Talk:Lagrangian point

Perhaps someone could generate a diagram of the various points?

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That would be lovely, but I'm restricted to ASCII here and I'm not about to start fiddling with slashes and backslashes and capital letter O's all afternoon :P -- Paul Drye

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The paragraph

The Earth's companion object Cruithne is in a somewhat Trojan-like orbit around the Earth, but not in the same manner as a true Trojan. It has a regular solar orbit that is bumped at times by Earth. When the asteroid approaches Earth, the asteroid takes orbital energy from Earth and moves into a larger, higher energy orbit. When the asteroid (in a larger and slower orbit) is caught up by Earth, Earth takes the energy back and so the asteroid falls into a smaller, faster orbit and eventually catches Earth to begin the cycle anew. Epimetheus and Janus, satellites of Saturn, have a similar relationship, though they are of similar masses and so actually exchange orbits periodically.

is fascinating information, but has nothing to do with Lagrangian points or Trojan objects. Whither should it be moved?

Maybe to asteroid? --AN

Which class of asteroids is Cruithne in and do we have a page for it?

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Put back in the "but differing" that Xaonon took out. It's a critical part of the definition!

Are you sure? I'm fairly certain that two equal masses orbiting each other would result in libration points as well -- a binary star system, for example. -- Xaonon

Well, technically you get them, but without the mass difference you lose the fundamental quality of an L-point: stability. Unless...

• Mass A is "substantially" larger than mass B -- by about a factor of 30.
• Mass C, at the libration point, has essentially no mass in comparison to both A and B.

...the points aren't linearly stable and can't hold anything. Basically, the centre of gravity of the system must be pretty close to A or it doesn't work. See J.M.A. Danby's "Fundamentals of Celestial Mechanics" (I think) where the ratio is discussed. -- Paul Drye

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Take "differing" out! Even if the above is true (which I doubt, at least for L1, L2, & L3), the intro makes no sense with "differing" when you consider the two "slightly differing" cases. It would have you believe the L points vanish at the point mass equality is passed. It's just nonsense.

And the idea in the intro that two masses combine to form L points is just lousy English, which amounts to more nonsense if you don't read between the lines.

And the intro fails to mention the important point that bodies at the L points are not at all in equilibrium, unless they have a certain velocity. Such bodies must be inserted into their L orbits as any orbital body must be.

This is my first and last contribution to the Wikipedia, as I see below that contributions must be licensed under the GNU FDL, which has proprietary features that require me to be less liberal than I normally care to be.

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Very good article, but I think the explanation as to why L1 L2 and L3 are unstable compared to L4 and L5 needs to be clearer. If you map the gradient fields for these points you'll notice that L1 L2 and L3 are at the top of hills but L4 and L5 are at the bottom of a depression, im not sure why that is but I think it is an expanation as to why objects would stay in their holes.--ShaunMacPherson 07:11, 10 Mar 2004 (UTC)

No, L1-3 are at saddle points in the pseudopotential field, while L4, L5 are at the tops of hills. (follow the external link to a pretty picture of the field.) Objects at 1-3 can just wander off, while staying at the same level. Objects at 4 and 5 fall down the hills, but then the Coriolis force kicks in, and keeps them in orbit around the Trojan points. –– wwoods 09:34, 25 Mar 2004 (UTC)

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In Lagrangian mechanics, a Lagrangian point is…

I changed this opening sentence to In celestial mechanics…, basically because:

1. The old definition suggested that Lagrange points emerge uniquely in Lagrangian mechanics. But Lagrange points are a physical phenomenon, independent from the theories or formalisms you use. They exist in Newtonian mechanics and in general relativity just as well.
2. While not actually a tautology to the insider, it may look confusingly so to an outsider. The old definition might have been true, but didn't really explain anything.
3. It makes sense to define a concept in the context of a wider, more generally known concept. Celestial mechanics meets that criterion better than Lagrangian mechanics does.

—Herbee 00:15, 2004 Mar 20 (UTC)

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An asteroid was discovered to be in Neptune's L4 point. I was wondering if someone could work it into the part talking about similar systems? The asteroid's name is 2001 QR322, and I just created a page for it. --Patteroast 16:50, 15 Jun 2004 (UTC)

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Can anyone answer a hypothetical question for me? If one had two super massive bodies of precisely equal mass orbiting about each other, would they generate Lagrange points as described here? I suspect that the positions of L1, L2 and L3 will be similar, but will L4 and L5 still be at the 60Deg Trojan points?

Thanks in advance, PBA

Yes, a system of two equal masses (they don't have to be "super massive") orbiting around their common center of mass will have all five Lagrange points. However the L4 and L5 points will be unstable. They will also be in a much higher orbit than the masses. If the masses are at a distance r from the CM, then the L4/5 points will be <math>\sqrt{3}r \approx 1.7 r<math> from the CM.
--wwoods 19:07, 23 Jun 2004 (UTC)

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So how does the rest of the solar system fit in? I mean it's all very well to speak of a 3 body system, but for all practical purposes, the other planetary bodies are going to interact with as well. How does that affect the relative stability of, for examaple, the Terra-Sol Lagrange points, or the Luna-Terra L-points? Is this possible effect the reason why one hears L5 advanced as a site for a sizable space habitat? Or is that due to some literary influence of which I am ignorant?

Yes, the presence of other masses in the real Solar System perturbs bodies at the L4 and L5 points, but obviously not too much, as evidenced by the presence of objects at various L4&5 points around the system.

L5 was proposed as a habitat site because the stability reduces the need for station-keeping, and because it was close to the Moon in terms of delta v (~0.7 km/s).

—wwoods 17:44, 21 Aug 2004 (UTC)