Hamilton's principle/Mathematics/The Euler-Lagrange equation - Revision history

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	second partials by hypothesis and <math>y \in C^2</math>, so		second partials by hypothesis and <math>y \in C^2</math>, so
	<math>\frac{d}{dx} L_{y'}</math> exists and is continuous. Equation (3) holds for		<math>\frac{d}{dx} L_{y'}</math> exists and is continuous. Equation (3) holds for
	every admissible <math>\eta</math>. ~~The fundamental lemma of~~		every admissible <math>\eta</math>. [[Hamilton's principle/Mathematics/The fundamental lemma\|The fundamental lemma]] therefore applies to
	[[Hamilton's principle/Mathematics/The fundamental lemma\|~~the~~ fundamental lemma]] therefore applies to
	the bracket, and it vanishes identically. That is (2).		the bracket, and it vanishes identically. That is (2).
	<math>\blacksquare</math>		<math>\blacksquare</math>
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	<math>y''</math> vanishes, (5) is not a second-order equation at all and the		<math>y''</math> vanishes, (5) is not a second-order equation at all and the
	problem is singular. The condition <math>L_{y'y'} \neq 0</math> is called		problem is singular. The condition <math>L_{y'y'} \neq 0</math> is called
	'''non-degeneracy''', and its signed version is ~~the Legendre condition of~~		'''non-degeneracy''', and its signed version is [[Hamilton's principle/Mathematics/The Legendre condition\|the Legendre condition]].
	[[Hamilton's principle/Mathematics/The Legendre condition\|the Legendre condition]].

	'''Solutions are called extremals, not extrema.''' A solution of (2) is an		'''Solutions are called extremals, not extrema.''' A solution of (2) is an

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			{{DISPLAYTITLE:The Euler-Lagrange equation}}
	{{Plebeian\|Hamilton's principle/Explainers/What the Euler-Lagrange equation says}}		{{Plebeian\|Hamilton's principle/Explainers/What the Euler-Lagrange equation says}}

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	<references />		<references />

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{{Plebeian|Hamilton's principle/Explainers/What the Euler-Lagrange equation says}}

The '''Euler-Lagrange equation''' is the differential equation satisfied by every
stationary path of a functional. It is the central result of the machinery track,
and everything in the applications track is an instance of it.

== Statement ==

[[File:OPL-HP-F-39.png|thumb|right|400px|The two rates the Euler-Lagrange equation balances.]]

'''Theorem.''' Let <math>L(x,y,y')</math> have continuous partial derivatives up
to second order, and let

:<math>J[y] = \int_a^b L\bigl(x,\, y(x),\, y'(x)\bigr)\, dx
\qquad\qquad\mathrm{(1)}</math>

on the admissible set <math>\mathcal{A}</math> of
[[Hamilton's principle/Mathematics/Functionals and the space of paths]]. If
<math>y \in \mathcal{A}</math> is stationary for <math>J</math>, then

:<math>\frac{\partial L}{\partial y}
- \frac{d}{dx}\!\left( \frac{\partial L}{\partial y'} \right) = 0
\qquad\qquad\mathrm{(2)}</math>

for every <math>x \in [a,b]</math>.

== Derivation ==

Stationarity means <math>\delta J[y;\eta] = 0</math> for every
<math>\eta \in \mathcal{V}</math>. By equation (6) of
[[Hamilton's principle/Mathematics/The first variation]],

:<math>\int_a^b \left[ L_y - \frac{d}{dx} L_{y'} \right] \eta \, dx = 0
\qquad\qquad\mathrm{(3)}</math>

The bracket is continuous on <math>[a,b]</math>: <math>L</math> has continuous
second partials by hypothesis and <math>y \in C^2</math>, so
<math>\frac{d}{dx} L_{y'}</math> exists and is continuous. Equation (3) holds for
every admissible <math>\eta</math>. The fundamental lemma of
[[Hamilton's principle/Mathematics/The fundamental lemma]] therefore applies to
the bracket, and it vanishes identically. That is (2).
<math>\blacksquare</math>

The derivation is three lines because the work was done on the two preceding
pages. That is the correct division: the integration by parts is where the
endpoints are used, and the lemma is where the arbitrariness of
<math>\eta</math> is used.

== Expanded form ==

Carrying out the total derivative in (2) with the chain rule,

:<math>\frac{d}{dx} L_{y'}
= L_{y'x} + L_{y'y}\, y' + L_{y'y'}\, y''
\qquad\qquad\mathrm{(4)}</math>

so that (2) becomes

:<math>L_{y'y'}\, y'' + L_{y'y}\, y' + L_{y'x} - L_y = 0
\qquad\qquad\mathrm{(5)}</math>

This is a second-order ordinary differential equation for <math>y</math>,
generally nonlinear. Three consequences follow immediately from its form.

'''It is second order, so it needs two conditions.''' The admissible set supplies
exactly two, <math>y(a) = y_a</math> and <math>y(b) = y_b</math>. The count
matches, which is why the fixed-endpoint problem is well posed. It is a boundary
value problem, not an initial value problem, and that difference is the source of
the non-uniqueness discussed on
[[Hamilton's principle/Mathematics/The Jacobi condition and conjugate points]].

'''Degeneracy when <math>L_{y'y'} = 0</math>.''' If the coefficient of
<math>y''</math> vanishes, (5) is not a second-order equation at all and the
problem is singular. The condition <math>L_{y'y'} \neq 0</math> is called
'''non-degeneracy''', and its signed version is the Legendre condition of
[[Hamilton's principle/Mathematics/The Legendre condition]].

'''Solutions are called extremals, not extrema.''' A solution of (2) is an
'''extremal'''. The word records only that the first variation vanishes. Whether
an extremal minimises anything is undecided by (2) and is settled, when it can be
settled, on [[Hamilton's principle/Mathematics/The second variation]].

== First integrals ==

Two cases reduce the order of (2) by one.

'''<math>L</math> free of <math>y</math>.''' Then <math>L_y = 0</math> and (2)
gives

:<math>\frac{d}{dx} L_{y'} = 0
\quad\Longrightarrow\quad
L_{y'} = \text{const} \qquad\qquad\mathrm{(6)}</math>

The conserved quantity <math>L_{y'}</math> is the '''conjugate momentum'''. Where
<math>x</math> is time and <math>y</math> a coordinate, (6) is conservation of
momentum.

'''<math>L</math> free of <math>x</math>.''' Then

:<math>y'\, L_{y'} - L = \text{const} \qquad\qquad\mathrm{(7)}</math>

which is the Beltrami identity, derived on
[[Hamilton's principle/Mathematics/The Beltrami identity]]. Where <math>x</math>
is time, (7) is conservation of energy.

Both cases are instances of one theorem. A symmetry of <math>L</math> — here,
independence of a variable — produces a conserved quantity. The general statement
is Noether's, on
[[Hamilton's principle/Mathematics/Noether's first theorem]]. Noether states it
in the form that <math>\rho</math> linearly independent combinations of the
Lagrange expressions become divergences.<ref>Noether, ''Invariante
Variationsprobleme'' (1918), Theorem I, in Tavel's translation. Verbatim.
Noether's Lagrange expressions <math>\psi_i</math> are the left-hand side of (2).</ref>

== Worked example: the shortest plane curve ==

Minimise arc length between <math>(a, y_a)</math> and <math>(b, y_b)</math>:

:<math>J[y] = \int_a^b \sqrt{1 + y'^2}\; dx \qquad\qquad\mathrm{(8)}</math>

Here <math>L = \sqrt{1 + y'^2}</math> is free of both <math>x</math> and
<math>y</math>. Using (6),

:<math>L_{y'} = \frac{y'}{\sqrt{1 + y'^2}} = c
\qquad\qquad\mathrm{(9)}</math>

Solving (9) for <math>y'</math> gives
<math>y'^2 (1 - c^2) = c^2</math>, so <math>y'</math> is a constant wherever
<math>\vert c \vert < 1</math>, and <math>y</math> is a straight line. The two
endpoint conditions fix its two parameters.

Note what has ''not'' been shown: that the line is the shortest curve. Equation
(2) is necessary, not sufficient. The line is here in fact the unique minimiser,
but establishing that requires the sufficiency theory, and the geodesic problem
on a sphere in
[[Hamilton's principle/Mathematics/The Jacobi condition and conjugate points]]
is the case where the corresponding extremal is stationary and is ''not'' a
minimum.

== Historical note ==

Euler reached the equation in 1744 by a limiting argument on a polygonal
approximation to the curve, in ''Methodus inveniendi lineas curvas maximi
minimive proprietate gaudentes''. Lagrange's variational derivation, close to the
one above, followed in the 1750s correspondence with Euler and reached print in
the ''Mécanique analytique'' of 1788.

<blockquote>
Provenance: this paragraph is editorial summary, not transcription. The Euler and
Lagrange source rows of the survey on
[[Hamilton's principle/Sources]] are not yet ESTABLISHED, and no equation on this
page is attributed to either. The two attributed equations, (2) as a Lagrange
expression and the Theorem I statement, come from Noether 1918, which is
ESTABLISHED.
</blockquote>

== Next ==

[[Hamilton's principle/Mathematics/Boundary terms and natural boundary conditions]]
relaxes the fixed-endpoint condition and recovers the boundary term dropped at
equation (5) of
[[Hamilton's principle/Mathematics/The first variation]].

== References ==

<references />

← Older revision		Revision as of 04:57, 16 September 2026
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	on the admissible set <math>\mathcal{A}</math> of		on the admissible set <math>\mathcal{A}</math> of
	[[Hamilton's principle/Mathematics/Functionals and the space of paths]]. If		[[Hamilton's principle/Mathematics/Functionals and the space of paths\|Functionals and the space of paths]]. If
	<math>y \in \mathcal{A}</math> is stationary for <math>J</math>, then		<math>y \in \mathcal{A}</math> is stationary for <math>J</math>, then

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	Stationarity means <math>\delta J[y;\eta] = 0</math> for every		Stationarity means <math>\delta J[y;\eta] = 0</math> for every
	<math>\eta \in \mathcal{V}</math>. By equation (6) of		<math>\eta \in \mathcal{V}</math>. By equation (6) of
	[[Hamilton's principle/Mathematics/The first variation]],		[[Hamilton's principle/Mathematics/The first variation\|the first variation]],

	:<math>\int_a^b \left[ L_y - \frac{d}{dx} L_{y'} \right] \eta \, dx = 0		:<math>\int_a^b \left[ L_y - \frac{d}{dx} L_{y'} \right] \eta \, dx = 0
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	<math>\frac{d}{dx} L_{y'}</math> exists and is continuous. Equation (3) holds for		<math>\frac{d}{dx} L_{y'}</math> exists and is continuous. Equation (3) holds for
	every admissible <math>\eta</math>. The fundamental lemma of		every admissible <math>\eta</math>. The fundamental lemma of
	[[Hamilton's principle/Mathematics/The fundamental lemma]] therefore applies to		[[Hamilton's principle/Mathematics/The fundamental lemma\|the fundamental lemma]] therefore applies to
	the bracket, and it vanishes identically. That is (2).		the bracket, and it vanishes identically. That is (2).
	<math>\blacksquare</math>		<math>\blacksquare</math>
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	value problem, not an initial value problem, and that difference is the source of		value problem, not an initial value problem, and that difference is the source of
	the non-uniqueness discussed on		the non-uniqueness discussed on
	[[Hamilton's principle/Mathematics/The Jacobi condition and conjugate points]].		[[Hamilton's principle/Mathematics/The Jacobi condition and conjugate points\|the Jacobi condition and conjugate points]].

	'''Degeneracy when <math>L_{y'y'} = 0</math>.''' If the coefficient of		'''Degeneracy when <math>L_{y'y'} = 0</math>.''' If the coefficient of
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	problem is singular. The condition <math>L_{y'y'} \neq 0</math> is called		problem is singular. The condition <math>L_{y'y'} \neq 0</math> is called
	'''non-degeneracy''', and its signed version is the Legendre condition of		'''non-degeneracy''', and its signed version is the Legendre condition of
	[[Hamilton's principle/Mathematics/The Legendre condition]].		[[Hamilton's principle/Mathematics/The Legendre condition\|the Legendre condition]].

	'''Solutions are called extremals, not extrema.''' A solution of (2) is an		'''Solutions are called extremals, not extrema.''' A solution of (2) is an
	'''extremal'''. The word records only that the first variation vanishes. Whether		'''extremal'''. The word records only that the first variation vanishes. Whether
	an extremal minimises anything is undecided by (2) and is settled, when it can be		an extremal minimises anything is undecided by (2) and is settled, when it can be
	settled, on [[Hamilton's principle/Mathematics/The second variation]].		settled, on [[Hamilton's principle/Mathematics/The second variation\|the second variation]].

	== First integrals ==		== First integrals ==
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	which is the Beltrami identity, derived on		which is the Beltrami identity, derived on
	[[Hamilton's principle/Mathematics/The Beltrami identity]]. Where <math>x</math>		[[Hamilton's principle/Mathematics/The Beltrami identity\|the Beltrami identity]]. Where <math>x</math>
	is time, (7) is conservation of energy.		is time, (7) is conservation of energy.

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	independence of a variable — produces a conserved quantity. The general statement		independence of a variable — produces a conserved quantity. The general statement
	is Noether's, on		is Noether's, on
	[[Hamilton's principle/Mathematics/Noether's first theorem]]. Noether states it		[[Hamilton's principle/Mathematics/Noether's first theorem\|Noether's first theorem]]. Noether states it
	in the form that <math>\rho</math> linearly independent combinations of the		in the form that <math>\rho</math> linearly independent combinations of the
	Lagrange expressions become divergences.<ref>Noether, ''Invariante		Lagrange expressions become divergences.<ref>Noether, ''Invariante
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	but establishing that requires the sufficiency theory, and the geodesic problem		but establishing that requires the sufficiency theory, and the geodesic problem
	on a sphere in		on a sphere in
	[[Hamilton's principle/Mathematics/The Jacobi condition and conjugate points]]		[[Hamilton's principle/Mathematics/The Jacobi condition and conjugate points\|the Jacobi condition and conjugate points]]
	is the case where the corresponding extremal is stationary and is ''not'' a		is the case where the corresponding extremal is stationary and is ''not'' a
	minimum.		minimum.
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	Provenance: this paragraph is editorial summary, not transcription. The Euler and		Provenance: this paragraph is editorial summary, not transcription. The Euler and
	Lagrange source rows of the survey on		Lagrange source rows of the survey on
	[[Hamilton's principle/Sources]] are not yet ESTABLISHED, and no equation on this		[[Hamilton's principle/Sources\|Sources]] are not yet ESTABLISHED, and no equation on this
	page is attributed to either. The two attributed equations, (2) as a Lagrange		page is attributed to either. The two attributed equations, (2) as a Lagrange
	expression and the Theorem I statement, come from Noether 1918, which is		expression and the Theorem I statement, come from Noether 1918, which is
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	== Next ==		== Next ==

	[[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]]
	relaxes the fixed-endpoint condition and recovers the boundary term dropped at		relaxes the fixed-endpoint condition and recovers the boundary term dropped at
	equation (5) of		equation (5) of
	[[Hamilton's principle/Mathematics/The first variation]].		[[Hamilton's principle/Mathematics/The first variation\|the first variation]].

	== References ==		== References ==

	<references />		<references />