The Three-Body Problem

Newton solved the two-body problem exactly: two gravitating masses follow ellipses, parabolas, or hyperbolas. Add a third body and the problem becomes — in general — unsolvable in closed form. Poincaré proved in 1889 that the three-body problem is chaotic: small changes in initial conditions lead to wildly different long-term orbits. Yet special solutions exist: the Lagrange points — five equilibrium positions where a third small mass can orbit stably. Space telescopes including James Webb orbit the L2 Lagrange point, 1.5 million km from Earth — mathematics carving stable niches out of chaos.