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	''Mathematical counterpart: [[Noether's theorems/Mathematics/Jet space and prolongation\|Jet space and prolongation]].''		''Mathematical counterpart: [[Noether's theorems/Mathematics/Jet space and prolongation\|Jet space and prolongation]].''
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Noether theorems volume, initial import (B-21)

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{{DISPLAYTITLE:Why Noether needed a third symbol}}

''Mathematical counterpart: [[Noether's theorems/Mathematics/Jet space and prolongation|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
[[Noether's theorems/Explainers/What happens at the edges|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.''

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