Noether's theorems/Explainers/Why energy and momentum are conserved

Noether's theorems: map
Reading the volume How to read this volume Notation Glossary Index of symmetries Figure gallery
Reading by level Start here: the plain-language primer The first theorem, stated and proved The second theorem, stated and proved The two theorems side by side The mathematics: Lie groups and Lie algebras The applications: General relativity and the energy question The history: Göttingen and the energy problem
Sources Sources Bibliography Chronology Further reading
Related Hamilton's principle On Physical Lines of Force

Mathematical counterpart: 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 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 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 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 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.