Noether's theorems/Explainers/What an action is

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: The action and the variational setup.

A varied path against a symmetry-transformed path

Both of Noether's theorems are about one thing. This page says what that thing is.

A score for a whole history

Throw a ball across a room. It follows one path. But you can imagine many other paths it might have taken between the same two points: higher, lower, wobblier, faster at the start.

Physics has a rule that gives each of those imagined paths a single number. The rule looks at the whole path from beginning to end and returns one score. That score is called the action.

The real path is the one whose score does not change when you nudge the path slightly. That is the whole of what physics asks of the action, and it is enough to determine how the ball moves.

Two features of this matter for everything that follows.

The score belongs to a whole history, not to a moment. You cannot ask what the action is at three o'clock. You can only ask what it is for a complete story with a beginning and an end.

The rule is separate from the score. The rule is a recipe: take the position, take the speed, combine them like this. The score is what you get when you feed a particular path into the recipe and add everything up. The same recipe gives different scores for different paths.

What it means to say the action does not change

Now suppose you change something. Move the whole experiment ten metres to the left. Turn it forty degrees. Wait an hour and do it again.

Each of these is a change you can make in a smooth, adjustable amount — ten metres or eleven, forty degrees or forty-one. Physics calls a change like this a transformation.

The condition both theorems start from is this: after the transformation, the score is the same.

Read that carefully, because it is easy to hear as something slightly different. It does not say the recipe gives the same answer at every point along the way. It says the total, added up over the whole path, comes out the same.

That distinction sounds fussy. It is not. Two things change when you transform: what the recipe returns at each point, and where the points are. Those two changes can cancel. When they do, the total is unchanged even though the recipe's output changed everywhere along the way.

Almost every mistake made about Noether's theorems traces back to this one point.

Three things worth fixing now

The condition is on the action, not on the motion. There are transformations that turn real motions into other real motions but change the score. Those look like symmetries and are not the kind either theorem uses. They give nothing.

Neither theorem tells you which transformations leave the score alone. You have to find those yourself, by looking at the recipe. The theorems tell you what follows once you have found one.

There is a weaker condition that also works, and Noether did not use it. It allows the score to change by an amount that depends only on the two endpoints. That version came three years later, from someone else, and this volume keeps the two apart. See Two ways a thing can stay the same.

Why start here

Everything in this volume is downstream of the action. The first theorem takes a symmetry of the action and returns a quantity that does not change during the motion. The second takes a different kind of symmetry of the same action and returns something else entirely.

Neither says anything at all about a system that does not come from an action. That is not a small restriction, and it is why this page comes first.

Why a score, and not a push

There is an older way to do all of this. Newton's way: work out the forces acting at each instant, and step the system forward moment by moment.

Both ways give the same answers. So why keep a second one?

Because the two ways make different things obvious. The force picture makes it obvious what is happening right now. The score picture makes it obvious what is symmetric — and symmetry is invisible in the force picture, because forces are local and symmetry is a statement about the whole.

Noether's theorems are only available in the second picture. There is no version of them that starts from forces, because there is nothing for a symmetry to be a symmetry of. That is not a limitation anyone chose. It is the reason the score picture, which for two centuries looked like an elegant alternative, turned out to be the one modern physics is written in.

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