Three-Body Problem
Three massive bodies interact via Newtonian gravity in a famously chaotic system. Tiny changes in initial conditions lead to wildly different trajectories. Start with a famous periodic solution, then drag a body and watch determinism give way to chaos.
The Mathematics of Chaos
The three-body problem asks: given three masses interacting only through gravity, what are their future positions? Newton solved the two-body problem analytically, but three bodies resist any closed-form solution. Henri Poincaré proved this impossibility in 1889, inadvertently founding chaos theory.
Key Formula
F = Gm₁m₂/r² — Every pair of bodies attracts with a force proportional to the product of their masses and inversely proportional to the square of their distance. With three bodies, each feels two such forces simultaneously, creating a coupled system of 18 first-order differential equations.
Periodic Solutions
Despite the general chaos, special initial conditions produce periodic orbits. The figure-eight solution, discovered by Cris Moore in 1993 and proved by Chenciner and Montgomery in 2000, has all three equal-mass bodies chasing each other along a single figure-eight path. Lagrange’s equilateral triangle solution (1772) places three bodies at the vertices of a rotating equilateral triangle.
Sensitivity to Initial Conditions
Drag any body slightly away from its periodic-orbit position and watch the ordered dance dissolve into chaotic wandering. This sensitivity is the hallmark of chaos — deterministic yet unpredictable. The system’s Lyapunov exponent is positive, meaning nearby trajectories diverge exponentially.