Math · chapter 10 of 13

Crashes, and the kind of trouble that is not a crash

Solutions can stop existing. With three bodies the only way is a collision, and a three-way collision needs the whole system to have no spin. With four or more, something stranger is possible.

The equations have a division in them, by the distance cubed. Let a distance go to zero and the arithmetic stops meaning anything. That is a singularity: a moment beyond which the solution does not continue.

Two bodies hitting each other does it. So does all three arriving at the same point at the same instant. Paul Painlevé proved in 1895 that for three bodies those are the only ways — every singularity is a collision. No other kind of breakdown is available.

Sundman added the condition on the three-way case, and it is a strange and beautiful one. A triple collision requires the total angular momentum to be exactly zero. Give the system the faintest overall spin and all three can never meet. Not unlikely — impossible. The spin is conserved, and it cannot be carried by a single point.

Painlevé also guessed that for more bodies there would be a singularity that is not a collision: a solution that ceases to exist in finite time with nothing having hit anything. It took until 1992, when Zhihong (Jeff) Xia built one with five bodies — an arrangement in which distances and speeds run to infinity in a finite time, each body having only ever passed near the others. Joseph Gerver later did it with four. For three, Painlevé's proof stands: a crash is the only way out.

This is where the three-body problem earns its position. Two bodies are solvable. Four or more can misbehave in ways three cannot. Three is the narrowest place where the trouble starts and the last place it is still fully mapped.

L = 0 ⟺ triple collision possible

Reading it: All three can meet at one point only if the whole system has no spin at all.

Sundman's theorem, 1907. The one-line reason: a point has no room for angular momentum, and angular momentum never changes.

I = Σ m_i r_i², d²Idt² = 4E − 2U

Reading it: Take the spread of the system. How its spread speeds up or slows depends only on the energy and the pull.

Lagrange–Jacobi. If the total energy is positive the spread must grow without limit: a system with too much energy cannot stay together, whatever anybody arranges.

Try it. Set up a near-collision in the sandbox: two bodies almost on top of each other. Watch the energy readout jump and the step size warning appear. The computer is not wrong about the physics there — it is being asked to divide by nearly nothing, which is the same problem Painlevé was writing about with a pen.

Sources

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