[Elrac]

Yes, but that's irrelevant to your parent commenter's point. I'll explain that in simpler terms: If the object has the slightest bit of (linear) momentum then it also has a non-zero velocity, which will over time move it away from said Lagrange point.

I'll add that even with zero velocity, the Lagrange points are stable only in an ideal system where the only gravitational influence is from the two bodies between which the Lagrange points sit. An object sitting on a Lagrange point could still be perturbed by attraction from other planets or passing asteroids. Given enough time, no orbit stays stable.

[deepsun]

No, you're confusing with unstable Lagrange points and Lissajous orbits. Stable Lagrange points have forces that move back the object if it's perturbed. So, perturbed slightly, the object can move around the exact Lagrange point in various ways, but come back.

[maxerickson]

So how come there are all the natural trojans then?

[jfoutz]

There are a few things to keep in mind - and I'm probably forgetting a bunch of important factors.

First think about solar wind. light shining provides a real force on everything the sun shines on, not much, but not zero - an 800m^2 sail is about 5 newtons. about as much as holding a full pint glass above the bar, constantly, forever. It takes some effort to overcome that force. Smaller things, smaller forces - but the force is always there.

Second, think about a marble in a bowl - the object at the Lagrange point. it'll sit at the bottom of the bowl. but if you constantly move the bowl in little circles the marble will little by little pick up momentum, go a little faster and faster, then start to climb the walls of the bowl. (this is like the sun shining on the object at the Lagrange point. eventually, it'll go so fast the marble gets kicked out.

A big heavy ball bearing will take a whole lot longer to pick up that momentum, and may never get enough to escape. I'm not sure how much friction there is in space (or stuff that acts like friction) but I'd believe the big heavy rocks really can't get kicked out by just solar wind, thus natural trojans.

Anyway - this is pretty hand-wavy but the math is the same. The sun is constantly applying force, you gotta deal with that somehow. small profile from the point of view of the sun, and high density probably work out great.