[idlewords]

I'm having some trouble with the opening salvo here:

"Now imagine placing a marble somewhere on that slippery sheet -- it is guaranteed to fall into one of the funnels. "

This holds for the case where there are two objects initially at rest, but I don't see it as obviously true if there are more than two objects in the universe.

[jholman]

I see two aspects to your question/comment: at-rest, and two-objects.

The whole point of that comment was to apply only to at-rest things (he goes on to contrast it with objects with velocity), though I think maybe you got that.

If you have a marble and two other objects, the other objects make two funnels, and there is a saddle (itself curved) where the two objects's funnels are equipotent. That's the three-body case. If it were possible to balance perfectly on this line of equipotence, you'd just slide to the lowest point on that line, and stay there, out of the funnels. This, IIUC, is one of the Lagrange points... L1, I think. In theory it's possibly-stable, but its stability exhibits negative feedback, so in practice it's impossible (although with station-keeping rockets, it's cheaper to hover there than most places).

As you add more funnels, you just get more of these saddle lines intersecting. There are many infinitely-fussy places in the universe where you could in-theory-but-not-in-practice hover without falling into a funnel (if it weren't for brownian motion and maybe some other quantum effects that disturb your infinitely-difficult equilibrium).

Of course, this whole analogy doesn't account for the fact that all of the bodies are acting on each other, not just the funnels acting on the marble.

[idlewords]

Right, the fact that all the bodies are acting on the other is what makes me feel the analogy is not kosher. The funnels are going to start moving around based on their mutual attraction, and I could see the marble being ejected to infinity, for example. It's less clear to me (but possible) that it could enter a stable orbit.

[lips]

This was lovely writing, and successfully enlightened me on a few points, but my mind immediately saw the marble deform the sheet. I'd suggest a ping-pong ball and a frictionless sheet for a less perplexing thought experiment.

[robotresearcher]

The marble can deform the sheet and it still works.

[mrfusion]

Isn't the universe expanding? If you start in the right place couldn't everything be moving away from you faster than you'd be pulled in?

[fixermark]

Yes. He's simplifying the description (though technically, he might still be correct; if only you and your pet existed in space and had no relative motion, it's entirely possible that dark energy would not exist, since we don't yet understand its origins or nature).

[4ad]

You're right, it's not obviously true. In fact it's quite complicated. But you can prove it mathematically.