Talk:Lagrange point/Archive 2

Archive 1

Archive 2

Exact position of L3, plus minor amendments

Beginning

The L-points are not necessarily in interplanetary space.

Corrected to "in orbital configuration".

> OK ("in an orbital ..." ??)

History and concepts

His name was hyphenated : Joseph-Louis Lagrange.

Good catch.

It has "It took hundreds of years before his mathematical theory was observed". His theory was published around 1772; Trojans were observed around 1905. Thet's not "hundreds of years" later.

"Over a hundred years"?

> OK

Diagrams

Recreated contour plot with n=25

The first, "... contour plot ...", diagram shows Earth, L3, L4 & L5 on a Sun-centred circle, and L1 & L2 reasonably close to Earth. That's satisfactory.

Actually, it shows L3 just outside the circle. It may not be all that clear.

> Agreed, agreed.

The problem is that the contour plot clearly shows a system where the ratio of masses primary:secondary is of the order of 10:1-50:1. In that case L3, L4 and L5 will be visibly off the secondary's orbit. I recreated the diagram here. –EdC 00:36, 5 February 2007 (UTC)

The second, "... far more massive ...", diagram shows L3 outside the circle. But if Earth, L4 & L5 all lie (as far as can be seen) on a primary-centred circle, then L3 should be similarly on that circle; not outside it.

The circle should be the orbital path of the secondary (centred on the barycentre), in which case L3 lies outside it. The diagrams should be fixed by moving the primary away from the barycentre, and L4 and L5 outside the circle.

> Doubt. Could be better to have "very much more massive" with Moon L3 L4 L5 on a circle centred on Earth, and L1 L2 very near Moon, AND also "considerably more massive" with everything properly shown. If the latter is a bit bigger, it will serve also for the L4 L5 geometrical srgument.

Yes, that could work. In that case the "very much more massive" diagram should have L1 and L2 pulled in as far as practicable. –EdC 00:36, 5 February 2007 (UTC)

Done, now we need to decide how to fix the contour plot. –EdC 02:16, 5 February 2007 (UTC)

The blue triangles (showing the gradient to be downhill going away from the points) indicate that L4 and L5 are unstable equilibria, whereas they are actually stable equilibria. —Preceding unsigned comment added by 208.71.200.91 (talk) 05:26, 24 February 2010 (UTC)

Yes L4 and L5 are the stable equilibria. But they really are the maxima of the pseudopotential; it takes the Coriolis force to keep objects from falling away from those points.

—WWoods (talk) 09:21, 24 February 2010 (UTC)

The triangles are equilateral and do not act effectively as arrows. It is therefore difficult if not impossible to be certain which direction they are intended to be pointing.

—thereaverofdarkness (talk) 20:47, 10 August 2015 (UTC)

Section "L₃"

The page says : "L₃ in the Sun-Earth system exists on the opposite side of the Sun, a little farther away from the Sun than the Earth is" - my italics. That wording will naturally be taken as saying that L3 is further from the centre of the Sun than the centre of the Earth is.

The better calculations measure distances from the barycentre, and show that L3 is a little further from the barycentre than the centre of the Earth is. But it seems that L3 is a little nearer to the centre of the Sun than the centre of the Earth is.

Hm. Yes, it is, isn't it?

> Not a lot of people know that, though.

Fixed - I hope. –EdC 01:05, 5 February 2007 (UTC)

New point: the article says "Example: L3 in the Sun–Earth system exists on the opposite side of the Sun, a little outside the Earth's orbit but slightly closer to the Sun than the Earth is." But how can it be OUTSIDE the Earth's orbit but CLOSER to the sun??

Because the Sun also orbits the barycenter – hence the Sun is closer to the far side of the Earth's orbit (if we ignore eccentricity, perturbation from other planets, and possibly a bunch of other things I forgot).  :) — the Sidhekin (talk) 21:48, 24 May 2008 (UTC)

External Links

Should include the paper itself, via Gallica.

Which paper?

> the one for which the reference in the main page is missing ...

> Lagrange, Joseph-Louis, ESSAI SUR LE PROBLÈME DES TROIS CORPS

> via <a href="http://www.merlyn.demon.co.uk/gravity4.htm#Refs">.

> It's written in Maths, slightly diluted with French.

Added as a reference. –EdC 02:15, 5 February 2007 (UTC)

See ...

<a href="http://www.merlyn.demon.co.uk/gravity4.htm#GLP">The Geometry of the Lagrange Points</a>.

Looks useful.

Added as an external link. –EdC 02:16, 5 February 2007 (UTC)

82.163.24.100 23:05, 2 February 2007 (UTC)

Thanks for your comments. –EdC 04:49, 3 February 2007 (UTC)

> 82.163.24.100 15:52, 3 February 2007 (UTC)

L_n or Ln ?

I know of no other work in which L1 is written as L₁, etc.; I suggest that the suffixing should go. 82.163.24.100 (talk) 13:45, 10 August 2009 (UTC)

I am relocating this section in an attempt to resurrect it. It seems like this fundamental question was never resolved, and the article remains inconsistent with respect to the format of the labels. I have no educated opinion on which is correct; however, I do suggest we should pick one format and stick to it throughout. It's a really simple point but it impacts the credibility of the article. – Wdchk (talk) 04:39, 11 June 2015 (UTC)

I left a comment here: User_talk:Ohms_law#Lagrange_templates --S Philbrick(Talk) 14:40, 15 December 2015 (UTC)

Footnotes four and five; same or different?

I noted that both footnote 4 and footnote 5 were dead links. I fixed footnote 4 with a link to the Internet archive source. I also looked at the source associated with footnote 5: Wayback link. At first glance, it appears to be the identical content as that in footnote 4 except for the header comment. Does anyone disagree if not we should simply replace footnote 5 with footnote 4. If it actually is different we should create a reference which picks up the Internet archive source.--S Philbrick(Talk) 14:43, 15 December 2015 (UTC)

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Why aren't they simply called balance points?

The intentional confusion and obfuscation around Mathematics in English is tiresome and childish. — Preceding unsigned comment added by 2600:1700:EA10:E5D0:A0A7:570F:5F5E:8A (talk) 06:53, 12 February 2018 (UTC)

Changes made to the Intuitive Explanation

The Intuitive Explanation as presented was simply wrong. The outward force sensed by the hand twirling the string is a real physical force, namely the tension. It is not the centrifugal force. Further, the fact that when released the revolving mass travels on a straight tangential trajectory has nothing to do with the centrifugal force being 'fictitious'. When the string is cut, there is no longer any centrifugal nor centripetal force, so we are in an entirely new dynamical situation which is unrelated to the previous state. In addition to fixing these problems I have made the section less verbose and replaced the word 'weight' (which actually means 'gravity force') by a specific item, the stone. PlantTrees (talk) 20:26, 12 March 2008 (UTC)

Possibly related question

Is there a stable orbit which runs through L4 L5, but into the 3rd dimension in the diagram? Or any pair of L1 L2 L3 ? Or any other pair? Just a mental rambling on my part, L4 L5 looks fairly obvious, but I am not able to do the math. —Preceding unsigned comment added by 78.32.144.39 (talk) 18:26, 10 April 2009 (UTC)

If you are still here 9 years later, then Yes and Yes. There are a number of excellent YouTube videos that show actual orbits of the Trojan (at the L4 & L5 points) and Hilda (at the L3 point) asteroids around Jupiter. It shows them in a sort of a "perspective" projection, which means you get some idea of "vertical" motion. You can see all combinations. Some asteroids stay at L4 and L5, some move between L4 and L5, some move through all three points. Be aware that the "easiest to understand" visualisations are shown as rotating with Jupiter, so Jupiter itself appears stationary. Search YouTube for "Trojan asteroids". As part of "background", these usually show the normal asteroid belt in green. — Preceding unsigned comment added by 58.166.224.167 (talk) 00:19, 3 March 2018 (UTC)

Here it is: https://www.youtube.com/watch?v=yt1qPCiOq-8 — Preceding unsigned comment added by 121.212.147.109 (talk) 03:26, 3 March 2018 (UTC)

Kidney bean-shaped orbit round L₄ and L₅?

The paper by Cornish cited in the article shows that orbits around L₄ and L₅ are characterised by two frequencies. Since these are not in general harmonically related, it seems unlikely that these orbits will have a simple closed form.

Suppose that we rotate the coordinate axes in Cornish's paper so that the x-axis is parallel to the 'ridge' in the potential 'hill'. The effect of this on the evolution matrix is to diagonalise the 2x2 submatrix forming its bottom left-hand corner. It then becomes

${\begin{bmatrix}0&0&1&0\\0&0&0&1\\{\frac {3}{4}}\left(2-{\sqrt {3\kappa ^{2}+1}}\right)\Omega ^{2}&0&0&2\Omega \\0&{\frac {3}{4}}\left(2+{\sqrt {3\kappa ^{2}+1}}\right)\Omega ^{2}&-2\Omega &0\\\end{bmatrix}}.$

The eigenvalues (which indicate the characteristic frequencies of the system when the orbit is stable) are of course unaffected by this rotation. If $\epsilon$ is an eigenvalue, it can be shown that

${\frac {\delta y}{\delta x}}={\frac {2\Omega \epsilon }{{\frac {3}{4}}\left(2+{\sqrt {3\kappa ^{2}+1}}\right)\Omega ^{2}-\epsilon ^{2}}}$

Substituting the eigenvalues in turn into this, we obtain

${\frac {\delta y}{\delta x}}=\pm i{\frac {\sqrt {2-{\sqrt {27\kappa ^{2}-23}}}}{2-{\frac {1}{4}}{\sqrt {27\kappa ^{2}-23}}+{\frac {3}{4}}{\sqrt {3\kappa ^{2}+1}}}}$

and

${\frac {\delta y}{\delta x}}=\pm i{\frac {\sqrt {2+{\sqrt {27\kappa ^{2}-23}}}}{2+{\frac {1}{4}}{\sqrt {27\kappa ^{2}-23}}+{\frac {3}{4}}{\sqrt {3\kappa ^{2}+1}}}}$

That these are pure imaginary is a consequence of having diagonalised part of the evolution matrix. It follows that the orbit has the general form

$\delta x(t)=A_{1}\sin \left({{\frac {1}{2}}{\sqrt {2-{\sqrt {27\kappa ^{2}-23}}}}\Omega t+\phi _{1}}\right)+A_{2}\sin \left({{\frac {1}{2}}{\sqrt {2+{\sqrt {27\kappa ^{2}-23}}}}\Omega t+\phi _{2}}\right)$

$\delta y(t)=A_{1}{\frac {\sqrt {2-{\sqrt {27\kappa ^{2}-23}}}}{2-{\frac {1}{4}}{\sqrt {27\kappa ^{2}-23}}+{\frac {3}{4}}{\sqrt {3\kappa ^{2}+1}}}}\cos \left({{\frac {1}{2}}{\sqrt {2-{\sqrt {27\kappa ^{2}-23}}}}\Omega t+\phi _{1}}\right)$

$+A_{2}{\frac {\sqrt {2+{\sqrt {27\kappa ^{2}-23}}}}{2+{\frac {1}{4}}{\sqrt {27\kappa ^{2}-23}}+{\frac {3}{4}}{\sqrt {3\kappa ^{2}+1}}}}\cos \left({{\frac {1}{2}}{\sqrt {2+{\sqrt {27\kappa ^{2}-23}}}}\Omega t+\phi _{2}}\right)$

where $A_{1}$, $A_{2}$, $\phi _{1}$ and $\phi _{2}$ are constants chosen to match a particular set of initial conditions. The orbit is thus a kind of epicycle in which the circular motions have been replaced by elliptical ones; it is not, however, a squashed epicycle since the eccentricities of the ellipses differ.

The relationships between the shapes of these elliptical components and that of an equipotential are shown in the above. The innermost (solid) ellipse is an equipotential while the middle (dashed) ellipse shows the shape of the slower of the two elliptical components and the outer (dotted) one that of the faster.

The above shows part of an orbit for a system with an earth-moon mass ratio. The periods of the elliptical components are about 3.35 and 1.05 times that of the two-mass system. It is fairly apparent from the picture how, as the body falls down the potential hill, the Coriolis force changes its direction of motion to bring it back up again.

Can the orbit ever be kidney bean-shaped? Cornish's paper - and thus the above - deal with approximations to orbits which lie 'close' to a Lagrangian point. Presumably, if (i) the size of the orbit became such that the curvature of the potential 'ridge' were large enough to 'bend' the ellipses and (ii) if we had a situation analogous to that in which either $A_{1}$ or $A_{2}$ were 0, we would obtain a kidney bean-shaped orbit rather than an elliptical one but the probability of the second condition occurring naturally would seem to be 0.

--IanHH (talk) 17:27, 16 March 2009 (UTC)

My impression is that when they talk about a "kidney bean" shaped orbit, they usually mean with respect to a frame co-rotating with the smaller mass. So if you revolve your Spirograph, then as you said, depending on your A1/A2/A3, you may well be able to get a "mutant" kidney shape.

Finally, a number of simulations shown on YouTube of specifically the Trojan asteroids, show several that visit around L3, L4 and L5. It's a bit hard to tell, though whether they are "kidney beaning". I would have made the asteroids different colours :D — Preceding unsigned comment added by 121.212.147.109 (talk) 03:31, 3 March 2018 (UTC)

L1

The distance of L1 from the smaller body should be added. I suppose it is 1/C^2 the distance between the two large bodies, where C is the quotient of their masses? --Roentgenium111 (talk) 13:23, 10 December 2009 (UTC)

I rather doubt that 1/C² can be accurate, though it may be a first approximation. http://www.merlyn.demon.co.uk/gravity4.htm may help. 82.163.24.100 (talk) 20:01, 2 February 2010 (UTC)

For Sun/Earth the distance ratio is near 100, and the mass ratio is near 300,000. The distances for L1 & L2 are almost equal for the Sun/Earth case, both ~1.5e6 km from Earth. The animation figure (Lagrangianpointsanimated.gif) shows the distance to L2 is much larger than to L1, which is wrong and needs to be fixed. Both of those distances are shown as several times too large, >>0.01 AU; which would be hard to see if rendered correctly, but this could be fixed by dropping the caption that refers to the Sun and Earth. Wwheaton (talk) 17:16, 6 May 2010 (UTC)

Can't possiby be 1/C^2. Jupiter is about 1/1000 the mass of the Sun, and the table shows 1-L1/SMA to be 6% . — Preceding unsigned comment added by 121.212.147.109 (talk) 03:37, 3 March 2018 (UTC)

How stable is "stable"?

I don't really understand this article. Would something placed at a Lagrange point really stay there, or would it need to fire thrusters every now and then to really stay there? Is it more like a gravitational hole where things placed there are pulled towards the center of a lagrange point, or is it just that you wouldn't feel any gravity there, but since you can't set your inertia precisely you would inevitably drift away from that point? 89.12.78.225 (talk) 19:09, 24 July 2010 (UTC)

L1, L2, L3 are unstable equilibria: there is no net force (in a rotating frame) at those precise points, but a body will fall away from them along the line between the major masses (but back toward that line) – like a ball balanced on a saddle. L4, L5 are stable equilibria, effectively attractors: a body perturbed from either will continue to wander in the neighborhood, like a ball in a bowl. Or so I (mis)understand. —Tamfang (talk) 22:13, 25 July 2010 (UTC)

Not exactly. Yes, the initial force will be along that line, but because the radius of the orbit changes, then due to conservation of angular momentum, the velocity will change, and the object will move perpendicular to that line. So if it moves "inward" the velocity will increase; conversely, if it moves "outward", the velocity will decrease. — Preceding unsigned comment added by 58.167.55.219 (talk) 07:43, 7 March 2018 (UTC)

With active and sufficiently accurate control, the amount of propellant needed to station-keep at the equilibrium point can be made arbitrarily small. The acceleration (read thrust) needed can also be very small. A small ion drive unit, as are now used on spacecraft at GSO, should be entirely sufficient, I think. Small perturbations by the Sun (for Earth/Moon), Moon (for Sun/Earth), planets, solar radiation pressure, etc, may be the limiting factors. Wwheaton (talk) 19:41, 8 August 2010 (UTC)

L4 and L5 are completely stable, provided that the primary/secondary mass ratio exceeds about 25 and external perturbations are minor. Otherwise, only L1 L2 L3 require station-keeping. 94.30.84.71 (talk) 11:18, 9 February 2011 (UTC)

Special Cases

If there is no rotation and the third body has negligible mass, both L4 and L5 become a circle around the mid-point of the line joining the other two bodies. 94.30.84.71 (talk) 13:21, 4 July 2012 (UTC)

If the two large masses are equal, then (L2 L3) and (L4 L5) are two indistinguishable pairs. 94.30.84.71 (talk) 13:21, 4 July 2012 (UTC) Uhh, that can't be right, surely? (L2 L3) are collineat with the two large masses, (L4 L5) are equilaterally triangular. How can they possibly be the same? Or have I misunderstood this sentence, and if so, does that mean the sentence is "bad" even if it is correct??? — Preceding unsigned comment added by 58.167.55.219 (talk) 07:51, 7 March 2018 (UTC)

Contradiction

The issue of whether L4 and L5 are stable has been raised several times above. Could this be sorted out please. At the moment there's a straight contradiction between the contour plot, showing blue arrows leading "downhill" from L4 and L5, and the statement "the triangular points (L4 and L5) are stable equilibria ...". I can't fix it myself because I don't know which one is right. Occultations (talk) 11:37, 24 February 2008 (UTC)

I came to this Talk page for the same reason. L4 and L5 would have to be low points in the potential in order for them to be stable (according to this NASA page) and have objects orbit those points. The fact that it's stable only if conditions regarding the M1/M2 ratio should come later as, presumably, the "islands" of stability just get smaller as the ratio decreases until they eventually disappear. bcwhite (talk) 12:09, 31 May 2012 (UTC)

The points are dynamically unstable (an object perturbed from L4 will continue to move away) but form stable equilibria (Coriolis force will curve an object's path back to L4). EdC (talk) 15:46, 24 February 2008 (UTC)

The plot's caption used to say, "Counterintuitively, the L₄ and L₅ points are the high points of the potential." Maybe something about the stability of L4 and L5 should be added to the introduction?

—WWoods (talk) 17:23, 24 February 2008 (UTC)

Stability of L4 & L5 depends on the mass ratio of the primary and secondary bodies. For Earth/Moon they have been analytically proved to be stable (ca ~1980 I think; in the sense that a small test mass near either L4 or L5, moving at low speed with respect to the Lagrange point, will remain in its vicinity), and I believe this result also applies to Sun/Earth & Sun/Jupiter, as ratio m₁/m₂ is even larger (~80 for Earth/Moon). However, the stability depends on the Coriolis force, as an object a small distance from the L4 (say) point will at first move away, but then move back towards it, looping I think in a rosette sort of path. Keith Symon's old textbook Mechanics (2nd edition, 1960, Addison Wesley) discusses this problem fairly extensively at an advanced undergraduate level, although in 1960 the long-term question was still open. I am not certain that the Earth/Moon proof actually applies to the real physical situation, with a somewhat (~0.05) eccentric orbit, perturbed by the Sun, but I believe long accurate numerical integrations suggest they are stable. Wwheaton (talk) 16:11, 6 May 2010 (UTC)

I'm still confused. This article doesn't mention "Coriolis" anywhere, yet that appears to be critical to stability. Obviously, L₄ and L₅ look unstable in terms of effective potential. Understanding that earth is orbiting the sun and therefore that L₄ and L₅ are orbiting the sun, I can see the potential for orbiting-like behavior around those points, but I haven't worked out the details. Also, I've never seen "effective potential" before. It looks like effective potential drops off at a distance as centrifugal force takes over. In a two-body problem, I think the effective potential for a given angular momentum would be an annular trough, the bottom of which would indicate the the radius of the stable circular orbit (and the trough shape indicating stability).

I think this article needs a description of the effects of perturbation of a mass near all Lagrange points, particularly L₄ and L₅. I'd like to see something like this: "Suppose a small object is placed at L₄ with the same angular velocity as earth. Were earth not there, it would be in orbit around the sun, but with earth there, it slows down (falls back toward the earth), causing it to drop to a lower orbit, speed up angularly, move forward (away from the earth), etc., orbiting L₄." Is that even right? —Ben FrantzDale (talk) 14:49, 4 March 2015 (UTC)

All we need is a simple analogy. L4 and L5 are stable. Like a marble in the centre of a valley. If it is moved away from the centre a tiny bit, it rolls back to the centre. The other points are unstable. Like a marble on the tiny flat spot on the top of a hill. If it is moved away, even a tiny bit, it will then roll away from the hill. — Preceding unsigned comment added by 121.212.53.120 (talk) 03:47, 1 February 2018 (UTC)

I added the following clarification to the stability section, which I'm hoping resolves this issue: "Although the L₄ and L₅ points are found at the top of a "hill", as in the effective potential contour plot above, they are nonetheless stable, as such a diagram ignores Coriolis acceleration (which depends on the velocity of an orbiting object and cannot be modeled as a contour map)." expensivehat (talk) 22:11, 5 September 2018 (UTC)

L1/2 equations possibly wrong

In "mathematical details", in the starting equations, the last two terms come from omega*(R+r) and omega*(R-r) and then the R cancels one power of R in the denominator in the first of the two last terms - so why is the numerator only M1 and not M1+M2? Yes, later there is a discussion of the case when M2<<M1, but the starting equation are written as exact, so that's not the reason. 109.239.67.153 (talk) 20:15, 11 February 2020 (UTC)

Removing clarification needed from the "L1 point" section

The unnecessary clarification was requested by Etymographer on 7 June 2020. The reason stated for clarification was an incorrect intuition that is already explained in the paragraph's text. The smaller large mass M2 orbits the larger large mass M1, and a relatively minute mass (say, a spacecraft) occupies L1, a point between M1 and M2. Thus, M1 pulls the craft one direction and M2 pulls it the opposite direction, but the pulls are unequal (M2's is considerably smaller). The net effect on the craft is that M1 cannot pull on the craft as strongly as it would in the absence of M2. Thus, the net gravitational force at L1 is less than that of M1 alone. The craft thus orbits M1 in a lower orbit which takes exactly as long as M2 takes to orbit M1. It is the reduced net gravity at point L1 that is responsible for changing the orbital mechanics at that point enough to reduce the orbit of the craft, which is exactly what makes L1 special. The sentences following the clarification notice say just the same thing in other words. The clarification needed flag is removed. Evensteven (talk) 07:29, 8 July 2020 (UTC)

Just a comment: this article takes fewer pains towards being understandable in favor of greater strains in trying to be technically accurate and mathematically complete. I have a Bachelor's degree in math, and like a good formula as well as any, but I see the use in the applications of things too, and this article could do a better job of communicating those things. For a more practical type of presentation, check out [1]. They manage to get a lot of basic ideas across without technical overkill. Wikipedia could learn from them. Unless, of course, the object here really is to train post-graduate students in orbital mechanics. Evensteven (talk) 22:41, 8 July 2020 (UTC)

Mars terraforming

In the "future / proposed mission," I think mention should be made of the idea of stationing a magnet at the Mars-Sol L1 point to create an artificial magnetosphere on Mars. See also the Terraforming_of_Mars page.

Oops. I see that there is a mention of that in the article, but I still think it ought to be added to the table, as it has been "proposed."

Nsayer (talk) 22:39, 22 July 2020 (UTC)

Requested move 18 August 2020

The following is a closed discussion of a requested move. Please do not modify it. Subsequent comments should be made in a new section on the talk page. Editors desiring to contest the closing decision should consider a move review after discussing it on the closer's talk page. No further edits should be made to this discussion.

The result of the move request was: Consensus to move page. (non-admin closure) — YoungForever^(talk) 20:23, 25 August 2020 (UTC)

---

Lagrangian point → Lagrange point – "Lagrangian point" give 141,000 results, while "Lagrange point" give 664,000 results. Therefore, "Lagrange point" is WP:COMMONNAME. Soumya-8974 ^talk _contribs ^subpages 18:12, 18 August 2020 (UTC)

• Support per nom, common name, and the now inconsistent Lagrange point colonization. It's already the prevalent name used in the article, and the usage of 'Lagrange' in External links is clear. Randy Kryn (talk) 19:22, 18 August 2020 (UTC)
• Support per above. --Ab207 (talk) 14:28, 20 August 2020 (UTC)
• Support per nom Cas Liber (talk · contribs) 10:48, 23 August 2020 (UTC)

The discussion above is closed. Please do not modify it. Subsequent comments should be made on the appropriate discussion page. No further edits should be made to this discussion.

L3 calculations

" r now indicates the distance of L3 from the position of the smaller object" In Mathematical details of L3. It should be mentioned that positive r implies L3 is closer to the larger body than the smaller body. It is so confusing right now. Also, the reference uses different origin which does not correspond to the result mentioned on Wikipedia — Preceding unsigned comment added by Garlicbeaver (talk • contribs) 13:41, 23 January 2021 (UTC)

I'm afraid this and the recent edit on the subject do not make sense to me. r in the derivation is along the line between the two bodies (and their center of mass.) Since the L3 point is on the far side from the smaller body, both the greater and smaller bodies are in the same direction from the L3 point. So a positive r implies a closer distance to both bodies. But the current text does define r as "position of the smaller object, if it were rotated 180 degrees about the larger object." So maybe someone was saying a positive r means closer to the large body than where the smaller body would be if it were rotated 180 degrees about the larger body. But that's hopelessly confusing. Could we just define R as equal to the the distance from smaller body and the center of mass, and r as the distance from a point R away from the center of mass and in the opposite direction from the smaller mass? (And towards the center of mass) Maybe adding a diagram with labels would help. Fcrary (talk) 03:36, 24 January 2021 (UTC)

Do we need a lengthy "in fiction" list of trivia here?

It is a leading question, I know. My opinion is that lengthy lists of every science fiction book and cartoon series that mention a given topic are not very useful. Look at the size of this list in relation to the article as a whole. I am trying to imagine the reader that comes to this article that would find such a list helpful. I am tempted to delete the entire section unless somebody wants to talk me out of it. CosineKitty (talk) 00:58, 22 March 2010 (UTC)

I concur. But I don't see much, if any, of a trivia section left. Maybe someone already deleted it. N2e (talk) 04:10, 25 July 2010 (UTC)

Personally, I liked it and am sorry to see it gone. There should at least be some sort of link to such a list, at Wikipedia or elaewhere. 94.30.84.71 (talk) 21:18, 8 February 2011 (UTC)

There is a list of references, fiction and non-fiction, mainly early ones, at http://www.merlyn.demon.co.uk/gravity4.htm#R3 . 94.30.84.71 (talk) 17:42, 4 July 2012 (UTC)

On trying http://www.merlyn.demon.co.uk/gravity4.htm#R3 the response was: "This site can’t be reached". I was looking for The Planet of Junior Brown by Virginia Hamilton. The planet is a fictitious one sharing Earth's orbit. Sounds like it's at one of the Lagrange points. Whether Hamilton knew about them is anyone's guess. How about starting a section "In Popular Culture" to document such instances? Particularly now that the merlyn.demon.co.uk page is defunct.2603:6010:4E42:500:3413:383D:A6CC:6AAE (talk) 21:28, 23 September 2021 (UTC)

Site www.merlyn.demon.co.uk disappeared when Demon Internet's user-sites all disappeared.

The index to an unauthorised out-of-date copy of that site, taken in 2010 by Wen-Nung Tsai of NCTU in Taiwan, can still be found at http://www.cs.nctu.edu.tw/~tsaiwn/introcs/sisc/runtime_error_200_div_by_0/www.merlyn.demon.co.uk/index.htm - it includes links to the relevant material.

More recent versions, from 2015, with much more Euler/Lagrange material can be found on The Wayback Machine, by starting at http://web.archive.org/web/20150514050046/http://www.merlyn.demon.co.uk/contents.htm - including translations of Lagrange's famous Essai and of E.304 & E.327.

94.30.84.71 (talk) 22:04, 13 October 2021 (UTC)

Once again from the top

There is no such thing as a 'centrifugal force'. GMS. — Preceding unsigned comment added by 92.12.18.129 (talk) 15:04, 24 December 2021 (UTC)

Mention of the Hilda Asteroids?

Whilst of course these are not AT Jupiter'Lagrange[ian] points per se, they do occasionally visit these points. So perhaps a SHORT reference to them might be appropriate? With then a link to the article about them? — Preceding unsigned comment added by 2001:8003:E422:3C01:7C5F:A8BC:333A:7D2D (talk) 23:08, 26 December 2021 (UTC)

I'm not sure about how to keep the reference short. Strictly speaking, the Hildas are on one of many types of stable, resonant orbits. Lagrange orbits are the stable 1:1 resonances, Hildas are on a particular sort of 3:2 resonance, and there are lots of other stable, resonant orbits. If we mention the Hildas, we should probably also mention other stable, resonant orbits. But describing that wouldn't be easy to keep short. Maybe just add Hildas and resonant orbits to the "See also" section? Fcrary (talk) 00:50, 27 December 2021 (UTC)

Orphaned references in Lagrange point

I check pages listed in Category:Pages with incorrect ref formatting to try to fix reference errors. One of the things I do is look for content for orphaned references in wikilinked articles. I have found content for some of Lagrange point's orphans, the problem is that I found more than one version. I can't determine which (if any) is correct for this article, so I am asking for a sentient editor to look it over and copy the correct ref content into this article.

Reference named "P Society Luyuan Xu":

• From List of missions to the Moon: How China's lunar relay satellite arrived in its final orbit. Luyuan Xu, The Planetary Society. 15 June 2018.
• From Yutu-2: Xu, Luyuan (15 June 2018). "How China's lunar relay satellite arrived in its final orbit". The Planetary Society. Archived from the original on 17 October 2018.

I apologize if any of the above are effectively identical; I am just a simple computer program, so I can't determine whether minor differences are significant or not. AnomieBOT⚡ 16:07, 30 December 2021 (UTC)