Hamilton's principle/Applications/Classical field theory - Revision history

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	well posed. Higher derivatives generally produce instabilities, which is why		well posed. Higher derivatives generally produce instabilities, which is why
	essentially every field theory in use is first-order in <math>\mathcal{L}</math>.		essentially every field theory in use is first-order in <math>\mathcal{L}</math>.
	~~The Einstein-Hilbert action of~~		[[Hamilton's principle/Applications/The Einstein-Hilbert action\|The Einstein-Hilbert action]] is the
	[[Hamilton's principle/Applications/The Einstein-Hilbert action\|~~the~~ Einstein-Hilbert action]] is the
	significant exception, and it escapes the problem for a reason specific to it.		significant exception, and it escapes the problem for a reason specific to it.

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			{{DISPLAYTITLE:Classical field theory}}
	{{Plebeian\|Hamilton's principle/Explainers/Fields}}		{{Plebeian\|Hamilton's principle/Explainers/Fields}}

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	<references />		<references />

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{{Plebeian|Hamilton's principle/Explainers/Fields}}

'''Classical field theory''' is the variational method applied to systems whose
degrees of freedom are labelled by position in space rather than by an index. The
vibrating string of
[[Hamilton's principle/Applications/Continuum systems: the vibrating string]] is
the one-dimensional case; everything structural was already visible there. This
page states the general construction and the two features that only appear once
there is more than one space dimension: relativistic invariance, and the
stress-energy tensor.

== The construction ==

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

A '''field''' is a function <math>\phi_i(x)</math> of the spacetime coordinates
<math>x = (x^0, x^1, x^2, x^3)</math>. The action is the integral of a
'''Lagrangian density''' over a spacetime region:

:<math>S[\phi] = \int_\Omega
\mathcal{L}\bigl( \phi_i,\, \partial_\alpha\phi_i,\, x \bigr)\, d^4x
\qquad\qquad\mathrm{(1)}</math>

By [[Hamilton's principle/Mathematics/Several independent variables]], the
stationarity condition with variations vanishing on <math>\partial\Omega</math>
is

:<math>\frac{\partial\mathcal{L}}{\partial\phi_i}
- \partial_\alpha\!\left(
\frac{\partial\mathcal{L}}{\partial(\partial_\alpha\phi_i)} \right) = 0
\qquad\qquad\mathrm{(2)}</math>

one equation per field. That is the whole formalism. What makes it powerful is
not (2) but the constraints one can put on <math>\mathcal{L}</math> before
writing it down.

== Locality and its consequences ==

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

Requiring <math>\mathcal{L}</math> to depend on the fields at a single point and
on finitely many derivatives there is the assumption of '''locality'''. Two
consequences follow immediately.

'''Field equations are partial differential equations, not integral
equations.''' Influence propagates through neighbouring points rather than acting
at a distance.

'''Conservation laws are local.''' Noether's theorem on
[[Hamilton's principle/Mathematics/Noether's first theorem]] gives
<math>\partial_\alpha j^\alpha = 0</math> — a density and a current obeying a
continuity equation — rather than a global constant. A quantity does not simply
stay constant; it flows. In one space dimension this was the energy density and
flux (8) of the string page.

Restricting to first derivatives is a further choice, and it is not innocent: it
makes (2) second order, which is what keeps the initial-value problem
well posed. Higher derivatives generally produce instabilities, which is why
essentially every field theory in use is first-order in <math>\mathcal{L}</math>.
The Einstein-Hilbert action of
[[Hamilton's principle/Applications/The Einstein-Hilbert action]] is the
significant exception, and it escapes the problem for a reason specific to it.

== Building a Lagrangian from symmetry ==

The practical method is to write the most general local density consistent with
the symmetries required, and let (2) produce the dynamics.

For a real scalar field with Lorentz invariance, the available scalars from
<math>\phi</math> and one derivative are <math>\phi</math> itself and
<math>\partial_\alpha\phi\,\partial^\alpha\phi</math>. So

:<math>\mathcal{L} = \tfrac{1}{2}\,\partial_\alpha\phi\,\partial^\alpha\phi
- V(\phi)
\qquad\qquad\mathrm{(3)}</math>

is the general form. Applying (2),

:<math>\partial_\alpha\partial^\alpha\phi + V'(\phi) = 0
\qquad\qquad\mathrm{(4)}</math>

With <math>V = \tfrac{1}{2}m^2\phi^2</math> this is the Klein-Gordon equation,
and with <math>V = 0</math> it is the wave equation in four dimensions — the
string's (6) with three space dimensions instead of one.

'''The symmetry chose the theory.''' Nothing was assumed about the dynamics; the
requirement of Lorentz invariance plus locality plus first derivatives left
almost no freedom. This is the characteristic method of modern physics and it is
a direct consequence of taking the action rather than the equations of motion as
the primitive object.

== The stress-energy tensor ==

If <math>\mathcal{L}</math> has no explicit dependence on <math>x</math>, the
action is invariant under the four translations
<math>x^\alpha \to x^\alpha + \varepsilon^\alpha</math>. That is a
four-parameter group, so
[[Hamilton's principle/Mathematics/Noether's first theorem]] gives four conserved
currents. Assembled, they form the '''canonical stress-energy tensor'''

:<math>T^{\alpha}{}_{\beta}
= \frac{\partial\mathcal{L}}{\partial(\partial_\alpha\phi_i)}\,
\partial_\beta\phi_i
- \delta^\alpha{}_\beta\,\mathcal{L}
\qquad\qquad\mathrm{(5)}</math>

satisfying <math>\partial_\alpha T^\alpha{}_\beta = 0</math>.

Compare (5) with the Hamiltonian <math>H = p\dot q - L</math> of
[[Hamilton's principle/Mathematics/The Hamiltonian and Hamilton's equations]]:
the <math>\beta = 0</math> component is exactly that expression with the momentum
replaced by the momentum density. '''The stress-energy tensor is the Legendre
transform of the Lagrangian density, indexed by direction.''' Energy density,
momentum density, energy flux and stress are its components.

Noether's own §6 concerns precisely these quantities. She calls the relations
arising from the displacement group the energy relationships and their
<math>B^{(\lambda)}</math> the energy components, and asks when they are proper
rather than improper — a question that becomes decisive on
[[Hamilton's principle/Applications/The Einstein-Hilbert action]].

'''(5) is not the last word.''' The canonical tensor is generally neither
symmetric nor gauge invariant, and it can be improved by adding a term whose
divergence vanishes identically — which changes no conservation law. The
symmetric improvement is the one that couples to gravity. Noether's classification
anticipates this: adding an identically divergence-free quantity is exactly the
freedom her improper divergence relations describe.

== Fields with internal symmetry ==

For a complex scalar, <math>\mathcal{L}</math> may be required invariant under

:<math>\phi \to e^{i\alpha}\phi
\qquad\qquad\mathrm{(6)}</math>

with <math>\alpha</math> constant. This is a one-parameter group acting on the
fields rather than on spacetime, and Noether gives one conserved current:

:<math>j^\alpha = i\left(
\phi^*\,\partial^\alpha\phi - \phi\,\partial^\alpha\phi^* \right)
\qquad\qquad\mathrm{(7)}</math>

which is the electric charge current up to normalisation. Charge conservation is
a symmetry of the action, and there is no other reason for it.

'''If <math>\alpha</math> is allowed to depend on position''', (6) becomes a
group of arbitrary functions, and the theory falls under
[[Hamilton's principle/Mathematics/Noether's second theorem]] rather than the
first. That change is the subject of the next page, and it forces the existence
of a new field.

<blockquote>
Provenance: every equation on this page is derived here or is standard. The
characterisations of Noether §6 are paraphrased from the ESTABLISHED source under
the translation rule stated on
[[Hamilton's principle/Mathematics/Noether's first theorem]]. See
[[Hamilton's principle/Sources]].
</blockquote>

== Next ==

[[Hamilton's principle/Applications/The electromagnetic Lagrangian]] makes the
phase in (6) local.

== References ==

<references />

← Older revision		Revision as of 04:58, 16 September 2026
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	degrees of freedom are labelled by position in space rather than by an index. The		degrees of freedom are labelled by position in space rather than by an index. The
	vibrating string of		vibrating string of
	[[Hamilton's principle/Applications/Continuum systems: the vibrating string]] is		[[Hamilton's principle/Applications/Continuum systems: the vibrating string\|Continuum systems: the vibrating string]] is
	the one-dimensional case; everything structural was already visible there. This		the one-dimensional case; everything structural was already visible there. This
	page states the general construction and the two features that only appear once		page states the general construction and the two features that only appear once
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	\qquad\qquad\mathrm{(1)}</math>		\qquad\qquad\mathrm{(1)}</math>

	By [[Hamilton's principle/Mathematics/Several independent variables]], the		By [[Hamilton's principle/Mathematics/Several independent variables\|Several independent variables]], the
	stationarity condition with variations vanishing on <math>\partial\Omega</math>		stationarity condition with variations vanishing on <math>\partial\Omega</math>
	is		is
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	'''Conservation laws are local.''' Noether's theorem on		'''Conservation laws are local.''' Noether's theorem on
	[[Hamilton's principle/Mathematics/Noether's first theorem]] gives		[[Hamilton's principle/Mathematics/Noether's first theorem\|Noether's first theorem]] gives
	<math>\partial_\alpha j^\alpha = 0</math> — a density and a current obeying a		<math>\partial_\alpha j^\alpha = 0</math> — a density and a current obeying a
	continuity equation — rather than a global constant. A quantity does not simply		continuity equation — rather than a global constant. A quantity does not simply
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	essentially every field theory in use is first-order in <math>\mathcal{L}</math>.		essentially every field theory in use is first-order in <math>\mathcal{L}</math>.
	The Einstein-Hilbert action of		The Einstein-Hilbert action of
	[[Hamilton's principle/Applications/The Einstein-Hilbert action]] is the		[[Hamilton's principle/Applications/The Einstein-Hilbert action\|the Einstein-Hilbert action]] is the
	significant exception, and it escapes the problem for a reason specific to it.		significant exception, and it escapes the problem for a reason specific to it.

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	<math>x^\alpha \to x^\alpha + \varepsilon^\alpha</math>. That is a		<math>x^\alpha \to x^\alpha + \varepsilon^\alpha</math>. That is a
	four-parameter group, so		four-parameter group, so
	[[Hamilton's principle/Mathematics/Noether's first theorem]] gives four conserved		[[Hamilton's principle/Mathematics/Noether's first theorem\|Noether's first theorem]] gives four conserved
	currents. Assembled, they form the '''canonical stress-energy tensor'''		currents. Assembled, they form the '''canonical stress-energy tensor'''

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	Compare (5) with the Hamiltonian <math>H = p\dot q - L</math> of		Compare (5) with the Hamiltonian <math>H = p\dot q - L</math> of
	[[Hamilton's principle/Mathematics/The Hamiltonian and Hamilton's equations]]:		[[Hamilton's principle/Mathematics/The Hamiltonian and Hamilton's equations\|the Hamiltonian and Hamilton's equations]]:
	the <math>\beta = 0</math> component is exactly that expression with the momentum		the <math>\beta = 0</math> component is exactly that expression with the momentum
	replaced by the momentum density. '''The stress-energy tensor is the Legendre		replaced by the momentum density. '''The stress-energy tensor is the Legendre
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	<math>B^{(\lambda)}</math> the energy components, and asks when they are proper		<math>B^{(\lambda)}</math> the energy components, and asks when they are proper
	rather than improper — a question that becomes decisive on		rather than improper — a question that becomes decisive on
	[[Hamilton's principle/Applications/The Einstein-Hilbert action]].		[[Hamilton's principle/Applications/The Einstein-Hilbert action\|the Einstein-Hilbert action]].

	'''(5) is not the last word.''' The canonical tensor is generally neither		'''(5) is not the last word.''' The canonical tensor is generally neither
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	'''If <math>\alpha</math> is allowed to depend on position''', (6) becomes a		'''If <math>\alpha</math> is allowed to depend on position''', (6) becomes a
	group of arbitrary functions, and the theory falls under		group of arbitrary functions, and the theory falls under
	[[Hamilton's principle/Mathematics/Noether's second theorem]] rather than the		[[Hamilton's principle/Mathematics/Noether's second theorem\|Noether's second theorem]] rather than the
	first. That change is the subject of the next page, and it forces the existence		first. That change is the subject of the next page, and it forces the existence
	of a new field.		of a new field.
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	characterisations of Noether §6 are paraphrased from the ESTABLISHED source under		characterisations of Noether §6 are paraphrased from the ESTABLISHED source under
	the translation rule stated on		the translation rule stated on
	[[Hamilton's principle/Mathematics/Noether's first theorem]]. See		[[Hamilton's principle/Mathematics/Noether's first theorem\|Noether's first theorem]]. See
	[[Hamilton's principle/Sources]].		[[Hamilton's principle/Sources\|Sources]].
	</blockquote>		</blockquote>

	== Next ==		== Next ==

	[[Hamilton's principle/Applications/The electromagnetic Lagrangian]] makes the		[[Hamilton's principle/Applications/The electromagnetic Lagrangian\|the electromagnetic Lagrangian]] makes the
	phase in (6) local.		phase in (6) local.