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The mathemetician credited with figuring this out is Joseph Louis Lagrange (1736-1813). In 1764, the Paris Academy of sciences for the best essay on the libration of the moon. Lagrange won the award with his analysis of a special case of the three-body problem.
Earth, the moon, and L4 form a regular 60-degree triangle; ditto for L5.
L1, L2, and L3 are saddle points: stable in the theta direction but unstable in the r direction. Objects located at these points would need to use fuel to maintain their orbital position.
L4 and L5 are theoretical stable points. An object placed at these points, 60 degrees ahead of and behind the moon at the radius of its orbit, will remain at the same point with respect to the moon.
This phenomenon can be observed in the orbiter of Jupiter; 60 degrees ahead of and behind Jupiter, in its orbit around the sun, are the Trojan asteroids. The Trojan asteroids are clusters of asteroids that have become trapped in Jupiter-Sun libration points. They are visible through a small telescope, and the larger Trojan asteroids even have names.
The lunar libration points, however, aren't really points. Because of the inclination of the moon's orbit around the earth and the influence of the sun's gravity field, L4 and L5 are not stable points. The good news is that there is a stable orbit around those points. The orbit is shaped like a kidney bean and about half a million miles long. Objects in these orbits would circulate around once every 89 days.
The size of the orbits about L4 and L5 make them less attractive as a location for communication satellites, or whatever, to support operations on the moon. A satellite in these orbits will wander over a good part of the lunar sky.
This is where the late, great L5 Society got its name. The L5 Society's concept was to locate large space habitats made from lunar material in the orbits about the Lagrange points. There are lots of arguments in favor of this location for space colonies, but that's getting just a tad bit ahead of the immediate goals of the Artemis Project.
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