The three-body problem

Two things pulling on each other by gravity are easy. The Sun and a single planet trace the same ellipse over and over, and you can say where that planet will sit a thousand years from now. Add a third body and the certainty collapses. Three objects pulling on each other have no general formula for where they go. You can step the motion forward on a computer, one tiny slice of time after another, but there is no equation that skips ahead to the answer.

And the motion is chaotic. Move one body by a hair at the start and the whole path ends up somewhere completely different. That is the heart of it: with three or more bodies, predicting the far future is, in general, impossible. It looks like a small step up from two bodies. It is not.

The same problem decides how spacecraft thread between worlds, whether a cluster of stars holds together, and how safe our own solar system really is over billions of years.

Two bodies are solvable because the problem quietly shrinks. Switch to the separation between them and you are left with a single body moving around a fixed point, with just enough conserved quantities (energy, momentum, angular momentum) to pin the motion down completely. The orbit is always a conic section: an ellipse, a parabola, or a hyperbola.

Three bodies do not shrink that way. Each one feels the pull of the other two,

\[ \mathbf{F}_i = \sum_{j \ne i} \frac{G\,m_i m_j\,(\mathbf{r}_j-\mathbf{r}_i)}{\lvert \mathbf{r}_j-\mathbf{r}_i\rvert^{3}} \]

and there are no longer enough conserved quantities to untangle the result. In 1889 Henri Poincaré went after exactly this while competing for a prize set by King Oscar II of Sweden. He proved that the clean general solution everyone was hunting for does not exist. Along the way he found something stranger and more important: the trajectories could depend so delicately on their starting point that the tiniest difference would blow up into a completely different future. That discovery is now called chaos, and the three-body problem is where it was born.

Sensitive dependence is the precise reason long-term prediction fails. Two nearby starting states drift apart roughly exponentially, like \(e^{\lambda t}\) for a positive Lyapunov exponent \(\lambda\), so any small error in your measurement, and there is always some error, grows until the prediction is worthless.

None of this means the problem is featureless. There are special, exact solutions hiding inside it. Lagrange found a configuration where three bodies sit at the corners of an equilateral triangle and rotate together forever. Euler found ones strung out in a line. In the restricted version, where one mass is too small to affect the other two, these give the five Lagrange points, the parking spots in space where we actually put telescopes. And in 2000, mathematicians found a delicate orbit where three equal masses chase each other forever around a figure eight. You can watch several of these in the simulation above.

The exact statement. Three bodies in three dimensions live in an eighteen-dimensional phase space: three positions and three velocities each. The classical conserved quantities give only ten constraints: six fix the centre of mass and its drift, three fix total angular momentum, one fixes energy. In the 1880s, Bruns and then Poincaré proved there are no further independent algebraic or analytic conserved quantities. Without them, the system cannot be reduced to a sequence of integrals, what the older literature calls solving by quadratures. That is the rigorous meaning of "no closed-form solution." It is a statement about missing conserved structure, not a failure of cleverness.

Solved, in a useless sense. In 1912 Karl Sundman produced a convergent power series, in powers of \(t^{1/3}\), that genuinely represents the motion of the general three-body problem, provided the angular momentum is nonzero so a triple collision is ruled out. Wang Qiudong extended the idea to \(n\) bodies in 1991. The catch is the rate of convergence: to reach even modest accuracy you would need a number of terms so vast it has no practical use at all. A solution exists on paper and is worthless in practice.

Collisions and escapes. Bring two bodies close and the force law nearly diverges. Real collisions are singularities of the equations, handled by regularization, coordinate changes such as the Levi-Civita and Kustaanheimo-Stiefel transforms that tame the blow-up. Stranger are non-collision singularities: Painlevé asked whether a system could break down without anything actually colliding, and Xia showed in 1992 that with five bodies you can fling a particle to infinity in finite time.

Order inside the chaos. Chaos is not the whole story. The KAM theorem, after Kolmogorov, Arnold and Moser, shows that many regular, quasi-periodic orbits survive small perturbations, so phase space is a marbled mix of stable islands and chaotic seas. This is why the solar system is best called marginally stable: numerical integrations give it a Lyapunov time of only a few million years, so detailed prediction fails beyond that, yet a catastrophic rearrangement stays unlikely within the Sun's remaining lifetime.

How it is actually done. Since formulas are out, orbital mechanics runs on numerical integration, usually with symplectic methods that respect the underlying geometry and keep energy errors bounded over long runs, such as the Wisdom-Holman map for the planets. The figure-eight orbit is a clean test case: equal masses, zero total angular momentum, and the exact starting values used in the simulation above. It is one of a whole zoo of periodic three-body choreographies found since.