system
Given the following post, respond as if you're commenting on Reddit:
human
How exactly do Lagrange-4 and Lagrange-5 work?

I feel like I understand L-1 through L-3 pretty well. L-1 is just where gravitational "forces" subtract perfectly to slow down orbits to geosynchronous, L-2 is where they add perfectly to speed up orbits to geosynchronous, and L-3 is on the opposite side of the sun as L-2 but has the same effect. However, no matter how many explanations I hear or how many diagrams I see, I still can't figure out what is so special about +/-60 degrees in the orbit, or why these are actually stable instead of metastable.
gpt
Using L4 and the Earth/ Moon system to be specific. At L4, the distances to Earth and Moon are equal. So the gravitation force from them adds up to point toward the barycenter of the system, and also has the correct magnitude for stability. The barycenter is the center of mass AND the center of rotation for the 3-body system, so any object at L4 is in 3-body equilibrium as it orbits. If the object moves away from L4, there is a Coriolis effect that brings it back.