restricted three-body problem
Two bodies orbiting each other can be solved exactly. Add a third and the clean answer vanishes — three bodies pulling on one another generally have no neat formula and can move chaotically forever. The restricted version tames this just enough to make progress: take two big bodies (say the Sun and a planet) on a simple circular orbit, then ask how a third, tiny body — too light to disturb the other two — drifts through their combined gravity.
By assuming the third body is massless, the problem freezes the two heavy bodies into a fixed, rotating pattern, and you study a single test particle moving in that pattern. This is still hard — the test particle's path can be regular or wildly chaotic depending on where it starts — but it is tractable enough to map the landscape of gravity. Out of it come some of the most useful results in celestial mechanics, including the five Lagrange points where the test body can sit at rest relative to the two giants.
The restricted three-body problem is the everyday tool of spaceflight and Solar System dynamics. It describes a spacecraft moving in the combined pull of Earth and Sun, the orbits of asteroids trapped near a planet's Lagrange points (the Trojan asteroids of Jupiter), and the gravitational pathways — low-energy 'tubes' between bodies — that mission designers exploit to move probes around the Solar System cheaply. It is the bridge from the solvable two-body world to the genuine, chaotic complexity of many bodies.
The James Webb Space Telescope parks itself near the Sun–Earth L2 Lagrange point, a quiet spot predicted by the restricted three-body problem where it can shade itself and orbit in lockstep with Earth using almost no fuel.
A century-old idealization tells modern telescopes where to sit.
'Restricted' is doing real work: the test body is assumed massless and the two big bodies move on a fixed circular orbit. The full three-body problem, where all three masses matter, has no general closed-form solution and is famously chaotic.