Hamilton's principle/Explainers/Electromagnetism from an action - Revision history

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	{{DISPLAYTITLE:Electromagnetism from an action}}		{{DISPLAYTITLE:Electromagnetism from an action}}
	{{Technical\|Hamilton's principle/Applications/The electromagnetic Lagrangian}}		{{Technical\|Hamilton's principle/Applications/The electromagnetic Lagrangian}}
			{{Hamilton's principle map}}

	This page explains how electromagnetism can be derived rather than discovered. It		This page explains how electromagnetism can be derived rather than discovered. It

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	If only differences matter, you should be able to turn all the dials by the same		If only differences matter, you should be able to turn all the dials by the same
	amount everywhere and change nothing. That is an ordinary symmetry, and by		amount everywhere and change nothing. That is an ordinary symmetry, and by
	[[Hamilton's principle/Explainers/Symmetry and conservation]] it gives a conserved		[[Hamilton's principle/Explainers/Symmetry and conservation\|Symmetry and conservation]] it gives a conserved
	quantity. That quantity is electric charge.		quantity. That quantity is electric charge.

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	That is the '''redundancy''' described at		That is the '''redundancy''' described at
	[[Hamilton's principle/Explainers/When symmetry is local]]. The description carries		[[Hamilton's principle/Explainers/When symmetry is local\|When symmetry is local]]. The description carries
	spare parts.		spare parts.

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	And the tidy reformulation of		And the tidy reformulation of
	[[Hamilton's principle/Explainers/The Hamiltonian]] does not apply cleanly, because		[[Hamilton's principle/Explainers/The Hamiltonian\|the Hamiltonian]] does not apply cleanly, because
	it needs a description without spare parts.		it needs a description without spare parts.

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	The same construction, applied to the choice of coordinates rather than a dial,		The same construction, applied to the choice of coordinates rather than a dial,
	gives gravity. It is at		gives gravity. It is at
	[[Hamilton's principle/Explainers/Gravity from an action]].		[[Hamilton's principle/Explainers/Gravity from an action\|Gravity from an action]].

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	{{Technical\|Hamilton's principle/Applications/The electromagnetic Lagrangian}}		{{Technical\|Hamilton's principle/Applications/The electromagnetic Lagrangian}}

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{{Technical|Hamilton's principle/Applications/The electromagnetic Lagrangian}}

This page explains how electromagnetism can be derived rather than discovered. It
uses no mathematics. The technical page it goes with is named above.

== A theory that was found twice ==

[[File:OPL-HP-F-26.png|thumb|right|400px|Counting physical polarisations from gauge redundancy.]]

Electromagnetism was worked out over a century, from experiments. Coulomb measured
forces between charges. Ørsted found that currents deflect compasses. Faraday found
that moving magnets make currents. Maxwell assembled it.

That is how it happened, and it took a hundred years.

There is a second route. Start from a single requirement about what a theory must
ignore, and electromagnetism comes out. Nothing about electricity or magnetism has
to be put in.

This page follows the second route. It is not how the subject was discovered, and
it is how the subject is now taught to people who will build the next theory.

== The requirement ==

[[File:OPL-HP-F-46.png|thumb|right|400px|Maxwell's equations split into dynamics and identities.]]

Start with a description of matter that involves a quantity with no direct physical
meaning — something like the setting of a dial, where only '''differences''' in
the setting matter, not the setting itself.

If only differences matter, you should be able to turn all the dials by the same
amount everywhere and change nothing. That is an ordinary symmetry, and by
[[Hamilton's principle/Explainers/Symmetry and conservation]] it gives a conserved
quantity. That quantity is electric charge.

Now make the harder demand. '''You should be able to turn each dial by a different
amount, independently at every point in space, and still change nothing.'''

== Why that demand cannot be met alone ==

It fails immediately, and the reason is easy to see.

Physics cares about how things change from place to place. If you turn the dial by
different amounts at neighbouring points, you have manufactured a change between
them. The theory notices.

So the demand is impossible as it stands.

Unless something else changes to compensate.

== The field appears ==

Suppose there is another quantity, spread through space, whose job is to cancel
exactly that manufactured difference. When you turn the dials by varying amounts,
this quantity shifts in a matching way, and the two effects cancel.

Then the demand can be met.

'''That compensating quantity is the electromagnetic potential.''' The electric
and magnetic fields are built from it.

The logic is worth stating slowly, because it is easy to skate past. You did not
put in electricity. You did not put in magnetism. You demanded that a certain
adjustment have no physical effect, discovered that the demand was impossible, and
found that it becomes possible if a new field exists.

'''The field is forced into existence by the requirement.'''

== Its own behaviour ==

The new field needs its own rules. Those are fixed the same way: write the simplest
cost rule that respects the same requirement.

There is very little freedom. Essentially one possibility survives, and turning the
handle produces '''Maxwell's equations'''.

Or rather, half of them. And the split is interesting.

== Two halves of Maxwell, two kinds of statement ==

Maxwell's equations come in two groups, and the standard presentation treats them
as four equations of equal standing. They are not.

'''One group is dynamics.''' It says how the field responds to charges and
currents. It comes out of the cost rule, and it could have been otherwise.

'''The other group is bookkeeping.''' It says there are no magnetic monopoles, and
it gives the law of induction. These follow '''automatically''' from the fact that
the field was built out of the compensating quantity. They are true by
construction, for any conceivable field, before any physics is put in.

That asymmetry looks arbitrary when the four equations are simply listed. Seen this
way it is not arbitrary at all. Two of them are laws and two of them are
consequences of how the description is built.

== Charge conservation is not optional ==

One more result comes free, and it is a strong one.

If you couple the field to a flow of charge, the whole construction only works if
that charge is '''conserved'''. A theory in which charge could appear or vanish
fails the requirement outright. It is not a theory at all.

So charge conservation is not an experimental fact that the theory happens to
respect. It is a precondition for the theory existing.

== The price of the description ==

The compensating quantity is not unique. Several different versions of it give
exactly the same electric and magnetic fields, and hence the same physics.

That is the '''redundancy''' described at
[[Hamilton's principle/Explainers/When symmetry is local]]. The description carries
spare parts.

Two consequences, both predicted by Noether's second theorem.

Some of the equations are not independent — they follow from the others as a matter
of arithmetic.

And the tidy reformulation of
[[Hamilton's principle/Explainers/The Hamiltonian]] does not apply cleanly, because
it needs a description without spare parts.

The redundancy also explains a number. Counting the description gives four
quantities. Removing the redundancy and the dependent equation leaves '''two''' —
and light has exactly two independent polarisations, which is why sunglasses work.

That count came out of an argument about what a theory must ignore.

== Why this route matters ==

Electromagnetism did not need deriving. It was already known.

The point is that the method '''generalises'''. Make the same kind of demand about
a different quantity and you get a different force. The strong and weak nuclear
forces were constructed this way in the twentieth century, and neither could have
been found by the patient experimental route that worked for electricity.

Electromagnetism is the worked example that showed the method is trustworthy.

== Where to go next ==

The same construction, applied to the choice of coordinates rather than a dial,
gives gravity. It is at
[[Hamilton's principle/Explainers/Gravity from an action]].