Mathematics 1890

On the Problem of the Three Bodies and the Equations of Dynamics

Henri Poincaré

Three bodies, one law of gravity — and no formula can foretell their path.

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In depth · the introduction

Two planets and a sun trace orbits you can predict for a billion years. Add a third heavy body, and the future slips out of reach — not because we lack a computer, but because no formula can ever exist.

The big idea

Newton had already solved the two-body problem: a planet and a star swing around each other on a perfect, repeating ellipse, forever predictable. The natural next question was the three-body problem — what happens when three masses all pull on one another? For two centuries, the best mathematicians hunted for the formula. Poincaré proved there isn't one.

He did something cleverer than solving the equations: he studied the shape of all possible motions at once. And he found that the paths can knot together in an infinitely fine tangle, so that two starts which differ by a hair end up in wildly different places. The motion is completely determined by the law of gravity, yet completely unpredictable in the long run. That is the idea we now call chaos — and Poincaré found it inside the night sky.

How it came about

In 1885 the King of Sweden, Oscar II, offered a prize for progress on exactly this question. Poincaré, already France's foremost mathematician, submitted a long memoir and won. It was set in type for the journal Acta Mathematica — and then the trouble began.

The journal's editor, Edvard Phragmén, kept asking about one delicate step. Poincaré checked, and to his horror found a real mistake: he had assumed certain curves closed up neatly when in fact they did not. He stopped the presses by telegram and paid out of his own pocket to destroy the printed copies and reset the whole thing — more than the prize money. But fixing the error was the making of him. Only by tracing what the curves really did could he see the endless tangle, and with it, chaos.

Why it mattered

Before Poincaré, “determined by exact laws” and “predictable” were treated as the same thing. He showed they are not. A system can obey perfectly fixed rules and still defeat all long-range prediction, because tiny uncertainties in where it starts blow up into huge uncertainties later. That single insight reshaped physics, weather forecasting, ecology, and economics — anywhere the future depends sensitively on the present. And it began the modern habit of studying not one solution but the whole landscape of possible motions.

A way to picture it

Think of two marbles released side by side at the top of a smooth slide: they land in nearly the same spot, every time. Now picture a pinball machine. Release two balls a hair apart and after a few bumpers they take completely different routes — yet nothing was random; each obeyed the same bumpers and gravity. The Solar System with three heavy bodies is the pinball machine, not the slide. The rules are exact; the outcome, past a certain horizon, is anyone's guess.

Where it sits

Newton (1687) had tamed two bodies; Poincaré showed why three resist taming. His qualitative method launched dynamical-systems theory, and the chaos he glimpsed lay half-forgotten until Edward Lorenz rediscovered it in a 1963 weather model — the famous “butterfly effect.” The thread runs on to the figure-eight orbit found in 2000, to the chaotic routes that now steer real spacecraft, and to every weather service that hedges its bets with an ensemble of forecasts instead of one.

The original document

Original source text

The problem and the prize

H. Poincaré · Acta Mathematica 13 (1890): 1–270 · Introduction (pp. 5–7)

In 1885, to honour the sixtieth birthday of King Oscar II of Sweden, Gösta Mittag-Leffler announced a prize competition. The first of its four questions, set by Weierstrass, asked, in effect: for a system of point masses attracting one another by Newton's law (no two ever colliding), find a series expansion of the coordinates of each body in known functions of time, converging uniformly for all time.

Poincaré did not solve that problem — no one has, in the form it was asked. Instead he studied the restricted three-body problem and the general structure of the equations of dynamics, and showed why a tidy convergent-series solution was not to be expected. The jury — Mittag-Leffler, Weierstrass, and Hermite — awarded him the prize all the same.

The error, and the acknowledgment

Revised introduction · on the correction of the memoir before publication

While the prize memoir was being typeset for Acta Mathematica, the journal's editor Lars Edvard Phragmén pressed Poincaré on a delicate point. Re-examining it in late 1889, Poincaré found a serious error: he had wrongly assumed certain asymptotic surfaces closed up smoothly. He telegraphed to halt the presses, and paid for the printed run to be destroyed and the memoir reset — a sum larger than the prize itself.

C'est aussi lui qui, en attirant mon attention sur un point délicat, m'a permis de découvrir et de rectifier une erreur importante. — “It is also he [Phragmén] who, by drawing my attention to a delicate point, enabled me to discover and to correct an important error.”

[ … ]

Correcting the error is what led him to the doubly-asymptotic (homoclinic) trajectories: curves that cross themselves infinitely often in a tangle so intricate that, he wrote, one would not even attempt to draw it. That tangle is the mathematical seed of what we now call chaos.

Sensitive dependence (stated later, 1908)

H. Poincaré · Science et méthode (1908) — not the 1890 memoir, but his clearest statement of the idea it uncovered

Il peut arriver que de petites différences dans les conditions initiales en engendrent de très grandes dans les phénomènes finaux. — “It may happen that small differences in the initial conditions produce very great ones in the final phenomena. … Prediction becomes impossible.”

Henri Poincaré · Paris · memoir 1890, restated 1908