Three bodies admit no formula. Make the third light enough, though, and the situation changes. A light body does not affect the motion of the heavy pair at all, so it merely moves on a fixed stage.
View that stage while turning with the two. In this frame the heavy pair stands still, and what appears in its place is the outward effect of the turning.
Look for where the pull and that outward effect balance and there are exactly five places. Lagrange found them in , and they carry his name.
Three lie on the line joining the two bodies; the remaining two form equilateral triangles with them. From the inside they are called L1, L2, L3, and then L4 and L5.
L1, L2 and L3 lie inside the orbit of the planet, outside it, and opposite it across the Sun.
They do balance, in the way a pencil stood on its point balances. Nudge one and it drifts away.
Nudged inward, the pull of the Sun wins and it falls; nudged outward, the turning wins and it leaves. Nothing acts to bring it back.
Spacecraft are placed there nonetheless. A small burn every few weeks is enough to hold station, costing only a few metres per second of velocity change in a year.
L4 and L5 are different. Bodies stay there with no holding at all.
On the balance of forces alone they too are unstable. But once something starts to drift, the turning produces a sideways effect, which bends the drift so that it circles the point instead of leaving it.
There is a condition: the two masses must differ by more than about to . Every Sun and planet pair satisfies it comfortably.
So does the Earth and Moon, and there are observations of faint clouds of dust at the L4 and L5 of that pair.
Asteroids crowd the L4 and L5 of the orbit of Jupiter, degrees ahead and degrees behind, going round the Sun at the same rate as Jupiter itself.
Nothing has to hold them, so they have been accumulating since the solar system formed. More than ten thousand have been found.
The leading camp took names from the Greek side of the Trojan war and the trailing camp from the Trojan side. Collectively they are the Trojans.
Jupiter is not alone. Over thirty are known at Neptune, a few at Mars, and two at the Earth.
The L2 of the Sun and the Earth lies million km from the Earth, directly away from the Sun.
From there the Sun, the Earth and the Moon all stay on the same side, so one sunshield blocks the lot. That is how the James Webb telescope stays within a few tens of degrees of absolute zero.
L2 is an unstable point, so the telescope does not sit on it. It rides a wide orbit around the point, with a small burn about every three weeks to hold it.
Lagrange was not thinking of telescopes while solving the three-body problem. It simply turned out that way, years later.