Why Noether needed a third symbol

Mathematical counterpart: Jet space and prolongation.

Noether uses three different symbols for "the change in something". Most later accounts use two. She never explains why she needed the third. This page gives the reason, and it is a good one.

The three, briefly

They were introduced on the edges page as thermometers.

The drifting one floats downstream and records everything that changed, including the fact that it has arrived somewhere new.

The fixed one is bolted to the bridge and records the change at one place.

The nudge is not a thermometer at all — it is you poking the system to see if it pushes back.

Two of these get kept in modern accounts. The fixed one is usually the casualty.

The reason

Here is what the fixed thermometer can do that the drifting one cannot.

Ask how fast a temperature is changing along the river — the difference between here and a metre downstream. Now ask two questions in the other order: first work out that difference, then apply the change; or first apply the change, then work out the difference.

For the fixed thermometer, you get the same answer either way. It is comparing readings at the same place, so nothing about the comparison cares which order you did things in.

For the drifting thermometer, you do not. It moved, so the two places you are differencing between are not the two places you started with, and a correction appears.

One of the two comparisons can be swapped with differentiation. The other cannot.

Why that decides the whole calculation

Noether's proof integrates by parts, over and over. Integration by parts is the manoeuvre that moves a rate of change from one factor to the other, and her argument does it dozens of times, sometimes to high order.

Every one of those moves requires swapping a change with a differentiation.

With the fixed-thermometer comparison, every swap is free. With the drifting one, every swap leaves a correction behind — and the corrections pile up, each carrying its own correction at the next step.

So the third symbol is not fastidiousness. It is what makes the calculation finite. Her proof written in two symbols would not be a slightly untidier proof; it would be an unmanageable one.

The tidy modern version

There is a device for handling all of this systematically.

Take a moving object and bundle it together with its speed, its acceleration, and however many further rates of change you need. Treat the whole bundle as a single point in a larger space. Then a transformation that mixes positions with speeds is just an ordinary transformation of that larger space, and the bookkeeping about which corrections appear takes care of itself.

The device is called a jet, and it is genuinely useful in modern work.

It is not needed here. Noether did the bookkeeping by hand and both theorems come out. This volume treats the device as its own subject rather than as something you have to learn first, and the reason her third symbol exists is a better thing to carry away than the apparatus that would have saved her the trouble.

The everyday version of the same distinction

You are on a train, walking towards the front.

Ask how fast you are moving. Relative to the carriage, a walking pace. Relative to the ground, a walking pace plus the train's.

Both are correct and they answer different questions. Now ask how fast your speed is changing — are you accelerating? Relative to the carriage, no. Relative to the ground, only if the train is.

The two answers can be worked out from each other, but the correction is not the same at the level of position as it is at the level of speed, and it is different again for acceleration. Every time you differentiate, you pick up another piece of bookkeeping about the moving frame.

That is the drifting thermometer, and it is why calculations done in a moving frame are fiddly rather than merely different.

The fixed thermometer has no such problem, because nothing is moving. Its corrections are zero at every level of differentiation, forever.

Noether's proof differentiates constantly. Working in the frame where the corrections are zero is not a preference; it is the difference between a proof that fits in a section and one that does not fit at all.

No mathematics is used on this page. The mathematical treatment is on the counterpart page named above.