Three-Body Problem

The three-body problem is the problem of predicting the motion of three gravitationally interacting bodies given their initial positions and velocities. Unlike the two-body problem, which Newton solved in closed form, the three-body problem has no general analytical solution. Henri Poincaré proved in 1887 that the system is non-integrable and can exhibit chaotic behavior: infinitesimally small changes in initial conditions produce exponentially diverging trajectories, making long-term prediction impossible in practice despite perfect determinism.

The problem is not merely a technical difficulty in celestial mechanics; it is the canonical example of how determinism and predictability diverge. It has become a proving ground for dynamical systems theory, numerical integration methods, and the limits of computational tractability in physical prediction. The restricted three-body problem — where one body is negligible in mass — admits special periodic solutions at the Lagrange points, but remains chaotic for generic initial conditions.