Talk:Two-body problem

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After the section "Change of variables", the solving of the equation is not pursued. A detailed discussion on the subject may be found at Eric Weisstein's world of physics. 130.236.182.61 12:07, 25 October 2005 (UTC)

Huh?

I followed this link from another page, and from the first sentence, I was lost. I'm not a physicist or a scientist of any sort. Can someone who understands this stuff please write a layman's description for the first sentence? (I've noticed this happens a lot with physics-related articles, but this is the first time I've mentioned it.)

Yeah, I know :( I've been meaning to fix this article, but I've been busy with a few others lately. Why don't you check back in a few days? Thanks for being patient. :) WillowW 16:44, 13 June 2006 (UTC)

P.S. If you find other physics articles that are too inscrutable, would you please let me know on my talk page? You shouldn't be shy about asking for more intelligibility. Sometimes it can't be helped; some topics are just hard to understand because they're so foreign to everyday life and also rely on lots of careful little definitions. But I'm sure that there are many others that need only a little help to make them clear. You could try some of the Physics articles mentioned on my User page; how about centripetal force? Thanks for your help! WillowW 16:44, 13 June 2006 (UTC)

Yuh huh! :D

Hi, rather than waiting, I decided to try my best at fixing the article now. Please let me know if you like it as is, or if there are still confusing parts. I didn't tinker with the stuff at the very end; maybe someone else can decide whether it's worth keeping? WillowW 19:43, 13 June 2006 (UTC)

Deletion of "Newtonian gravity" section

I apologize for having deleted the section on "Newtonian gravity", but I hope that the following will explain my reasoning, and that everyone will agree that the article is improved.

The author seems to have intended this section mainly as a collection of rules of thumb for working with two-body problems. However, given that the general formulae are given in the earlier text, such rules of thumb do not seem to be needed. Moreover, it is not clear how these particular rules of thumb are useful or explanatory, and they do not seem to be strongly organized. Finally, the topic of "two-body problem" is independent of the particular interaction potential between the two bodies; hence, rules of thumb specific to gravitation or to the Earth-Sun (or a hypothetical Sun-Sun) system seem not general enough for this article. Perhaps they would do better under Kepler's laws of planetary motion?

Therefore, since the main points of this section were covered elsewhere (e.g., in the general formulae preceding this section), it seemed to help the flow of the article to eliminate this section and go straight to the explanatory examples. Thanks for your patience with me, Willow 07:22, 6 August 2006 (UTC)

Some text from the first half of what you deleted is replicated at standard gravitational parameter#Two bodies orbiting each other, so I added the a^3/T^2 = M formula there, and changed the link at Alpha Centauri#System components. -- Paddu 16:16, 27 August 2006 (UTC)

I reproduce the deleted section here, in case anyone would like to glean from it

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Applying the gravitational formula we get that the position of the first body with respect to the second is governed by the same differential equation as the position of a very small body orbiting a body with a mass equal to the sum of the two masses, because $m_{1}m_{2}/\mu =m_{1}+m_{2}$.

Assume:

• the vector r is the position of one body relative to the other (above called x)
• r, v, the semi-major axis a, and the specific relative angular momentum h are defined accordingly (hence r is the distance)
• $\mu =G(m_{1}+m_{2})$, the standard gravitational parameter (the sum of those for each mass)

where:

• $m_{1}$ and $m_{2}$ are the masses of the two bodies.

Then:

• the orbit equation applies; recalling that the positions of the bodies are $m_{2}/(m_{1}+m_{2})$ and $-m1/(m1+m2)$ times r, respectively, we see that the two bodies' orbits are similar conic sections; the same ratios apply for the velocities, and, without the minus, for the angular momentum with respect to the barycenter and for the kinetic energies
• for circular orbits $rv^{2}=r^{3}\omega ^{2}=4\pi ^{2}r^{3}/T^{2}=\mu$
• for elliptic orbits: $4\pi ^{2}a^{3}/T^{2}=\mu$ (with a expressed in AU and T in years, and with M the total mass relative to that of the Sun, we get $a^{3}/T^{2}=M$)
• for parabolic trajectories $rv^{2}$ is constant and equal to $2\mu$
• h is the total angular momentum divided by the reduced mass
• the specific orbital energy formulas apply, with specific potential and kinetic energy and their sum taken as the totals for the system, divided by the reduced mass; the kinetic energy of the smaller body is larger; the potential energy of the whole system is equal to the potential energy of one body with respect to the other, i.e. minus the energy needed to escape the other if the other is kept in a fixed position; this should not be confused with the smaller amount of energy one body needs to escape, if the other body moves away also, in the opposite direction: in that case the total energy the two need to escape each other is the same as the aforementioned amount; the conservation of energy for each mass means that an increase of kinetic energy is accompanied by a decrease of potential energy, which is for each mass the inner product of the force and the change in position relative to the barycenter, not relative to the other mass
• for elliptic and hyperbolic orbits $\mu$ is twice the semi-major axis times the absolute value of the specific orbital energy

For example, consider two bodies like the Sun orbiting each other:

• the reduced mass is one half of the mass of one Sun (one quarter of the total mass)
• at a distance of 1 AU: the orbital period is ${1 \over 2}{\sqrt {2}}$ year, the same as the orbital period of the Earth would be if the Sun would have twice its actual mass; the total energy per kg reduced mass (90 MJ/kg) is twice that of the Earth-Sun system (45 MJ/kg); the total energy per kg total mass (22.5 MJ/kg) is one half of the total energy per kg Earth mass in the Earth-Sun system (45 MJ/kg)
• at a distance of 2 AU (each following an orbit like that of the Earth around the Sun): the orbital period is 2 years, the same as the orbital period of the Earth would be if the Sun would have one quarter of its actual mass
• at a distance of ${\sqrt[{3}]{2}}\approx 1.26$ AU: the orbital period is 1 year, the same as the orbital period of the Earth around the Sun

Similarly, a second Earth at a distance from the Earth equal to ${\sqrt[{3}]{2}}$ times the usual distance of geosynchronous orbits would be geosynchronous.

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I moved it to Gravitational two-body problem and am improving it.--Patrick 13:03, 23 June 2007 (UTC)

Deletion of "General Relativistic Gravity" section

My reasons for deleting this section are slightly different. Here, the author points out an interesting correction of general relativity to the planetary motion expected from solving a classical two-body problem of gravity.

This section, although covering a fascinating topic, does not seem at home in this article, since it is specific for the relativistic gravitational two-body problem, and not the general two-body problem. Therefore, it would seem to belong rather under the general theory of relativity, or under a more specific article about the motion of planets. It would be possible to cover the classical and GR gravitational two-body problems as illustrative examples in this article, but that would significantly increase the article's length. What does everyone think -- is it worth it? Willow 07:38, 6 August 2006 (UTC)

Here's the deleted text again, in case anyone wishes to glean from it

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In the general theory of relativity gravity behaves somewhat differently, but, to a first approximation for weak fields, the effect is to slightly strengthen the gravity force at small separations. Kepler's First Law is modified so that the orbit is a precessing ellipse, its major and minor axes rotating slowly in the same sense as the oribital motion. The law of conservation of angular momentum still applies (Kepler's Second Law). Kepler's Third Law would in principle be altered slightly, but in practice, the only way to measure the sum of the masses is by applying that Law as it stands, so there is effectively no change. These results were first obtained approximately by Einstein, and the rigorous two body problem was later solved by Howard Percy Robertson.

Too technical

This article needs more prose. It should be intelligible for those with and without expertise in math. Not that the math is necessarily so complicated, but mathematical physics, even when it deals only with algebra, is too technical to stand alone in an article. Perhaps someone who's invested in this article and has a strong understanding of the two-body problem can expound upon the topic... ask123 (talk) 16:35, 24 April 2008 (UTC)

Generalize

It is unclear from the article what applications there are for the general solution to the two-body problem other than for inverse-square laws like gravity. -- Beland (talk) 07:42, 2 December 2008 (UTC)

That's probably because the only two forces outside an object on a more than atomic scale are gravitational and electromagnetic, and they're both inverse square. However, if one object is allowed to get inside the other, the force exerted by the one in which the other is located is linear. Sslong (talk) 18:47, 4 May 2009 (EDT)

"social" Tow-body problem

I think there should be added a sentence or two on the problem that sometimes is referred to als Two-body problem. I am however not really sure wether this sould be added to this page (the name is obviously derived from this one) or to make a redirection/disambiguation page. The last option however would mean a page specified to the social problem while I guess one could suffice with something like:

Sometimes the term Two-body problem is used for the problem that it is hard two scientists which form a couple to both find a position at a distance which is such that they can still live together.

(I have to agree that this sentence may be splitted or so). --yanneman (talk) 08:25, 25 March 2009 (UTC)

TIDAL FORCE

I think it is important to recreate the GIF animations so that they also include a colored side of a possible result of tidal force resulting from the two-body system, especially when referring to planet-satellite or planet-planet situations.

____Ἑλλαιβάριος Ellaivarios____ 21:11, 25 February 2012 (UTC)

Derivative notation

Newton's notation for the derivative with respect to time (the dots over the x in this article) is seldom enough used that many readers will be at a loss to determine its meaning. Leibniz's notation (dx/dt, dv/dt, etc.) may be preferable. --Kent Heiner 10:39, 27 January 2013 (US/Pacific)

Laws of Conservation of Energy

I don't believe that the potential function U is defined in the article. — Preceding unsigned comment added by 98.201.153.4 (talk) 20:40, 18 January 2016 (UTC)

Structural hackery done, more needed

I just did a pretty major rewrite of the overview parts of this article, trying to provide a general explanation of what's going on for more casual readers.

The rewrite was initially motivated by my own inability to easily find all the material on the problem from this article. In particular, this article presented a second-order differential equation that's at the center of the most important and interesting case, promised to show how to solve it... and then didn't. The actual solution is in other articles with counterintuitive names that didn't have any described links, just general "see also" references.

There's still a lot of work to do, and I'm not really qualified to do it. In particular, pretty much the same math is presented here, in Classical central-force problem, and in Kepler problem. Of the three, Classical central-force problem looks to be most complete... but is misnamed, because it actually also covers the generalization to the full two-body problem. There's also Gravitational two-body problem, which gives solutions for the gravitational case but does not try to derive them.

I would suggest one of the following:

• Have two articles: split the math part from the popular part, but combine all of the math into one comprehensive article.
  • Make the "Two-body problem" article just a general, mostly non-mathematical introduction, along the lines of what I've put into the intro.
  • Update the "Classical central-force problem" article to cover any math that's in this article that it doesn't already cover, and remove that math from this article.
  • Also fold in anything that's missing from the "Kepler problem" article.
  • Rename "Classical central-force problem" to some more general title that makes it clear that it actually solves the general two-body problem, not just the central force problem.
  • Replace the majority of math this article with a link to that article, and make the "Kepler problem" article a redirect to it.
  • Merge Gravitational two-body problem, which is just a list of solutions without any explanation of how you get them, into one or the other article. I'm not sure which makes more sense.

• Merge all three into a single article under the title "Two-body problem", with a popular introduction at the top and everything else at the bottom. Present the central force problem as a stepping stone to the full two-body problem, and make "Classical central-force" problem a redirect to that section in the merged article.

76.10.176.53 (talk) 19:47, 6 June 2019 (UTC)