Hamilton's principle/Explainers/Fields - Revision history

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	{{Technical\|Hamilton's principle/Applications/Classical field theory}}		{{Technical\|Hamilton's principle/Applications/Classical field theory}}
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	This page explains how the method describes things that fill space. It uses no		This page explains how the method describes things that fill space. It uses no

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	[[Hamilton's principle/Explainers/Electromagnetism from an action]].		[[Hamilton's principle/Explainers/Electromagnetism from an action]].

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{{Technical|Hamilton's principle/Applications/Classical field theory}}

This page explains how the method describes things that fill space. It uses no
mathematics. The technical page it goes with is named above.

== What a field is ==

[[File:OPL-HP-F-29.png|thumb|right|400px|Local conservation as a flow across a boundary.]]

Some things in physics are located at a point. A ball has a position.

Other things are spread out. The temperature in a room has a value at every point.
So does the air pressure. So do the electric and magnetic fields, and the
gravitational field.

Anything with a value at every point of space is called a '''field'''.

Fields are not exotic. A temperature map on a weather forecast is a picture of a
field.

== The same method, more of it ==

[[File:OPL-HP-F-51.png|thumb|right|400px|Locality: influence passing from neighbour to neighbour.]]

[[Hamilton's principle/Explainers/More than one variable]] showed the method
handling several moving parts, and then infinitely many by treating a string as a
row of beads and using more and more of them.

Fields are that idea in three dimensions. Instead of a value at each point along a
line, a value at each point of space.

The recipe is unchanged. Write down a cost rule. Ask which arrangement makes the
total stop changing. Out come the equations.

What changes is what you get: not a trajectory but a whole pattern, evolving in
time.

== One assumption doing most of the work ==

Modern physics adds one requirement, and it is worth stating plainly because
almost everything follows from it.

'''The cost at a point may depend only on what is happening at that point and in
its immediate vicinity.'''

Nothing in the rule may refer to what is happening far away.

This is the assumption of '''locality'''. It rules out any theory in which
something here depends directly on something over there.

Two consequences follow at once.

'''Influence spreads.''' If nothing acts at a distance, disturbances must travel
from neighbour to neighbour. That is why fields carry waves. Light is a
disturbance of the electromagnetic field, passed from point to point.

'''Conserved quantities have to travel.''' For a ball, energy conservation means a
number stays the same. For a field it means more: energy cannot vanish here and
appear there. It has to '''flow''' across the space between.

That is why it makes sense to speak of energy travelling along a wire or arriving
from the Sun. Conservation became local, and local conservation gives things
routes.

== Building a theory from what it must not care about ==

The most striking thing about this approach is how little you have to put in.

The usual procedure is:

# Decide what the theory must be indifferent to. It should not matter where you are, which way you face, or how fast you are moving uniformly.
# Write down the most general cost rule consistent with those indifferences.
# Turn the handle.

Very often step two leaves almost no freedom. The requirements are so restrictive
that only a couple of possibilities survive, and the equations follow.

'''The symmetry chose the theory.''' Nothing about the dynamics was assumed. This
is the characteristic method of modern physics, and it is a direct consequence of
treating the cost rule, rather than the equations, as the thing you write down
first.

== Two examples of the method working ==

'''Electromagnetism''' comes out of demanding that a certain quantity can be
adjusted independently at every point without physical effect. Make that demand and
the electromagnetic field is forced into existence to make it possible. That is at
[[Hamilton's principle/Explainers/Electromagnetism from an action]].

'''General relativity''' comes out of demanding that nothing depend on your choice
of coordinates, anywhere. That is at
[[Hamilton's principle/Explainers/Gravity from an action]].

In both cases the requirement is about what the theory must '''ignore''', and the
content follows from it.

== Conserved quantities again ==

[[Hamilton's principle/Explainers/Symmetry and conservation]] applies here without
change, and produces something familiar.

Indifference to time gives conservation of energy. For a field, that means an
energy density with an accompanying flow.

Indifference to position gives conservation of momentum, similarly as a density
and a flow.

There is also a kind of symmetry that has nothing to do with space or time — an
indifference to a quantity internal to the field itself. That gives conservation of
'''electric charge'''.

Charge conservation is not an extra law. It is what a particular indifference
produces, and there is no other reason for it.

== Why energy is subtle in field theory ==

One honest complication.

The first expression for the energy of a field that the method produces is not
always the right one to use. It can behave badly under changes that ought to make
no difference.

It can be repaired, by adding something whose effect on the total is nothing at
all. The repaired version is the one that couples to gravity.

This is not a defect being patched over. Noether's work classifies exactly this
freedom, and identifies which parts of a conservation law carry information and
which are bookkeeping. The subject of the repair and the subject of her second
theorem are the same subject.

It becomes decisive for gravity, where the ambiguity cannot be repaired away at
all. That is at
[[Hamilton's principle/Explainers/Gravity from an action]].

== Where to go next ==

The first worked example is at
[[Hamilton's principle/Explainers/Electromagnetism from an action]].

← Older revision		Revision as of 04:59, 16 September 2026
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	[[File:OPL-HP-F-51.png\|thumb\|right\|400px\|Locality: influence passing from neighbour to neighbour.]]		[[File:OPL-HP-F-51.png\|thumb\|right\|400px\|Locality: influence passing from neighbour to neighbour.]]

	[[Hamilton's principle/Explainers/More than one variable]] showed the method		[[Hamilton's principle/Explainers/More than one variable\|More than one variable]] showed the method
	handling several moving parts, and then infinitely many by treating a string as a		handling several moving parts, and then infinitely many by treating a string as a
	row of beads and using more and more of them.		row of beads and using more and more of them.
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	adjusted independently at every point without physical effect. Make that demand and		adjusted independently at every point without physical effect. Make that demand and
	the electromagnetic field is forced into existence to make it possible. That is at		the electromagnetic field is forced into existence to make it possible. That is at
	[[Hamilton's principle/Explainers/Electromagnetism from an action]].		[[Hamilton's principle/Explainers/Electromagnetism from an action\|Electromagnetism from an action]].

	'''General relativity''' comes out of demanding that nothing depend on your choice		'''General relativity''' comes out of demanding that nothing depend on your choice
	of coordinates, anywhere. That is at		of coordinates, anywhere. That is at
	[[Hamilton's principle/Explainers/Gravity from an action]].		[[Hamilton's principle/Explainers/Gravity from an action\|Gravity from an action]].

	In both cases the requirement is about what the theory must '''ignore''', and the		In both cases the requirement is about what the theory must '''ignore''', and the
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	== Conserved quantities again ==		== Conserved quantities again ==

	[[Hamilton's principle/Explainers/Symmetry and conservation]] applies here without		[[Hamilton's principle/Explainers/Symmetry and conservation\|Symmetry and conservation]] applies here without
	change, and produces something familiar.		change, and produces something familiar.

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	It becomes decisive for gravity, where the ambiguity cannot be repaired away at		It becomes decisive for gravity, where the ambiguity cannot be repaired away at
	all. That is at		all. That is at
	[[Hamilton's principle/Explainers/Gravity from an action]].		[[Hamilton's principle/Explainers/Gravity from an action\|Gravity from an action]].

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

	The first worked example is at		The first worked example is at
	[[Hamilton's principle/Explainers/Electromagnetism from an action]].		[[Hamilton's principle/Explainers/Electromagnetism from an action\|Electromagnetism from an action]].