Newton solved the two-body problem completely. The Sun and a single planet trace closed ellipses set by their energy and angular momentum — give the starting positions and speeds and you can name the planet's place a thousand years on. Add just one more body and that clean, predictable world collapses.
In 1889 Henri Poincaré entered a prize competition held for the 60th birthday of King Oscar II of Sweden, hoping to prove the Solar System stable for all time. He found the opposite. There is no general formula for three gravitating bodies — and worse, their motion is sensitive to initial conditions: two starts that differ by an unmeasurably small amount peel apart into completely different futures. A mistake in his own prize essay, caught after it had been printed, forced a rewrite that is often called the birth of chaos theory.
The difficulty is almost countable. Three bodies moving in a plane carry an eighteen-number state — a position and a velocity for each. Physics hands you a fixed budget of conserved quantities to spend against that: total energy, momentum, the center of mass, angular momentum — ten in all. Two bodies have just enough such constraints to be solved by plain integration; three fall short, and Bruns and Poincaré proved no new algebraic conserved quantities exist to close the gap. Karl Sundman even found a convergent power-series solution in 1912 — yet it converges so glacially that summing enough terms to predict anything real is hopeless, and it fails outright at a triple collision.
Sensitive dependence is the practical wall, and it's what the Nudge button shows. Two nearby trajectories separate exponentially fast — the rate is called the Lyapunov exponent — so every extra digit of precision in your starting measurement buys only a fixed, small amount of extra prediction time. Start a twin a millionth of a unit away and it tracks the original perfectly, then suddenly tears off on its own path. No faster computer rescues you: the information needed to predict the far future simply isn't present in any finite measurement of the start.
Most three-body systems look nothing like the tidy presets. Drop three comparable masses near each other and they trade energy through close slingshots until one is flung away for good, leaving a tight pair behind — a binary plus an ejection is the near-universal ending. The triples that do last survive by being hierarchical: a close pair orbited by a distant third, the scales kept far enough apart that each piece sees the others as almost-simple. That's why the Sun–Earth–Moon system and most triple stars endure, and the KAM theorem (Kolmogorov–Arnold–Moser) explains which near-regular orbits hold together and which dissolve.
And yet exact repeating solutions do exist — they're just islands of measure zero in a sea of chaos. Euler (three bodies in a line) and Lagrange (an equilateral triangle) found the first in the 1760s. The figure-eight, in which three equal masses chase one another along a single looping curve, was spotted numerically by Cris Moore in 1993 and proven to exist by Chenciner and Montgomery in 2000; in 2013 Šuvakov and Dmitrašinović turned up thirteen new families, and the catalog of these "choreographies" keeps growing. Strip one mass down to nothing and you get the restricted problem, whose five Lagrange points are real enough to park a spacecraft — the James Webb Space Telescope orbits the Sun–Earth L2, a million miles out, riding a balance point this very problem predicts.
Key terms
- integrable system
- One with enough conserved quantities to be solved by integration, with predictable, non-chaotic motion. Two bodies qualify; three generally don't.
- sensitive dependence
- Tiny differences in the start grow into huge differences later. The hallmark of chaos — and the reason long-term prediction fails.
- Lyapunov exponent
- The rate at which nearby trajectories separate. Positive means exponential divergence; its inverse sets your prediction horizon.
- hierarchical stability
- How real triples survive: a close pair plus a far-off third, scales separated so the chaos stays gentle. The Sun–Earth–Moon recipe.
- Lagrange points
- Five balance points in the restricted three-body problem where a light object can ride along. JWST sits at the Sun–Earth L2.