Math · chapter 4 of 13
Ten things that never change
Eighteen numbers, ten of them pinned down by conservation laws. The eight left over are where the trouble lives — and two theorems say no further law is coming.
There is a way of solving equations by bookkeeping. Find a quantity that never changes, and you have traded one unknown for a known. Enough of those and the problem falls open.
Three bodies hand you ten of them for free. The total momentum is three numbers that hold steady. The balance point's coasting gives three more. The total energy is one. The total spin — angular momentum — is three. Ten quantities, fixed for all time, whatever the three rocks get up to.
Eighteen minus ten is eight. Then you can slide the clock, because the laws do not care what time it is, and you can turn the whole picture, because they do not care which way is north. Call it six. Six stubborn numbers.
For two hundred years the hope was that somebody would find another conserved quantity and then another, until the six were gone and the formula appeared. Two results ended that hope.
Heinrich Bruns, 1887. Beyond the ten, there are no more conserved quantities that are algebraic in the coordinates, the velocities and the time. Henri Poincaré, 1890. In the restricted case, with the mass ratio small and not special, no further conserved quantity exists that is analytic and single-valued.
Read those narrowly, the way they were proved. Neither says that no expression of any kind can exist — Sundman later wrote one down, and it is in a later chapter. They say that the bookkeeping route is closed. You cannot grind the six numbers away by finding more things that never change, because there are no more of the usable kind.
Reading it: Total momentum, three numbers. The balance point's straight-line drift, three more. Total spin, three.
Ten counting the energy below. These are the classical integrals, and they are all of them.
Reading it: Add up the motion of all three, then subtract the pull holding them together. That total never changes.
This one is the workhorse of the numbers chapter: a computer's answer is checked by watching whether its energy stays put.
Reading it: The bookkeeping gets you from eighteen unknowns down to six and stops there.
Bruns 1887 and Poincaré 1890 are the proofs that it stops.
This one runs in the browser, with JavaScript on. The words above it stand on their own.
Eighteen numbers, ten of them held fixed by conservation, then the clock and the compass. Six left, and no way through them by bookkeeping.