 11:43, 21 July 2009 (UTC)

tag removed

The 'unreferenced' template has been removed, because the article does indeed have some citations. This makes the tag inappropriate in view of WP:Unreferenced, which says that "Wikipedia articles that have no citations belong in this category".

Existing citations are mainly in support of one section (history). So there might easily be room for 'citation needed' tags elsewhere. I didn't have time to explore that aspect. Terry0051 (talk) 12:46, 22 July 2009 (UTC)

[From Terry0051] I've added some 'citation-needed' tags: this is on the basis that they replace the previous general 'unreferenced' tag that was removed a short time ago. A few references are present, but large sections of the pre-existing content seems in need of RS or perhaps any reasonable explanation at all for general comprehensibility. (What these sections very probably need is recasting into more encyclopedic form and language). Terry0051 (talk) 17:27, 24 July 2009 (UTC)

Solution?

So, has the damn' thing been solved for classical mechanics, or not? The article doesn't say. *Septegram*Talk*Contributions*

Re: Septegram 01:13, 8 November 2009 (UTC): it is by Sundman around 1913. That is described in detail in the article about the n-body problem. The solution however is useless for practical purposes.Oub (talk) 12:26, 13 December 2009 (UTC):

The article says there is no solution, and then immediately gives a solution (the video). — Preceding unsigned comment added by 84.66.21.64 (talk) 08:59, 23 February 2023 (UTC)

Problems with section n-body problem

It is stated

N-Body problems deal with the question of how n objects will move under one of the physical forces such as gravity. These problems don’t have an analytic solution for n greater than two (Except for special cases). Thus, for these kinds of problems using numerical solutions is unavoidable.

and there is a link to solutions in closed form. That is confusing and partially wrong.

1. The n-body problem has an analytical solution, expressed by a convergent power series. This was found by Sundman/Wang. The solution is useless for practical purposes and that is why numerical analysis is at the moment the only way to obtain quantitative results.

1. What is not known is a solution in closed form, but it is not proven that such a solution does not exist.

So I suggest to change the relevant text. I wait a day or two then I do it myself.Oub (talk) 12:32, 13 December 2009 (UTC):

Providing vs. possessing

Maybe there is some kind of specialized math terminology in use here that I'm not familiar with, but the last clause of this sentence doesn't make any sense to me:

'Solving' this problem means providing a generally applicable method for making this kind of determination of gravitational trajectories, or possessing such a method.

Does the bolded clause actually have a purpose--Subversive Sound (talk) 07:43, 8 April 2010 (UTC)

In Popular Culture

So this is what prevents a physicist having a threesome on xkcd! —Preceding unsigned comment added by 81.209.12.90 (talk) 12:56, 9 August 2010 (UTC)

where is this article

this article is just silly. Frankly I can't think of anything else much to say. coupling. non-linearity. smooth differentiable space. more than 2 variables. I can't see that here. I've not read newton in the original. I imagine he brought these things up in some oblique 17th century way, which would be very interesting to hear about from whomever it was who knew which propositions of book 3 are involved. A historian I suspect. Probably Oxbridge or ivy. No one else has read the principia for years. It's the maths stupid.Duracell (talk) 01:37, 31 January 2011 (UTC)

"Examples" Section and Others

Most of the "examples" section is lacking citations and poorly written, containing colloquialisms and unnecessary jargon. It may be best to completely scrap the section (except the "Circular restricted three-body problem" portion)and allow it to be completely rewritten with proper citations. Overall, I would consider the page to be in great need of work to fix colloquial speech and general grammar. — Preceding unsigned comment added by 65.110.254.76 (talk) 03:56, 28 January 2012 (UTC)

Constant-pattern solutions

I have added a new section "Constant-pattern solutions". In principle, it supersedes the core of the "Circular restricted" section. 94.30.84.71 (talk) 15:23, 4 March 2012 (UTC)

3-body symmetric orbit simulation

I suggest to add external link:

• 3-body symmetric orbit simulation

Do you mind? Breny47 (talk) 19:34, 18 October 2012 (UTC)

Since the case can quite easily be solved exactly, I see no need to show what appears to be an approximating numerical solution. 94.30.84.71 (talk) 12:20, 25 October 2012 (UTC)

Transpose 3.1 & 3.2

3.1 "Circular restricted three-body problem" and 3.2 "Constant-pattern solutions" should change places, as the latter is more general, and the former adds little of value. 94.30.84.71 (talk) 12:23, 25 October 2012 (UTC)

shades of Kovalevskaya?

What is meant by the parenthetical comment (shades of Kovalevskaya)? Is this supposed to be in the article? A Google search only returns a Ukrainian model of that name and this article. —193.157.210.88, 16:34, 14 November 2014

Sofia Vasilyevna Kovalevskaya (Russian: Со́фья Васи́льевна Ковале́вская) (15 January [O.S. 3 January] 1850 – 10 February [O.S. 29 January] 1891) was the first major Russian female mathematician, responsible for important original contributions to analysis, differential equations and mechanics, and the first woman appointed to a full professorship in Northern Europe. She was also one of the first women to work for a scientific journal as an editor.[1]

There are several alternative transliterations of her name. She herself used Sophie Kowalevski (or occasionally Kowalevsky), for her academic publications. After moving to Sweden, she called herself Sonya — Preceding unsigned comment added by Da5id403 (talk • contribs) 23:37, 15 May 2015 (UTC)

I added a wiki link to the Cauchy–Kovalevskaya theorem, which is probably what the long-vanished editor had in mind here. —72.68.81.122 (talk) 07:10, 17 March 2020 (UTC)

Please clarify or remove the following statement

In the section "Circular restricted three-body problem", the final sentence reads "It can be useful to consider the effective potential."

Because there is no clear mapping for what "it" refers to, this sentence is at best confusing and at worst misleading. It is impossible to tell if the author meant "It (the Circular restricted three-body problem itself) can be useful when considering the effective potential", which would still be a broken sentence; or meant 'It' idiomatically, as in "The user may find consideration of the Effective Potential useful when...", which would again still be an incomplete and broken sentence.

So: restate or remove, please. — Preceding unsigned comment added by 71.231.179.40 (talk) 22:14, 19 May 2016 (UTC)

That sentence is not present anymore, but in English, in such a sentence the pronoun "it" is usually a placeholder resending (in the presnt case) to "to consider the effective potential". (To refer to something just said one would rather use "this", which cannot be empty, instead of "it".) It would mean the same, but in a more awkward (and thus less idiomatic) way to say "Considering the effective potential can be useful" (and I notice a posteriori that the present sentence uses a similar construction with a placeholder "empty subject"). The same construction exists in French ("Il peut être utile de considérer le potentiel effectif" = "Considérer le potentiel effectif peut être utile") but not in every language and possibly not in the mother language of the above anonymous contributor. In English the sentence under consideraton is neither incomplete, broken, nor ungrammatical. Tonymec (talk) 14:24, 18 October 2019 (UTC)

adding Hill's lunar equations

I suggest that a reference is made in the article under the section periodic solutions to Hill's differential equations, since these equations give an (approximate) periodic solution for the three-body problem.

Hill's differential equations, first applied to lunar stability and therefore also known as Hill's lunar equations, are a method to approximate the periodic motion of the moon around the earth which in its turn has a periodic motion around the sun. Floquet proved convergence of Hill's differential equations, meaning that the Hill equations are suitable for solving the three-body problem. — Preceding unsigned comment added by 134.157.34.127 (talk) 12:55, 20 June 2016 (UTC)

Good review articles - 2014, 2015

• https://arxiv.org/pdf/1508.02312.pdf - "The three-body problem", Z.E. Musielak and B. Quarles

• Musielak, Z. E., & Quarles, B. (2014). The three-body problem. Reports on Progress in Physics, 77(6), [065901]. DOI: 10.1088/0034-4885/77/6/065901 (https://uta.influuent.utsystem.edu/en/publications/the-three-body-problem )

-- John Broughton (♫♫) 01:18, 7 February 2017 (UTC)

Numerical methods for integration

The following sentence appears at about the 33% point in the article.

The use of computers, however, makes solutions of arbitrarily high accuracy over a finite time span possible using Numerical methods for integration of the trajectories.

"Numerical methods for integration" is no longer the best term to describe the use of numerical methods to solve differential equations. In fact that term now redirects to the article titled "Numerical methods for ordinary differential equations." I suggest the sentence be changed roughly as follows:

The use of computers, however, makes solutions of arbitrarily high accuracy over a finite time span possible using successively more accurate numerical methods to find the trajectories.

Dratman (talk) 08:38, 10 February 2018 (UTC)

Jacobi constant

How about the Jacobi constant for the reduced problem: C = ω²r²-2Φ-v². The Jacobi constant is defined for the circular restricted three body problem (CRTBP) and is used to bild Hill surfaces with v=0. Ra-raisch (talk) 14:18, 24 June 2018 (UTC)

December 2019 possible solution?

Came across this article claiming a new solution to the problem:
"Researchers crack Newton's elusive three-body problem". Phys.org. Retrieved 19 December 2019.
Can someone more scientifically-proficient evaluate for potential inclusion into the article? 216.239.163.202 (talk) 16:47, 19 December 2019 (UTC)

«While the researchers stress that their findings do not represent an exact solution to the three-body problem, statistical solutions are still extremely helpful in that they allow physicists to visualize complicated processes.»

«More information: A statistical solution to the chaotic, non-hierarchical three-body problem, Nature (2019). DOI: 10.1038/s41586-019-1833-8, https://nature.com/articles/s41586-019-1833-8»

So this is only a statistical solution, not an exact solution. It may be interesting, but I'm not qualified to evaluate it. Anyone around? Tonymec (talk) 03:57, 20 December 2019 (UTC)

I think it's worth including. I don't have time at the moment, but eventually someone should put in a sentence or two. MaxwellMolecule (talk) 18:33, 20 December 2019 (UTC)

Clearing up "analytic" and "closed-form" terminology

As previously mentioned on this talk page, there's some confusion with the terms "analytic" and "closed-form", which have since been fixed on the n-body problem section, but was wrong on the general solution section. To clear up the confusion: the three-body problem has no closed-form solution - a solution involving finitely many standard operations - but there is an analytic solution - an exact and calculable solution, though we usually exclude limits/integrals - given by Sundman. I don't know what the original editor meant by There is no general analytical solution to the three-body problem given by simple algebraic expressions and integrals. I've corrected it now, but just putting this here so people are aware (and can discuss this change if need be). Volteer1 (talk) 17:58, 4 February 2021 (UTC)

According to the Wikipedia article for the notion of a “closed-form” solution, the notion changes when the tools allowed for finding expressions changes, so it’s not an absolute concept. This article should reflect that, very clearly. In particular, after saying there’s no “closed-form” solution to the 3 body problem, the next paragraph states that one can solve the problem using Puiseux Series. The only way that both can be true is if a Puiseux Series is declared to not be in the tool set for “closed-form”. However, without a rather sophisticated understanding of the ad hoc nature of a time-dependent or tool set dependent definition (which seems to have been misunderstood by Neil Degrasse Tyson, who, it is generally understood, fails to be severely un-sophisticated, but claims that the three-body problem is unsolvable, which naturally brings to mind possible confusion with things like the Entscheidungsproblem.), the average reader is getting nothing out of this except to conclude that whoever devised these terms is insane. I suggest listing on this page the set of tools allowed for this particular tool set dependent notion of “closed forms”. For example, a reasonable person might object that even the differential equation $y^{\prime }=y$ has no closed form solution, since Euler’s number requires limits in its definition (for most readers’ exposure to calculus and the role of the number $e$ in that subject), which, in the first few lines in the article on “closed-form” solutions, limits are (prima facie) disallowed. Matt Insall (talk) 00:30, 19 April 2024 (UTC)

approximate trajectories gif in the main page

I looked at the nice simulation of three bodies starting from rest in a scalene triangle in the first page. It correctly shows that the center of mass remains constant, but also the plane of the three points should remain the same, as proved in Siegel-Moser, Lectures on Celestial Mechanics, p. 28, since the angular momentum is initially zero and then must remain zero for all times. This doesn't seem the case in the animated gif, probably due to rounding errors or collisions.

A. Marzocchi, Dept. of Mathematics and Physics, Università Cattolica del Sacro Cuore, Brescia, Italy — Preceding unsigned comment added by 185.11.153.228 (talk) 09:21, 24 March 2021 (UTC)

AFAICT, the plane of the three points is the plane of the page, or something parallel to it; and that remains constant. The angular momentum is, IIRC, the sum of the products of the distances of each point from the center of mass by its mass (but all three masses being equal we can simplify the equation by that amount) by its signed angular velocity. Hard to tell by eye whether or not the momentum/momenta of the point(s) moving clockwise compensate that/those of the point(s) moving counterclockwise. — Tonymec (talk) 08:41, 25 March 2021 (UTC)

Sundman's Theorem

The section discussing Sundman's theorem could use further clarification on how the proof shows what the article asserts about it. — Preceding unsigned comment added by 2603:7000:8303:E89B:3DCE:6323:2BFA:9AD0 (talk) 18:15, 4 February 2022 (UTC)

Philosophical questions regarding Sundman's exact solution.

I think this article should address resulting existential questions as well. Since the Sundman theorem requires so much calculation (like 10**8million elements of the series for positions and speeds for each moment AND without giving any helpful clue about results for next or previous moment), how do celestial bodies know where and how fast they need to go at any time? I mean nature obviously uses an exact solution, as stars and planets move in an analog fashion (smoothly), rather than approximated or digital (i.e. abruptly jumping to their next position or moving slightly zig-zag). The laws of natural sciences are generally believed to be written in mathemathics, thus nature must be able to solve them well to last and run stably as we see it.

On the other hand, it is claimed if our entire Universe was transmutated into Silicon and melted to wafers for chipmaking and all used to build a mega-supercomputer, it still wouldn't have enough CPU power to keep solving the 3-body problem in real-time just for our Solar System, with equal precision to Planck-constant (i.e. a computed trajectory undistinguishable from smooth motion). If so, how does our Universe, containing billions of galaxies, with billions of stars in each of them know exactly who needs to go where, when and how fast? Is there a parallel, much larger Universe with enough computational power running our Universe or is our Universe a holographic projection of that larger universe (us being mere 3D projections of a higher-dimensional object). Probably there are scientific / philosophical theories worth mention, which are trying to address that question. 94.21.121.96 (talk) 11:03, 16 November 2022 (UTC)

Celestial bodies don't "know" where and how fast they need to go, they just go where their mass, momentum, and the forces acting on them (as a first approximation we could think of Newton's laws of motion, but these become insufficient for very massive, very small or very fast objects) pull them. We could say that the planets, solar systems and galaxies act like an analogue computer the size of the universe, continuously solving their own equations of motion, so they don't need any external digital computer made of wafers and chips to solve them. — Tonymec (talk) 20:11, 11 December 2022 (UTC)

"Solution"?

The article currently has this sentence in its lede: "In February 2024, several studies were reported which may suggest a solution." This is supported by a citation (and a plug) to a recent paper. But my (ignorant) reading of this is that it's a sophisticated way of analysing the problem rather than a "solution". And, as I understand it, "solutions" to chaotic problems are only going to ever going to be better numerical approximations. Which, while a good thing, are not some revolution as it implied by this statement. I'm deleting the statement as (a) it's very recent, and (b) it's unclear from my reading that the work it highlights is more than an improved approximation.

Happy to be corrected, however. But if restored there should be more of an attempt to explain what's special about this recent work (and in plain English, please). PLUMBAGO 09:59, 19 March 2024 (UTC)

@Plumbago: (and others) - Thank You for your comments - and concerns about my somewhat recent edit - and the related undoing/revert of the edit - the related references seemed relevant and worth adding to the main "Three-body problem" article but, for one reason or another, may not be in fact - others more knowledgeble about this than I are welcome to comment of course - in any case - hope this helps in some way - Stay Safe and Healthy !! - Drbogdan (talk) 10:30, 19 March 2024 (UTC)

References

Animation for gravitational 3-body problem under *Solutions* looks incorrect

In the animation, File:3bodyproblem.gif, the center of mass pretty clearly drifts toward the bottom right. The three bodies initially have zero momentum, so the momentum of the system is also zero. Per the law of conservation of momentum, the center of mass should not move. The Mathematica code which I assume generated the animation and given below it states: "Initial total momentum is zero, so the center of mass does not drift away".

I have noted this on the talk page for the animation as well. Billybass (talk) 16:05, 26 March 2024 (UTC)

Nice observation! I hope the creator of that animation can replace it with a "correction factor".

This can be done easily in software that frequently resets the center of mass at time t (call it c(t)) to the original center of mass c(0).

One way to do this is to do this: For small h > 0, at instants t = n*h:

Set each position x_j(t), 1 ≤ j ≤ 3, equal to a new value:

x_j_new(t) = x_j(t) + c(0) - c(t).

This will ensure that the center of mass stay constant.

(Assuming there is not a bug in the program, causing it to do the wrong calculation, and the problem is just one of numerical analysis.)

celestial classics

In the first sentence, should "classical mechanics" be instead "celestial mechanics"? —Tamfang (talk) 03:39, 20 April 2024 (UTC)

Animation

The animation of the 3 body problem would disturb quite a few people. It doesn't meet accessibility guidelines of the manual of style. I am not technically proficient enough to modify it so I ask another editor to do so. Wikipedia:Manual_of_Style/Accessibility#Images Humpster (talk) 04:14, 25 July 2024 (UTC)

The ideal solution is converting it into a video, correct? Remsense诉 04:16, 25 July 2024 (UTC)

I believe so. It must be controllable by the reader.

Further into the article, there are several more animations, making the article unreadable for some people. Humpster (talk) 04:20, 25 July 2024 (UTC)

Which dimension?

The beautiful set of 20 periodic solutions to the 3-body problem does not state if it is in 2 dimensions or in 3 dimensions.

I'm guessing it's 2 dimensions, but I am not sure.

I hope someone (like its creator PeRosselló) who knows the correct answer will add this information to the caption to that illustration.