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	{{DISPLAYTITLE:Why energy and momentum are conserved}}		{{DISPLAYTITLE:Why energy and momentum are conserved}}
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	''Mathematical counterpart: [[Noether's theorems/First theorem/Energy, momentum and angular momentum\|Energy, momentum and angular momentum]].''		''Mathematical counterpart: [[Noether's theorems/First theorem/Energy, momentum and angular momentum\|Energy, momentum and angular momentum]].''
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	''No mathematics is used on this page. The mathematical treatment is on the		''No mathematics is used on this page. The mathematical treatment is on the
	counterpart page named above.''		counterpart page named above.''

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NT>NoetherVolumeBuild: Noether theorems volume, initial import (B-11)

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Noether theorems volume, initial import (B-11)

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{{DISPLAYTITLE:Why energy and momentum are conserved}}

''Mathematical counterpart: [[Noether's theorems/First theorem/Energy, momentum and angular momentum|Energy, momentum and angular momentum]].''

Three famous conservation laws, three symmetries, one theorem. This page says
which goes with which and what the connection actually asserts.

== Start with what was already known ==

Energy, momentum and angular momentum were known to be conserved long before
1918. Pendulums, planets, colliding billiard balls — the evidence was centuries
old and nobody doubted it.

'''Noether did not discover these conservation laws.''' What she found was why
they hold, and that the answer is the same answer three times over.

== Time does not matter, so energy is conserved ==

Set up a pendulum and start it swinging. Come back tomorrow, set up the same
pendulum the same way, and start it again.

It does the same thing. The laws did not change overnight.

That is a symmetry: shifting when you run the experiment leaves the scoring rule
alone. One knob — how far you shift.

One knob, one conserved quantity. That quantity is energy.

Note what the symmetry is not. It is not that nothing changes with time; the
pendulum swings. It is that the '''rules''' do not change with time.

== Place does not matter, so momentum is conserved ==

Same experiment, moved across the room. Same result.

Three knobs, since you can move it three independent ways. Three conserved
quantities, and together they are momentum.

There is a version of this you meet earlier in physics without knowing it is the
same thing. If a coordinate never appears in the scoring rule — only how fast it
is changing does — then the quantity conjugate to it is conserved. That is exactly
this symmetry, and the theorem explains why the trick works.

== Direction does not matter, so angular momentum is conserved ==

Turn the whole experiment to face a different way. Same result.

Three knobs again, one for each axis. Three conserved quantities, which together
are angular momentum.

A spinning skater pulling her arms in speeds up. Nothing pushed her. The reason
she speeds up is that space does not care which way she is facing.

== The structural difference between the three ==

Energy is not quite like the other two, and it is worth seeing why.

Momentum and angular momentum come from moving the '''things''' — the positions,
the orientations. Energy comes from moving the '''clock'''.

Because energy comes from moving the clock, its flow picks up an extra
contribution that the other two do not have, from the frame having shifted. That
is the picture-frame effect from
[[Noether's theorems/Explainers/What happens at the edges|the edges page]], and it
is why a treatment that only ever moves the things can produce momentum but not
energy.

== What the theorem is really telling you ==

Not that energy is conserved. You knew that.

It is telling you that '''energy conservation and the uniformity of time are the
same fact'''. Not two facts that happen to agree. One fact, written two ways.

Which means the connection can fail. In a situation where the rules genuinely do
change with time, energy is not conserved — and this is not a breakdown of
physics, it is exactly what the theorem predicts. The expanding universe is such a
situation, and there is
[[Noether's theorems/Explainers/Where the light's energy goes|a page on it]].

== How many are there altogether ==

The theorem lets you count, and the count for ordinary physics in empty space is
worth knowing.

One for shifting the clock. Three for moving through space. Three for turning.
And three more for changing to a laboratory that is sliding at a steady speed,
which is the case on
[[Noether's theorems/Explainers/Drifting centres and electric charge|the next
page]] and needs a slightly weaker condition than the others.

Ten in total. Ten symmetries of empty space, ten conserved quantities, and no
eleventh. If somebody claims a new conservation law for an isolated system in
ordinary space, the theorem says it is either one of these ten in disguise or it
does not come from a symmetry of this kind at all.

== When the reasons stop applying ==

Each of the three on this page rests on a fact about the world that could fail,
and in at least one case does.

Time is uniform in the laboratory. It is not uniform across the history of the
universe, which has been expanding since it began. The rules governing light
travelling across that expansion genuinely do change with time — and light
travelling across it genuinely does lose energy. See
[[Noether's theorems/Explainers/Where the light's energy goes|Where the light's
energy goes]].

That is not an exception to the theorem. It is the theorem, read in the direction
people rarely read it.

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

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