How do LaGrange Points work?

Hmmm. Well, L5 and L4 are at the same distance as the moon. I would think their orbit depends on orbital speed, rather than any neutral gravity point. I know, that's an observation, not an answer.

Actually, rosborn, if I had wondered, I would have asked you.

An object at L1, L2, or L3 is meta-stable, like a ball sitting on top of a hill. A little push or bump and it starts moving away. A spacecraft at one of these points has to use frequent, small rocket firings or other means to remain in the area. Orbits around these points are called 'halo orbits'. The Solar and Heliospheric Observatory (SOHO) is in a halo orbit around the Sun-Earth L1 position, about a million miles Sunward from Earth, and the Microwave Anisotropy Probe (MAP), is in a halo orbit around the Sun-Earth L2 Position, about a million miles in the opposite direction. These are the first spacecraft to be positioned in Lagrange orbits.

An object at L4 or L5 is truly stable, like a ball in a bowl: when gently pushed away, it orbits the Lagrange point without drifting farther and farther, and without the need of frequent rocket firings. The Sun's pull makes any object in the Earth-Moon L4 and L5 locations "orbit" the Lagrange point in an 89-day cycle. These regions could be ideal for the Space habitats devised by Gerard K. O'Neill in 1969. Each habitat could house tens of thousands of people. Lagrange points

The first three Lagrangian points are stable only in the plane perpendicular to the line between the two bodies. This can be seen most easily by considering the L1 point. A test mass displaced perpendicularly from the central line would feel a force pulling it back towards the equilibrium point. This is because the lateral components of the two masses' gravity would add to produce this force, whereas the components along the axis between them would balance out. However, if an object located at the L1 point drifted closer to one of the masses, the gravitational attraction it felt from that mass would be greater, and it would be pulled closer. (The pattern is very similar to that of tidal forces.)

The L1 and L2 points have some practical value since, although a satellite would wander away if left to itself, a relatively modest effort at station-keeping can prevent it from doing so.

By contrast, L4 and L5 are stable equilibria (cf attractor), provided the ratio of the masses M1/M2 is > 24.96. When a body at these points is perturbed, it moves away from the point, but the Coriolis force then acts, and bends the object's path into a stable, kidney bean-shaped orbit around the point (as seen in the rotating frame of reference). In the Sun-Jupiter system several thousand asteroids, collectively referred to as Trojan asteroids, are in such orbits. Other bodies can be found in the Sun-Saturn, Sun-Mars, Jupiter-Jovian satellite, and Saturn-Saturnian satellite systems. There are no known large bodies in the Sun-Earth system's Trojan points, but clouds of dust surrounding the L4 and L5 points were discovered in the 1950s. Clouds of dust, even fainter than the notoriously weak gegenschein, are also present in the L4 and L5 of the Earth-Moon system.

http://www.worldhistory.com/wiki/L/Lagrangian-point.htm

http://www.worldhistory.com/wiki/L/Lagrangian-point.htm