Why speed has to come along for the ride
Mathematical counterpart: Prolongation to derivatives.
The recipe for the action does not only use where things are. It uses how fast they are moving. So when a transformation moves things around, it moves the speeds too — and it does not get to choose how.
You do not get a free choice
Film a cyclist from a parked car. You measure a certain speed.
Now film the same cyclist from a car that is itself moving. You measure a different speed, and you can work out exactly what it will be before you look: the cyclist's speed minus your own.
You had a free choice about how to move your camera. You had no choice at all about what that does to the measured speed. Once the first is settled, the second is forced.
Extending a transformation to the speeds in the way it is forced to is called prolongation. It is bookkeeping, and it is unavoidable bookkeeping.
Two corrections, not one
When a transformation shifts where you measure from, the change in a speed has two parts.
The first is obvious: the thing you are measuring moved, so its rate of change moved with it.
The second is easy to forget: you are now measuring against a different ruler. Rate of change means change per unit of something, and if that something has shifted, the ratio shifts as well.
If the transformation leaves the ruler alone and only moves the object, the second part vanishes and prolongation is barely visible. Most textbook accounts stay in that case, which is why the word rarely appears in them.
Noether does not stay in that case. Her transformations are allowed to move the ruler, and both corrections are live throughout.
The runaway
Noether allows something further, and it has a strange consequence she points out rather than hides.
She lets the transformation itself depend on speeds. Not just on where things are — on how fast they are going.
Now watch what happens when you build up a large change out of many tiny ones. Each tiny step needs to know the speeds. Working out the speeds after that step brings in the acceleration. The next step needs that, and working it out brings in the next rate of change after acceleration. And so on.
The list grows at every step and never stops growing. A large transformation built this way can depend on infinitely many rates of change.
This is not a flaw to be excluded. It is the price of the generality. The alternative is to forbid transformations that depend on speeds, which throws away cases the theorems otherwise cover. Noether pays the price and says so.
A word about jets
Modern mathematics has a tidy device for this. Bundle a position together with its speed, its acceleration and however many further rates of change you need, and treat the whole bundle as a single point in a bigger space. Then a prolonged transformation is just an ordinary transformation of that bigger space, and the bookkeeping takes care of itself.
The device is called a jet. It is genuinely useful and it is not needed here. Noether did the bookkeeping by hand and the theorems do not depend on the device. This volume treats it as its own subject rather than as something you must learn first.
What goes wrong if you skip the second correction
It is worth seeing the failure rather than being told about it.
Take the symmetry that gives you energy: shift the clock. Here the ruler absolutely does move, because the ruler is the clock.
Now suppose you keep only the first correction — the things moved — and forget that you are measuring rates against a shifted clock.
You do not get nonsense. You get a clean, plausible-looking quantity. It is simply not the energy. The piece you dropped is exactly the term that distinguishes energy from a partial sum of it, and nothing in the calculation warns you, because the calculation never noticed it was incomplete.
That is what makes this worth a page. A missing term that produces obvious rubbish gets caught immediately. A missing term that produces something tidy and wrong can survive for a long time.
The general rule is short. If the transformation touches only the things, one correction. If it touches the ruler as well — the clock, the coordinates, anything you measure against — two.
No mathematics is used on this page. The mathematical treatment is on the counterpart page named above.