The three-body problem

What just happened?

Three bodies attract one another gravitationally and weave an irregular dance. A tiny change of the initial conditions (slider $s$) leads to a completely different evolution — even though every step is fully determined by Newton’s laws. This is exactly deterministic chaos.

Two bodies — easy, three — not:

For two bodies (e.g. Sun–planet) Kepler and Newton showed that the orbit is a conic section — an ellipse described by a compact formula. It is governed by the law of universal gravitation:

But already for three bodies there is no general formula for the path of motion — the system has to be integrated numerically step by step, exactly as this simulation does.

Poincaré and the birth of chaos theory:

Studying the stability of the Solar System in a competition held by King Oscar II of Sweden, Henri Poincaré discovered (prize-winning work in 1889, published in 1890) that a three-body system can behave chaotically: it is fully deterministic, yet so sensitive to the initial conditions that long-term prediction is impossible. This was the first mathematical analysis of chaos — long before Edward Lorenz.

The invariant — what stays certain:

Although the path itself cannot be predicted, the total energy and the total momentum of the system are conserved (as is the position of the centre of mass when the momentum is zero):

In the plot on the right of the scene $E$ and $|\vec p|$ stay flat — they are invariants, even though the path is unpredictable.

Where do you see it?

The three-body problem has real astronomical significance — the stability of orbits, space missions using “gravity assists” — and a cultural one: it is the title of Liu Cixin’s famous science-fiction novel. It shows the limit of predictability even in the “deterministic” world of physics.

Why this matters for the exam:

This is a model example of the difference between a closed-form solution (two bodies → ellipse) and a numerical simulation (three bodies → chaos). Remember the law of gravitation $F = G\,m_1 m_2 / r^2$ and that determinism does not mean predictability.

Remember: for two bodies the orbit is an ellipse given by a formula; for three there is no general formula for the path — the motion can be chaotic (Poincaré, 1890), although energy and momentum always remain conserved.