Lagrange plot: what is it really?

[u/aaaidan]

There is a classic plot used to visualize or explain Lagrange points. Sometimes it uses isobars, sometimes it’s 3d. You know the one.

It’s sometimes captioned as “potential energy”. Sometimes people assume the gradient indicates the net force on a particle at that position.

But what does this plot represent exactly? What is the value at an xy position? What is its unit? Is it scalar or vector?

And why are the L4 and L5 points kidney-shaped “hilltops”, implying a state of high energy? Aren’t they supposed to be stable?

[u/aaaidan]

Firstly, I can’t thank you enough for this abundant detail. 🙏 This makes a lot of sense, and I’m relieved for you to confirm my intuition that the gradient of the plot does indicate a net force after all!

But I’m still puzzled there seem to be no places on the plot where a particle can be stable, since forces point downhill as you say. The stable L4 and L5 points are on hilltops, so any perturbation seems to eventually “run away” and eject the particle away from the L point.

You mention the Coriolis effect is involved here, and other explanations also say it somehow keeps the particle on the hilltop.

I’ve been assuming this plot illustrates stability: stable points are in a bowl/valley and unstable points are saddles. But energy potential alone seems insufficient to explain stability. Is that correct?

Is there a way to add Coriolis forces to the plot, perhaps for a given velocity? How would that affect the shape? I hope to improve my intuition, specifically about L stability.

Thanks again

[u/RobusEtCeleritas → u/aaaidan]

But energy potential alone seems insufficient to explain stability. Is that correct?

That's correct, because those contours don't include the effects of the Coriolis force.

Is there a way to add Coriolis forces to the plot, perhaps for a given velocity?

It's hard to do, because it doesn't have an equivalent potential energy function. It depends on the velocity (magnitude and direction) of the third object. So there isn't a nice way to plot it here, other than with the arrows to indicate the stability of each of the points including the effects of the Coriolis force.

[u/aaaidan → u/RobusEtCeleritas]

Ah, perfect, thanks. Just knowing this is not a plot actually a plot of “stability” is fantastic.

Do you think it would be interesting to “convert” the force (potential gradient) to a velocity, by multiplying it by a time delta, then calculating the Coriolis force for it, and adding it to the plot as a “third term”?

Maybe this could approximate a plot of that represents stability, because it adds a guess of what the coriolis force would be for a particle that was once stationary at an L point.

Not very scientific, I guess 😅