PMA Publisher: Published from the Principia Machina source

2026-10-15T19:30:22Z

Published from the Principia Machina source

← Older revision		Revision as of 19:30, 15 October 2026
Line 1:		Line 1:
	{{DISPLAYTITLE:Symmetry and conservation}}		{{DISPLAYTITLE:Symmetry and conservation}}
	{{Technical\|Hamilton's principle/Mathematics/Noether's first theorem}}		{{Technical\|Hamilton's principle/Mathematics/Noether's first theorem}}
			{{Hamilton's principle map}}

	This page explains why conserved quantities exist. It uses no mathematics. The		This page explains why conserved quantities exist. It uses no mathematics. The

BFP Rauch: Pipe internal links so cross-references read as titles, not paths

2026-09-16T04:58:57Z

Pipe internal links so cross-references read as titles, not paths

← Older revision		Revision as of 04:58, 16 September 2026
Line 138:		Line 138:

	It is at		It is at
	[[Hamilton's principle/Explainers/When symmetry is local]].		[[Hamilton's principle/Explainers/When symmetry is local\|When symmetry is local]].

	== Where to go next ==		== Where to go next ==

	The local case, and what it produces instead of conservation laws, is at		The local case, and what it produces instead of conservation laws, is at
	[[Hamilton's principle/Explainers/When symmetry is local]].		[[Hamilton's principle/Explainers/When symmetry is local\|When symmetry is local]].

BFP Rauch: Add DISPLAYTITLE: title the page by its section rather than its path

2026-09-16T04:47:00Z

Add DISPLAYTITLE: title the page by its section rather than its path

← Older revision		Revision as of 04:47, 16 September 2026
Line 1:		Line 1:
			{{DISPLAYTITLE:Symmetry and conservation}}
	{{Technical\|Hamilton's principle/Mathematics/Noether's first theorem}}		{{Technical\|Hamilton's principle/Mathematics/Noether's first theorem}}

Line 143:		Line 144:
	The local case, and what it produces instead of conservation laws, is at		The local case, and what it produces instead of conservation laws, is at
	[[Hamilton's principle/Explainers/When symmetry is local]].		[[Hamilton's principle/Explainers/When symmetry is local]].

BFP Rauch: 1 revision imported

2026-09-09T05:05:17Z

1 revision imported

← Older revision		Revision as of 05:05, 9 September 2026
(No difference)

hamiltons-principle>HPVBuild: Initial import of B-15

2026-09-08T12:00:00Z

Initial import of B-15

New page

{{Technical|Hamilton's principle/Mathematics/Noether's first theorem}}

This page explains why conserved quantities exist. It uses no mathematics. The
technical page it goes with is named above.

== Things that do not change ==

[[File:OPL-HP-F-09.png|thumb|right|400px|Symmetry to conservation, four examples.]]

Some quantities in physics stay the same no matter what happens.

Energy is one. Momentum is another. Angular momentum, the quantity that keeps a
skater spinning, is a third.

For a long time these were facts. Careful experiments showed the numbers did not
change, and physics recorded that they did not. Nobody could say '''why''' these
particular quantities and not others.

Emmy Noether answered that in 1918, and the answer is one of the most useful in
physics.

== The answer ==

'''Every continuous symmetry gives a conserved quantity.'''

A symmetry means something you can change about your description without changing
the physics.

Do your experiment a metre to the left. If it behaves identically, the laws do not
care about position. Do it tomorrow. If it behaves identically, the laws do not
care about time. Turn the whole apparatus to face north. If it behaves identically,
the laws do not care about direction.

Each of those indifferences produces a conserved quantity.

{| class="wikitable"
! the laws do not care about !! what is conserved
|-
| where you are || momentum
|-
| what time it is || energy
|-
| which way you face || angular momentum
|}

The three great conservation laws are three statements about what the universe is
indifferent to.

== Why it works ==

The argument, in outline and without any symbols, runs like this.

The whole of this volume rests on a total cost for a path. A symmetry means moving
the path — shifting it sideways, or later in time — leaves that total unchanged.

Now put those two facts together. Nudging a true path leaves the total unchanged,
because the path is stationary. Shifting the path leaves the total unchanged,
because of the symmetry.

Two different reasons for the same conclusion. Combining them produces a
constraint, and that constraint says a particular quantity cannot change as the
motion proceeds.

The full working is on the technical page. The shape of it is what matters here:
'''two independent reasons for one fact, combined.'''

== Why this changed things ==

Before 1918, conservation laws were experimental results. You checked them.

After, they became consequences. If you believe the laws of physics are the same
everywhere, you '''must''' believe momentum is conserved. They are not separate
claims.

The reverse direction is at least as useful. Find a new conserved quantity in an
experiment and you have found a symmetry you did not know about — and symmetries
tell you about the structure of the theory.

That reversal is how much of modern particle physics is done. Conservation of
electric charge, for instance, comes from a symmetry that has nothing to do with
space or time at all, but with a quantity that can be shifted without physical
effect. Charge is conserved because of that indifference.

== Where the volume already used it ==

This theorem explains things this volume derived earlier without knowing why.

A skater on flat ice keeps their momentum, because the ice is the same
everywhere. A pendulum keeps its energy, because the laws do not change from
moment to moment. A planet sweeps out equal areas in equal times, because space
has no preferred direction — that is Kepler's second law, and it is angular
momentum conservation wearing seventeenth-century clothes.

Each of those was found separately, by its own calculation. They are one theorem.

== Two cautions ==

'''The symmetry has to be a symmetry of the cost rule.'''

That is a stronger requirement than it sounds. The equations of motion can have
symmetries that the cost rule does not, and those produce nothing. Only
indifferences of the underlying rule count.

'''Independent symmetries need not give independent quantities.'''

Sometimes two symmetries yield conserved quantities that turn out to be related,
so the second tells you nothing new. Noether pointed this out herself. Counting
symmetries does not straightforwardly count useful conservation laws.

== The limits ==

The theorem needs the symmetry to be '''continuous''' — something you can do a
little bit of.

You can shift by a millimetre, or a micrometre, or any amount you like. That is
continuous, and the theorem applies.

Some symmetries are not like that. A snowflake looks the same after turning by
sixty degrees, but not after turning by one degree. That is a discrete symmetry,
and it gives no conserved quantity by this route. Discrete symmetries matter in
physics for other reasons.

The theorem also assumes the system has a cost rule at all. Systems with friction
usually do not, in the simple form, and their energy is genuinely not conserved —
it leaks into heat, which is a description of parts the model left out.

== A second theorem ==

Noether's paper contains '''two''' theorems, not one.

The second treats a different kind of symmetry: one where the change can be
chosen '''independently at each point in space''', rather than being the same
everywhere.

It gives a completely different conclusion, and it is the theorem she wrote the
paper to prove. It is far less known than the first, which misrepresents the paper.

It is at
[[Hamilton's principle/Explainers/When symmetry is local]].

== Where to go next ==

The local case, and what it produces instead of conservation laws, is at
[[Hamilton's principle/Explainers/When symmetry is local]].

Hamilton's principle/Explainers/Symmetry and conservation - Revision history

PMA Publisher: Published from the Principia Machina source

BFP Rauch: Pipe internal links so cross-references read as titles, not paths

BFP Rauch: Add DISPLAYTITLE: title the page by its section rather than its path

BFP Rauch: 1 revision imported

hamiltons-principle>HPVBuild: Initial import of B-15