School of Systems and Complexity · Orbital mechanics
Three masses of comparable size, governed by Newton's law of gravity. No randomness. No approximation. And no hierarchy to hide behind. For almost all starting conditions, the trajectories are impossible to predict beyond a short horizon. This is the three-body problem in its undisguised form.
Three bodies, comparable masses, mutual gravity only. Newton's law integrated with fourth-order Runge-Kutta. Same physics as planetary motion — but without the mass hierarchy that lets the solar system pretend to be stable.
Three classic configurations. Figure-8 is rare order. Pythagorean ends in ejection. Chaos shows the generic case. Reset any scenario to see how sensitive it is to tiny differences in initial conditions.
The result
The two-body problem has an exact analytical solution. Newton wrote it down in 1687. Given the initial positions and velocities of two gravitating masses, you can calculate where they will be at any future time — exactly, forever. The orbit is a perfect ellipse, repeating without deviation.
Add a third body of comparable mass and the problem changes category. Henri Poincaré proved in 1890 that no such solution exists. The system is chaotic: small differences in initial conditions grow exponentially. No measurement precision, however fine, gives infinite predictive horizon. The uncertainty always catches up.
"It may happen that small differences in the initial conditions produce very great ones in the final phenomena. A small error in the former will produce an enormous error in the latter. Prediction becomes impossible."
— Henri Poincaré, Science and Method (1908)This was the founding result of chaos theory — discovered not by meteorologists or economists, but by a mathematician trying to prove that the solar system was stable. He proved the opposite.
The three scenarios
Discovered by Chenciner and Montgomery in 2000 — only 110 years after Poincaré. Three equal masses chase each other along a single closed figure-of-eight curve. It is one of the rare known periodic solutions of the three-body problem: a vanishingly thin slice of initial conditions for which the chaos cancels itself out by symmetry. Press Reset and watch — eventually, tiny numerical differences will break the symmetry and the chaos will reassert itself.
Studied since Burrau (1913). Three masses in ratio 3 : 4 : 5, placed at rest at the vertices of a right triangle with sides in the same proportion. They pull on each other, accelerate, have close encounters, and eventually one body is ejected. The system always ends the same way: a tight binary plus a fugitive. The Pythagorean problem is the classic demonstration that three-body systems do not generally remain bounded.
Three equal masses in a generic starting configuration — no special symmetry, no special ratios. Close encounters, temporary captures, unpredictable ejections. This is what the three-body problem looks like for almost all initial conditions: the chaotic regime, which fills the parameter space, surrounding the rare islands of order like Figure-8.
Place three bodies anywhere on the canvas. Set their masses. Press Run. The bodies start at rest — gravity does all the work. Try placing them in a perfectly symmetric triangle and watch what happens. Try one heavy body and two light ones. Try three bodies in a near-collinear arrangement. Almost every configuration ends the same way: ejection, with a tight binary left behind. The exceptions are vanishingly rare. This is the parameter space Poincaré was talking about.
The inverse experiment. A stylised star system — one dominant Sun, a handful of light planets, a moon — sits in stable hierarchy and will happily orbit indefinitely. Then you break it on purpose. Raise Jupiter's mass until it rivals what holds the inner planets in place, and the system that looked permanent unravels in seconds. Raise the Moon's mass past its planet's and the Earth-Moon binary inverts. Masses and distances are stylised for visibility, not drawn to scale: the point is the mechanism, not the numbers. Stability was never structural — it was hierarchical, and the hierarchy was yours to dismantle.
The companion question
If three bodies are generically chaotic, why has the solar system — with one star, eight planets, dozens of moons, thousands of asteroids — been stable for 4.5 billion years? The answer is hierarchy: the Sun holds 99.86% of the system's mass, and each planet's moons are tiny compared to the planet. Each body orbits something much heavier than itself, and the scales separate cleanly. The chaos that Poincaré proved was present is suppressed — not removed — by the hierarchy.
This is the more interesting story: most stable real-world systems are not structurally stable in Poincaré's sense. They are hierarchically stable. The chaos is latent in the equations the whole time. When the hierarchy breaks — when a body is comparable in mass to what it orbits — the chaos arrives all at once.
The companion simulator "When the three-body problem hides" takes the Sun-Earth-Moon system and lets you crank up the Moon's mass. You can watch the moment the hierarchy collapses and the chaos shown here begins to assert itself. Together the two pages make a single point: chaos is generic, stability is the exception, and the exception is almost always hierarchical.
A note on the solar system itself
Numerical simulations going out billions of years show that the inner solar system is chaotic in the long run. There is roughly a 1% probability that Mercury's orbit will become so eccentric within the Sun's remaining lifetime that it collides with Venus, or is ejected. We exist in a window of quasi-stability, not eternal order. Poincaré's chaos is not absent from our system. It is merely operating on timescales longer than human attention.