Sorry, I don't do General Relativity. (And Ken might well contend that I should control my forays into the Newtonian world, too ...)
Perhaps Ken or hhEb09'1 can offer some GR insight.

Grant Hutchison

Yes, I did miss that question, and yes, that is what I meant.

Relativity applies, certainly, but the description would require a numerical calculation to solve the relevant equations, as analytic solutions only appear in simpler geometries than even this one. I don't do such calculations, so like Grant I can't offer much insight, except to say that an earlier thread suggested that NASA routinely uses general relativity corrections for Lagrange-point type calculations, but I don't think they are very substantial and they might not even matter at all to within the precision of the telemetry.

Ken G, you should be more careful in your quotes! Even I had trouble figuring out what I was supposed to have said! Hold up. I don't think you can criticize the use of the term just because you misread a contour map!

Besides, those maps are not the whole story, obviously.The term is fairly well understood, what people make of it is up to them. I think calling them not actually stable is even more misleading--that's why I objected when you agreed with Ken G about that.

Ah, what a sad place the world would be if we didn't learn from our own mistakes and try to transmit those lessons to others.

From which I deduce that you can't see the point in explaining to children that there's actually no lead in lead pencils. Let 'em work it out for themselves, lazy little creatures.

Tut. Then you'll need to explain in what way the points themselves are stable, in the knowledge that forces acting on the L4 and L5 points serve to pull objects away from those points, and that those forces increase in magnitude with increasing distance from the points.

Grant Hutchison

That's as much a mischaracterization of what I said, as saying that the L4 and L5 points are unstable.Wrong. I'm here, ain't I? Which forces do you mean? You are not talking about the gravitation of the two primary masses, surely. Do you mean, forces inferred from Cornish's contour map?

It is indeed.

The forces represented by Cornish's map, yes. That's the combined effect of the gravity of the two masses and the centrifugal force of the corotating reference frame. As Ken has already pointed out, that's an unstable equilibrium. If you move a little closer to the two masses, gravity wins over centrifugal force; a little farther away, centrifugal wins over gravity.
Cornish has marked the direction of the resultant forces with blue arrows (which would have prevented my long-ago confusion if they'd appeared on the equipotential maps I looked at), and is quite explicit about the effect of these forces: "When a satellite parked at L4 or L5 starts to roll off the hill it picks up speed."

Grant Hutchison

That map does not include the velocity-dependent components of the generalized potential U_Ω, and so the map itself is a little deceptive.

Picture this: In the diagram of the rotating frame, if there were a particle orbiting L4, wouldn't there be a force more or less perpendicular to the path of the particle, pointing towards L4? That's not exactly the same as the cubic example I gave of course (much more compicated) but I think the idea is similar. It's certainly why the points L4 and L5 are said to be stable, rather than unstable.The map does not even represent the velocity-dependent centrifugal force.That sentence is followed by "At this point the Coriolis force comes into play - the same force that causes hurricanes to spin up on the earth - and sends the satellite into a stable orbit around the Lagrange point." (I've colored the word "stable" red.)

Part of the problem in this debate is the terminology. There are two separate "effective potentials", and neither is really all that informative about the stability of Lagrange points. The first is the one Grant is talking about, gravity plus centrifugal, which is great for finding the Lagrange points but lousy for analyzing stability because it leaves out the Coriolis force. Then there's the effective potential that applies for orbit around a point mass, which is a purely r-dependent potential that includes conservation of angular momentum to replace the theta behavior. That potential has a reduced dimensionality, and is very useful for stability of an orbit, but it presupposes no intrinsic theta dependence to the forces-- i.e., it achieves reduced dimensionality by assuming a symmetry that is not present in the Lagrange situation. So I don't think that one is applicable either. Grant's approach was to use the first type of effective potential, which I mentioned was unstable, but then include the coriolis effect which tends to stabilize it modulo interesting effects like teardrop and horseshoe orbits.

Yes, that's the coriolis effect that Grant was talking about.

There is no velocity-dependent centrifugal force, the frame corotates with the two massive bodies.

The fact that all three of us agree with this statement, and have from the start, shows that we are really just arguing terminology.

One doesn't have to leave out the coriolis force, but Cornish does, in his map. Impossible to draw the map otherwise.Right, we were actually talking about a specific one, the one mentioned in Cornish's paper. I don't think the two "separate 'effective potentials'" has introduced any ambiguity.I was just pointing out that it acts similar to the "restorative force" that he associated with the term "stable."Oops, I went too far there. Centrifugal force is velocity dependent, but we're talking about a specific rotating frame.Right. You both have said that "stable" is an inappropriate term to apply to the L4/L5 points. I disagree.

PS: Looking back at Cornish's paper, I just noticed an interesting omission--his map of the potential is for an example with the central mass ten times the other mass. Of course, it must be greater than 25 times in order for the L4/L5 points to be stable. I doubt that there is much of a qualitative difference, but that is interesting.

Yes one does, in that there is no general prescription for including the coriolis force in a scalar potential. You'd need a vector potential for that, as the coriolis force acts like magnetism.

The point is, though it may be easy to stay in the general vicinity of L4 and L5 if you don't miss by too much, it is still very hard to stay at L4 or L5. So in one sense it is stable, in another it is unstable. There is no stable scalar potential, and it was in that sense I was using the term. I think we all agree now.

Cornish includes it in the effective potential, in his paper, but not in the map illustration.You just said that in this case there is no scalar potential period.

Are you conceding that the L4 and L5 points are truly stable? When the conditions are satisfied I mean.

I've just thought of an example that might be better for the purposes than my first cubic example of a stable point. Take a test tube and drop a BB in the bottom of it. There is a point where the BB comes to rest, kinda like the bottom of my cubic, but three-dimensional. If there were no friction, the BB would not come to a rest, but would rock back and forth. If it were spun a little, it would "orbit" the bottom of the tube. Now, mount the test tube on a lazy susan, and turn the lazy susan. Now the motion is even more complicated. If the BB is at rest at the bottom of the tube, and it is perturbed, it will not rock back and forth, but instead will go into an "orbit". Very much like particles at L4 and L5.

PS: Exercise: Cornish mentions that the restriction on stability of L4/5 is that M₁ is greater than 25M₂ times (1-sqrt(1 - 4/625))/2, whereas this World of Physics website says the restriction is that mu is less than 1/2 - sqrt(23/108). Show that these are exactly the same.

Again, no scalar potential can include it.
I said no such thing, I said there was no scalar potential that can include the coriolis force. You can still define a scalar potential and then tack the coriolis force on manually, as in Grant's description, i.e., you can have a potential, but it will not give a complete description by itself. That's the best you can do without using a vector potential, and what "stability" means for a vector potential is a rather different animal, as we are discussing.
If they were "truly" stable, you could remain at one. Hence the terminology issue.

But you are making my point-- the BB would stay at the bottom indefinitely if placed there, yes? That is stable. The quintessential example of a stable system is a simple harmonic oscillator, whether or not there are higher dimensions that would allow "orbital" type motions (like a Foucault pendulum, say). The L4 and L5 points are not that, which is exactly what I meant when I said there were unstable. Perhaps a better term would have been "not stable in the standard sense" rather than "unstable".
Not quite, the big difference is that the speed of the BB will not grow after you perturb it, though it may deflect in direction. Near the L4 and L5 points, if you give a tiny perturbation, the speed will grow until the coriolis term is strong enough to compete with what the unstable scalar potential is doing.

As Ken says, we're debating terminology.
From the above, it's evident that you're happy to call a point "stable" if there exists a stable orbit around it.
I was making a differentiation between the stable orbit (which is a property of the region around the point), and the point itself (to which I applied the phrase "stable isn't quite the right word").
To appreciate that it's the region and not the point which is decisive in the stability issue, you could ask yourself why Coriolis force doesn't settle objects into stable motion in the vicinity of L1, L2 or L3 in the same way as it does at L4, L5: it's having exactly the same effect on objects moving in those regions. The answer is that Coriolis force traps the objects on the slopes of the effective potential surface, which in the case of L1, L2 and L3 leads the object into orbit around one of the parent masses, but in the case of L4 and L5 leads the object into orbit around the point itself.
All the Lagrangian points are stable in a mathematical way, since forces precisely balance at those points; all are unstable if perturbed (since objects are pulled away from them if they become uncentred): the shape of the potential surface surrounding them then determines if stable orbits around the points are possible.
I think that's a useful differentiation to make, and a useful understanding to have: hence my reservations about using the phrase "stable point".

Grant Hutchison

Perhaps "unstable point, stable basin" or "stable orbits around an unstable equlibrium point" effectively captures the spirit of this discussion.

I'd want to avoid the word "basin" myself, since it conjures up visions of central restorative forces in the same way hhEb09'1's analogies with graph minima and test-tubes do.
"Stable orbits around an unstable equlibrium point" works for me. It's what I was trying to convey in the post which seems to have triggered this whole debate: But that description seems to have been unacceptable to hhEb09'1, so I suspect your renderings will be unacceptable, too.

Grant Hutchison

Hee hee. Yeah, I thought better about "basin" later, my guess is "stable orbits" is really the way to go. The key point is that the stability lies in the motion, not the potential, if we restrict to configuration space. If we want to generalize our coordinate system to include velocity, then we don't need a potential at all, as we can easily solve for a vector field that the trajectory must follow at all points in x,v space, since the components of the vector field are dx/dt,dv/dt.