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	{{Plebeian\|Hamilton's principle/Explainers/More than one variable}}		{{Plebeian\|Hamilton's principle/Explainers/More than one variable}}

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	<references />		<references />

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{{Plebeian|Hamilton's principle/Explainers/More than one variable}}

The last extension of the machinery replaces the interval by a domain and the
single independent variable by several. The Euler-Lagrange equation becomes a
'''partial''' differential equation, and the boundary term becomes a surface
integral handled by the divergence theorem. This is the form in which every field
theory in the applications track is written.

== Setup ==

[[File:OPL-HP-F-52.png|thumb|right|400px|The boundary term in one dimension and in a field.]]

Let <math>\Omega \subset \mathbb{R}^m</math> be a bounded domain with piecewise
smooth boundary <math>\partial\Omega</math>, with points written
<math>x = (x_1, \ldots, x_m)</math>. Let <math>u : \Omega \to \mathbb{R}</math>
be twice continuously differentiable, with prescribed boundary values

:<math>u = u_0 \quad \text{on } \partial\Omega
\qquad\qquad\mathrm{(1)}</math>

Write <math>u_{,\alpha} = \partial u / \partial x_\alpha</math>. Let
<math>L(x, u, u_{,\alpha})</math> have continuous second partials and define

:<math>J[u] = \int_\Omega L\bigl(x,\, u,\, u_{,\alpha}\bigr)\, d^m x
\qquad\qquad\mathrm{(2)}</math>

Variations <math>\eta \in C^2(\bar\Omega)</math> vanish on the whole boundary.

== Derivation ==

[[File:OPL-HP-F-56.png|thumb|right|400px|From one independent variable to several.]]

Differentiating <math>J[u + \varepsilon\eta]</math> at
<math>\varepsilon = 0</math>,

:<math>\delta J = \int_\Omega \left[
\frac{\partial L}{\partial u}\,\eta
+ \sum_{\alpha=1}^m \frac{\partial L}{\partial u_{,\alpha}}\,\eta_{,\alpha}
\right] d^m x
\qquad\qquad\mathrm{(3)}</math>

Integration by parts is now the product rule plus the divergence theorem. Write
<math>A_\alpha = (\partial L / \partial u_{,\alpha})\,\eta</math>. Then

:<math>\sum_\alpha \frac{\partial A_\alpha}{\partial x_\alpha}
= \sum_\alpha \frac{\partial L}{\partial u_{,\alpha}}\,\eta_{,\alpha}
+ \sum_\alpha \frac{\partial}{\partial x_\alpha}
\!\left( \frac{\partial L}{\partial u_{,\alpha}} \right)\eta
\qquad\qquad\mathrm{(4)}</math>

Integrating (4) over <math>\Omega</math> and applying the divergence theorem to
the left side,

:<math>\int_{\partial\Omega} \sum_\alpha
\frac{\partial L}{\partial u_{,\alpha}}\, \eta\, n_\alpha \, dS
= \int_\Omega \sum_\alpha
\frac{\partial L}{\partial u_{,\alpha}}\,\eta_{,\alpha}\, d^m x
+ \int_\Omega \sum_\alpha \frac{\partial}{\partial x_\alpha}
\!\left( \frac{\partial L}{\partial u_{,\alpha}} \right) \eta \, d^m x
\qquad\qquad\mathrm{(5)}</math>

with <math>n_\alpha</math> the outward unit normal. The surface integral vanishes
because <math>\eta = 0</math> on <math>\partial\Omega</math>. Substituting the
resulting expression into (3),

:<math>\delta J = \int_\Omega \left[
\frac{\partial L}{\partial u}
- \sum_{\alpha=1}^m \frac{\partial}{\partial x_\alpha}
\!\left( \frac{\partial L}{\partial u_{,\alpha}} \right)
\right] \eta \; d^m x
\qquad\qquad\mathrm{(6)}</math>

By the multidimensional form of the fundamental lemma, given on
[[Hamilton's principle/Mathematics/The fundamental lemma]], the bracket vanishes:

:<math>\frac{\partial L}{\partial u}
- \sum_{\alpha=1}^m \frac{\partial}{\partial x_\alpha}
\!\left( \frac{\partial L}{\partial u_{,\alpha}} \right) = 0
\qquad\qquad\mathrm{(7)}</math>

<math>\blacksquare</math>

Equation (7) is the Euler-Lagrange equation for a field. It is a second-order
partial differential equation. The structure is the same as the one-dimensional
case throughout: differentiate under the integral, move the derivative off the
variation, discard a boundary term, apply the lemma.

== Several fields ==

With <math>N</math> fields <math>u_1, \ldots, u_N</math>, independent variations
give one equation per field, by the same select-one-index argument as on
[[Hamilton's principle/Mathematics/Several dependent variables]]:

:<math>\frac{\partial L}{\partial u_i}
- \sum_{\alpha=1}^m \frac{\partial}{\partial x_\alpha}
\!\left( \frac{\partial L}{\partial u_{i,\alpha}} \right) = 0,
\qquad i = 1, \ldots, N
\qquad\qquad\mathrm{(8)}</math>

== The boundary term does not always vanish ==

Two situations leave the surface integral in (5) standing, and both matter later.

'''Free boundary.''' If <math>u</math> is unprescribed on part of
<math>\partial\Omega</math>, the argument of
[[Hamilton's principle/Mathematics/Boundary terms and natural boundary conditions]]
applies and yields the natural condition

:<math>\sum_\alpha \frac{\partial L}{\partial u_{,\alpha}}\, n_\alpha = 0
\quad \text{on that part}
\qquad\qquad\mathrm{(9)}</math>

'''Second derivatives in <math>L</math>.''' If <math>L</math> contains
<math>u_{,\alpha\beta}</math>, two integrations by parts are needed and the
surface term involves the normal derivative of <math>\eta</math>, which does not
vanish merely because <math>\eta</math> does. The Einstein-Hilbert action is
exactly this case, and the standard remedy is to add a compensating surface term.
That is taken up on
[[Hamilton's principle/Applications/The Einstein-Hilbert action]].

== Divergences and conservation ==

The combination appearing in (4),

:<math>\mathrm{Div}\, A = \sum_{\alpha=1}^m
\frac{\partial A_\alpha}{\partial x_\alpha}
\qquad\qquad\mathrm{(10)}</math>

is not an artefact of the derivation. It is the object Noether's theorems are
about. Noether writes the identity produced by partial integration as

:<math>\textstyle\sum \psi_i\, \delta u_i = \delta f + \mathrm{Div}\, A
\qquad\qquad\mathrm{(11)}</math>

where the <math>\psi_i</math> are the Lagrange expressions, that is, the left
sides of (8).<ref>Noether, ''Invariante Variationsprobleme'' (1918), equation
(3), in Tavel's translation. Verbatim in structure and notation.</ref>

Her first theorem states that invariance of <math>J</math> under a
<math>\rho</math>-parameter group makes <math>\rho</math> independent
combinations of the Lagrange expressions into divergences. Setting the field
equations <math>\psi_i = 0</math> then gives <math>\mathrm{Div}\, B = 0</math>,
and Noether notes that these divergence equations are what have lately been
called laws of conservation.<ref>Noether, ''Invariante Variationsprobleme''
(1918), §1 and §3, Tavel translation. Verbatim phrase.</ref>

In one independent variable a divergence is a total derivative and
<math>\mathrm{Div}\,B = 0</math> reduces to <math>B = \text{const}</math>, which
recovers the first integrals of
[[Hamilton's principle/Mathematics/The Euler-Lagrange equation]]. Conservation
laws in field theory and first integrals in mechanics are one result read in two
dimensions of the independent variable. This is developed on
[[Hamilton's principle/Mathematics/Noether's first theorem]].

== Worked example: the vibrating string ==

Let <math>u(x,t)</math> be the transverse displacement of a string of linear
density <math>\rho</math> under tension <math>\tau</math>. The Lagrangian density
is kinetic minus potential energy per unit length:

:<math>L = \tfrac{1}{2}\rho\, u_{,t}^2 - \tfrac{1}{2}\tau\, u_{,x}^2
\qquad\qquad\mathrm{(12)}</math>

<math>L</math> contains no <math>u</math>, so
<math>\partial L/\partial u = 0</math>. The two derivative slots give
<math>\partial L / \partial u_{,t} = \rho\, u_{,t}</math> and
<math>\partial L / \partial u_{,x} = -\tau\, u_{,x}</math>. Substituting into (7)
with <math>(x_1, x_2) = (t, x)</math>,

:<math>0 - \frac{\partial}{\partial t}\bigl( \rho\, u_{,t} \bigr)
- \frac{\partial}{\partial x}\bigl( -\tau\, u_{,x} \bigr) = 0
\qquad\qquad\mathrm{(13)}</math>

that is

:<math>\rho\, u_{,tt} = \tau\, u_{,xx}
\qquad\qquad\mathrm{(14)}</math>

the wave equation, with speed <math>c = \sqrt{\tau/\rho}</math>. The full
treatment, including a free end and the natural condition (9), is on
[[Hamilton's principle/Applications/Continuum systems: the vibrating string]].

== The machinery track is complete ==

Eight pages have carried the subject from the definition of a functional to a
field equation. Everything after this point either constrains the machinery
([[Hamilton's principle/Mathematics/Holonomic constraints]]), asks whether an
extremal is actually a minimum
([[Hamilton's principle/Mathematics/The second variation]]), restructures it
([[Hamilton's principle/Mathematics/The Legendre transform]]), or applies it.

No page in this track has established that any extremal minimises anything. That
is the correct state of the subject at this depth, and it is why this volume says
stationary rather than least.

== Next ==

[[Hamilton's principle/Mathematics/Holonomic constraints]] opens the second
mathematics track.

== References ==

<references />

← Older revision		Revision as of 04:57, 16 September 2026
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	By the multidimensional form of the fundamental lemma, given on		By the multidimensional form of the fundamental lemma, given on
	[[Hamilton's principle/Mathematics/The fundamental lemma]], the bracket vanishes:		[[Hamilton's principle/Mathematics/The fundamental lemma\|the fundamental lemma]], the bracket vanishes:

	:<math>\frac{\partial L}{\partial u}		:<math>\frac{\partial L}{\partial u}
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	With <math>N</math> fields <math>u_1, \ldots, u_N</math>, independent variations		With <math>N</math> fields <math>u_1, \ldots, u_N</math>, independent variations
	give one equation per field, by the same select-one-index argument as on		give one equation per field, by the same select-one-index argument as on
	[[Hamilton's principle/Mathematics/Several dependent variables]]:		[[Hamilton's principle/Mathematics/Several dependent variables\|Several dependent variables]]:

	:<math>\frac{\partial L}{\partial u_i}		:<math>\frac{\partial L}{\partial u_i}
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	'''Free boundary.''' If <math>u</math> is unprescribed on part of		'''Free boundary.''' If <math>u</math> is unprescribed on part of
	<math>\partial\Omega</math>, the argument of		<math>\partial\Omega</math>, the argument of
	[[Hamilton's principle/Mathematics/Boundary terms and natural boundary conditions]]		[[Hamilton's principle/Mathematics/Boundary terms and natural boundary conditions\|Boundary terms and natural boundary conditions]]
	applies and yields the natural condition		applies and yields the natural condition

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	exactly this case, and the standard remedy is to add a compensating surface term.		exactly this case, and the standard remedy is to add a compensating surface term.
	That is taken up on		That is taken up on
	[[Hamilton's principle/Applications/The Einstein-Hilbert action]].		[[Hamilton's principle/Applications/The Einstein-Hilbert action\|the Einstein-Hilbert action]].

	== Divergences and conservation ==		== Divergences and conservation ==
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	<math>\mathrm{Div}\,B = 0</math> reduces to <math>B = \text{const}</math>, which		<math>\mathrm{Div}\,B = 0</math> reduces to <math>B = \text{const}</math>, which
	recovers the first integrals of		recovers the first integrals of
	[[Hamilton's principle/Mathematics/The Euler-Lagrange equation]]. Conservation		[[Hamilton's principle/Mathematics/The Euler-Lagrange equation\|the Euler-Lagrange equation]]. Conservation
	laws in field theory and first integrals in mechanics are one result read in two		laws in field theory and first integrals in mechanics are one result read in two
	dimensions of the independent variable. This is developed on		dimensions of the independent variable. This is developed on
	[[Hamilton's principle/Mathematics/Noether's first theorem]].		[[Hamilton's principle/Mathematics/Noether's first theorem\|Noether's first theorem]].

	== Worked example: the vibrating string ==		== Worked example: the vibrating string ==
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	the wave equation, with speed <math>c = \sqrt{\tau/\rho}</math>. The full		the wave equation, with speed <math>c = \sqrt{\tau/\rho}</math>. The full
	treatment, including a free end and the natural condition (9), is on		treatment, including a free end and the natural condition (9), is on
	[[Hamilton's principle/Applications/Continuum systems: the vibrating string]].		[[Hamilton's principle/Applications/Continuum systems: the vibrating string\|Continuum systems: the vibrating string]].

	== The machinery track is complete ==		== The machinery track is complete ==
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	Eight pages have carried the subject from the definition of a functional to a		Eight pages have carried the subject from the definition of a functional to a
	field equation. Everything after this point either constrains the machinery		field equation. Everything after this point either constrains the machinery
	([[Hamilton's principle/Mathematics/Holonomic constraints]]), asks whether an		([[Hamilton's principle/Mathematics/Holonomic constraints\|Holonomic constraints]]), asks whether an
	extremal is actually a minimum		extremal is actually a minimum
	([[Hamilton's principle/Mathematics/The second variation]]), restructures it		([[Hamilton's principle/Mathematics/The second variation\|the second variation]]), restructures it
	([[Hamilton's principle/Mathematics/The Legendre transform]]), or applies it.		([[Hamilton's principle/Mathematics/The Legendre transform\|the Legendre transform]]), or applies it.

	No page in this track has established that any extremal minimises anything. That		No page in this track has established that any extremal minimises anything. That
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	== Next ==		== Next ==

	[[Hamilton's principle/Mathematics/Holonomic constraints]] opens the second		[[Hamilton's principle/Mathematics/Holonomic constraints\|Holonomic constraints]] opens the second
	mathematics track.		mathematics track.