View source for Noether's theorems/Explainers/What an action is

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''Mathematical counterpart: [[Noether's theorems/First theorem/The action and the variational setup|The action and the variational setup]].''

[[File:NT-F-02.png|thumb|right|360px|alt=Three curves between endpoints. A solid curve is the true path; a dashed curve with the same fixed endpoints is a variation; a third curve offset from both, with its own moved endpoints, is a symmetry-transformed path. A note distinguishes testing the action from mapping the whole problem.|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
[[Noether's theorems/Explainers/Two ways a thing can stay the same|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.''