Hamilton's principle/Mathematics/The fundamental lemma - Revision history

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	The '''fundamental lemma of the calculus of variations''' closes the gap left at		The '''fundamental lemma of the calculus of variations''' closes the gap left at
	the end of		the end of
	[[Hamilton's principle/Mathematics/The first variation]]. It states that a		[[Hamilton's principle/Mathematics/The first variation\|the first variation]]. It states that a
	continuous function whose weighted integral vanishes for every admissible weight		continuous function whose weighted integral vanishes for every admissible weight
	must itself be zero everywhere. Without it, stationarity yields an integral		must itself be zero everywhere. Without it, stationarity yields an integral
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	route to the Euler-Lagrange equation when the smoothness of <math>y</math> is		route to the Euler-Lagrange equation when the smoothness of <math>y</math> is
	not given. It is used on		not given. It is used on
	[[Hamilton's principle/Mathematics/The Weierstrass excess function]], where		[[Hamilton's principle/Mathematics/The Weierstrass excess function\|the Weierstrass excess function]], where
	extremals with corners are admitted.		extremals with corners are admitted.

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	proof is the same, with the bump (3) built from		proof is the same, with the bump (3) built from
	<math>\delta^2 - \Vert x - x_0 \Vert^2</math>. This is the version used on		<math>\delta^2 - \Vert x - x_0 \Vert^2</math>. This is the version used on
	[[Hamilton's principle/Mathematics/Several independent variables]].		[[Hamilton's principle/Mathematics/Several independent variables\|Several independent variables]].

	== Remark on generality ==		== Remark on generality ==
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	== Next ==		== Next ==

	[[Hamilton's principle/Mathematics/The Euler-Lagrange equation]] applies the		[[Hamilton's principle/Mathematics/The Euler-Lagrange equation\|the Euler-Lagrange equation]] applies the
	lemma to equation (7) of		lemma to equation (7) of
	[[Hamilton's principle/Mathematics/The first variation]].		[[Hamilton's principle/Mathematics/The first variation\|the first variation]].

	== References ==		== References ==

	<references />		<references />

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

The '''fundamental lemma of the calculus of variations''' closes the gap left at
the end of
[[Hamilton's principle/Mathematics/The first variation]]. It states that a
continuous function whose weighted integral vanishes for every admissible weight
must itself be zero everywhere. Without it, stationarity yields an integral
identity and nothing more; with it, stationarity yields a differential equation.

== Statement ==

[[File:OPL-HP-F-03.png|thumb|right|400px|A localised test function.]]

'''Lemma.''' Let <math>E</math> be continuous on <math>[a,b]</math>. Suppose

:<math>\int_a^b E(x)\,\eta(x)\, dx = 0 \qquad\qquad\mathrm{(1)}</math>

for every <math>\eta \in C^2[a,b]</math> with
<math>\eta(a) = \eta(b) = 0</math>. Then <math>E(x) = 0</math> for every
<math>x \in [a,b]</math>.

The strength of the lemma lies in how little it asks of the test class. It is
enough that (1) hold for smooth <math>\eta</math> vanishing at the ends; the
conclusion is then pointwise and holds on the closed interval.

== Proof ==

The argument is by contradiction, and its shape is worth naming before the
details: ''if <math>E</math> is nonzero somewhere, build an <math>\eta</math>
that is supported exactly where <math>E</math> keeps its sign, and the integral
cannot cancel.''

Suppose <math>E(x_0) \neq 0</math> for some <math>x_0 \in (a,b)</math>. Without
loss of generality take <math>E(x_0) > 0</math>; otherwise replace
<math>E</math> by <math>-E</math>, which changes nothing in (1).

'''Step 1. Isolate a subinterval where <math>E</math> keeps its sign.'''
<math>E</math> is continuous, so there is <math>\delta > 0</math> with
<math>[x_0 - \delta,\, x_0 + \delta] \subset (a,b)</math> and

:<math>E(x) > \tfrac{1}{2} E(x_0) > 0 \quad \text{for}\quad
\vert x - x_0 \vert < \delta \qquad\qquad\mathrm{(2)}</math>

'''Step 2. Build a test function supported there.''' Put

:<math>\eta(x) = \begin{cases}
\bigl[ \delta^2 - (x - x_0)^2 \bigr]^3, & \vert x - x_0 \vert \leq \delta \\
0, & \text{otherwise}
\end{cases} \qquad\qquad\mathrm{(3)}</math>

This <math>\eta</math> is admissible. It vanishes at <math>a</math> and
<math>b</math>, being zero outside a subinterval of <math>(a,b)</math>. It is
non-negative, and strictly positive on
<math>(x_0 - \delta,\, x_0 + \delta)</math>. The cube is what buys the
smoothness: the bracket vanishes to first order at
<math>x = x_0 \pm \delta</math>, so the cube vanishes to third order there and
<math>\eta \in C^2[a,b]</math>, as the test class requires.

'''Step 3. Evaluate.''' Because <math>\eta</math> vanishes outside the
subinterval,

:<math>\int_a^b E\,\eta\, dx
= \int_{x_0 - \delta}^{x_0 + \delta} E(x)\,\eta(x)\, dx
> \tfrac{1}{2} E(x_0) \int_{x_0 - \delta}^{x_0 + \delta} \eta(x)\, dx > 0
\qquad\qquad\mathrm{(4)}</math>

using (2) on the first factor and <math>\eta \geq 0</math>, not identically
zero, on the second. This contradicts (1).

Therefore <math>E</math> vanishes on the open interval
<math>(a,b)</math>, and by continuity at the endpoints as well.
<math>\blacksquare</math>

== Where each hypothesis is used ==

{| class="wikitable"
! hypothesis !! used in !! what fails without it
|-
| <math>E</math> continuous || Step 1 || sign cannot be propagated from a point to a neighbourhood; the conclusion drops to "almost everywhere"
|-
| <math>\eta</math> arbitrary in the class || Step 2 || a fixed or finite family of test functions cannot localise, and cancellation is not excluded
|-
| <math>\eta</math> vanishes at the endpoints || Step 2 || not needed for the lemma, but required for <math>\eta</math> to be an admissible variation
|-
| <math>\eta</math> in <math>C^2</math> || Step 2 || the bracket must be cubed rather than squared to meet the class
|}

The middle row is the load-bearing one. The lemma is a statement about the
''richness'' of the test class, and every proof of it is a construction of a
sufficiently localised bump.

== Variants ==

'''Du Bois-Reymond form.''' If instead

:<math>\int_a^b F(x)\,\eta'(x)\, dx = 0 \qquad\qquad\mathrm{(5)}</math>

for every admissible <math>\eta</math>, with <math>F</math> continuous, then
<math>F</math> is constant. This variant avoids assuming in advance that
<math>L_{y'}</math> is differentiable along the extremal, and it is the honest
route to the Euler-Lagrange equation when the smoothness of <math>y</math> is
not given. It is used on
[[Hamilton's principle/Mathematics/The Weierstrass excess function]], where
extremals with corners are admitted.

'''Higher order.''' If (1) holds with <math>\eta</math> replaced by
<math>\eta^{(k)}</math>, then <math>E</math> is a polynomial of degree less than
<math>k</math>. This is the form needed for integrands containing higher
derivatives.

'''Several dimensions.''' On a bounded domain
<math>\Omega \subset \mathbb{R}^n</math>, if
<math>\int_\Omega E\,\eta \, d^n x = 0</math> for every smooth <math>\eta</math>
of compact support in <math>\Omega</math>, then <math>E \equiv 0</math>. The
proof is the same, with the bump (3) built from
<math>\delta^2 - \Vert x - x_0 \Vert^2</math>. This is the version used on
[[Hamilton's principle/Mathematics/Several independent variables]].

== Remark on generality ==

The lemma is the point at which the calculus of variations makes contact with
distribution theory. In modern language it says that a continuous function which
is zero as a distribution is zero as a function. Nothing in this volume requires
that language, but it explains why the lemma feels like an identity rather than a
theorem: it is the statement that testing against enough smooth functions
determines a continuous function completely.

== Next ==

[[Hamilton's principle/Mathematics/The Euler-Lagrange equation]] applies the
lemma to equation (7) of
[[Hamilton's principle/Mathematics/The first variation]].

== References ==

<references />