Math · chapter 5 of 13
A hair's difference, and a different sky
Move one rock by the width of an atom and an hour later the whole arrangement is different. The equations are exact; the answer is still out of reach.
Here is the thing that makes three bodies different in kind, not in difficulty.
Start two copies of the same three rocks. In the second copy, nudge one of them by a millionth of a millimetre. Run both. For a while the two copies agree. Then they part, and the gap does not grow by a little each time — it doubles, and doubles again, on a fixed schedule. Twenty doublings turns a millionth into a whole. After that the two copies have nothing to do with each other.
The time it takes to grow by a factor of e — about three — has a name: the Lyapunov time. It is the shelf life of a prediction. Past a few dozen Lyapunov times, knowing the starting numbers to any precision you can name buys you nothing.
This is not sloppiness in the equations and it is not randomness. Run the same start twice and you get the same path twice, to the last decimal. The trouble is that you never have the start to the last decimal. Nobody does. The rocks are not measured that well, and cannot be.
Poincaré found this in 1889 while competing for a prize from Oscar II of Sweden, and he found it by finding a mistake in his own winning entry. The corrected memoir describes two curves that were supposed to meet cleanly and instead cross each other an infinite number of times, folded into what he called a tangle so complicated he would not attempt to draw it. He paid to have the printed copies replaced. That reversal is where the study of chaos begins, thirty years before anyone had a computer to see it with.
The cleanest demonstration is the oldest test case in the subject. Carl Burrau, 1913: weights of three, four and five, at rest, at the corners of a three-four-five triangle. Nothing is moving and nothing is arranged. It falls together, misses, swings, misses again for about sixty time units, and then throws one body out for good and leaves the other two paired up — and which body that is depends on the last decimal place you kept.
Our own solar system is in this condition, mildly. Track the inner planets and the predictions go soft after around five million years — fine for a calendar, useless for asking where Mercury will be in a hundred million.
Reading it: The gap between two nearly identical starts multiplies itself as time goes on, instead of adding.
τ is the Lyapunov time. The demo below measures it for whatever start you give it, by fitting a straight line to the gap on a log scale.
Reading it: How long a prediction lasts: the Lyapunov time, times the logarithm of how much better your starting numbers are than the error you will accept.
The logarithm is the bad news. A thousand times better measurements buy about seven more Lyapunov times. Not a thousand times longer — seven times the constant.
Reading it: The solar system's own shelf life, from Laskar's calculations.
Which is why the standard result is a probability — about a 1% chance of Mercury's orbit going unstable within five billion years — and not a date.
This one runs in the browser, with JavaScript on. The words above it stand on their own.
Burrau's problem run 12 times over, each copy moved by 1e-09 at the start — a billionth of the distance between the bodies — and stepped together with a step taken from the closest pair. The bars are which body ended up thrown out. The copies do not agree. The first ejection happened at t = 62 and the last at t = 63; the median energy error over the whole run was 1e-06. Some of the disagreement is the arithmetic rather than the physics, which is the same point said twice — see So you step it. Run by tools/ghost_tally.py.
The gap between two runs of the same start, on a log scale, for four sizes of nudge — computed while this page was built. The straight climb is the exponential; its slope gives a Lyapunov time of about 3.51 time units, so a starting error grows roughly a thousandfold every 8.1. The flattening at the top is the gap running out of room.