Hamilton's principle/Mathematics/The Legendre condition - Revision history

BFP Rauch: Pipe internal links so cross-references read as titles, not paths

2026-09-16T04:57:48Z

Pipe internal links so cross-references read as titles, not paths

BFP Rauch: Add DISPLAYTITLE: title the page by its section rather than its path

2026-09-16T04:46:36Z

Add DISPLAYTITLE: title the page by its section rather than its path

← Older revision		Revision as of 04:46, 16 September 2026
Line 1:		Line 1:
			{{DISPLAYTITLE:The Legendre condition}}
	{{Plebeian\|Hamilton's principle/Explainers/Is it really a minimum}}		{{Plebeian\|Hamilton's principle/Explainers/Is it really a minimum}}

Line 172:		Line 173:

	<references />		<references />

BFP Rauch: 1 revision imported

2026-09-09T05:05:14Z

1 revision imported

← Older revision	Revision as of 05:05, 9 September 2026
(No difference)

hamiltons-principle>HPVBuild: Initial import of M2-05

2026-09-08T12:00:00Z

Initial import of M2-05

New page

{{Plebeian|Hamilton's principle/Explainers/Is it really a minimum}}

The '''Legendre condition''' is the local necessary condition for a minimum:
the coefficient of <math>\eta'^2</math> in the second variation must not be
negative anywhere on the extremal. It is cheap to check and it is the first thing
to check. It is also decisively insufficient, and this page is as much about the
gap it leaves as about the condition itself.

== Statement ==

[[File:OPL-HP-F-16.png|thumb|right|400px|A needle variation.]]

'''Theorem (Legendre's necessary condition).''' If <math>y</math> is a weak local
minimum of <math>J</math>, then

:<math>P(x) = L_{y'y'}\bigl(x, y(x), y'(x)\bigr) \geq 0
\qquad\qquad\mathrm{(1)}</math>

for every <math>x \in [a,b]</math>. For a maximum the inequality reverses.

The '''strengthened Legendre condition''' is the strict form
<math>P(x) > 0</math> on <math>[a,b]</math>. It is the hypothesis every
sufficiency theorem in this track assumes.

== Derivation ==

[[File:OPL-HP-F-49.png|thumb|right|400px|The four conditions of the sufficiency theory.]]

Suppose <math>P(x_0) < 0</math> at some interior <math>x_0</math>. Build a
variation that makes the second variation negative.

By continuity there is <math>\delta > 0</math> with
<math>P(x) < \tfrac{1}{2}P(x_0) < 0</math> on
<math>I = (x_0 - \delta,\, x_0 + \delta) \subset (a,b)</math>. For
<math>0 < h \leq \delta</math> define the '''needle''' variation

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

This is admissible: it is <math>C^2</math>, vanishes outside a subinterval of
<math>(a,b)</math>, and therefore vanishes at both endpoints.

The scaling is the point of the construction. Direct integration gives

:<math>\int \eta_h^2\, dx = O(h^{-1}),
\qquad
\int \eta_h'^2\, dx = O(h^{-3})
\qquad\qquad\mathrm{(3)}</math>

so as <math>h \to 0</math> the derivative term grows two powers of
<math>h</math> faster than the amplitude term. In the Legendre form of
[[Hamilton's principle/Mathematics/The second variation]],

:<math>\delta^2 J[y;\eta_h] = \int_I \left[ P\,\eta_h'^2 + Q\,\eta_h^2 \right] dx
\qquad\qquad\mathrm{(4)}</math>

the first integral is bounded above by
<math>\tfrac{1}{2}P(x_0)\int\eta_h'^2\,dx</math>, which is negative and of order
<math>h^{-3}</math>. The second is bounded in magnitude by
<math>\max\vert Q\vert \int \eta_h^2\, dx</math>, of order <math>h^{-1}</math>.
For <math>h</math> small enough the first dominates and

:<math>\delta^2 J[y;\eta_h] < 0
\qquad\qquad\mathrm{(5)}</math>

contradicting the necessary condition (7) of
[[Hamilton's principle/Mathematics/The second variation]]. Hence
<math>P \geq 0</math> on the interior, and at the endpoints by continuity.
<math>\blacksquare</math>

'''The mechanism in one sentence:''' a narrow, tall-sloped wiggle changes the
path very little but its derivative enormously, so wherever <math>P</math> is
negative there is a variation that lowers <math>J</math>.

== Reading the condition ==

<math>L_{y'y'} \geq 0</math> says <math>L</math> is convex in its
<math>y'</math> slot along the extremal. Convexity in the velocity is exactly
what makes going fast expensive, and a functional that rewards high speed cannot
have a minimum: the wiggle exploits it.

Three connected facts, all consequences of the same coefficient:

{| class="wikitable"
! <math>P = L_{y'y'}</math> !! consequence !! where treated
|-
| <math>P < 0</math> somewhere || no minimum || this page
|-
| <math>P = 0</math> somewhere || Euler-Lagrange degenerates; corners possible || [[Hamilton's principle/Mathematics/The Euler-Lagrange equation]]
|-
| <math>P > 0</math> throughout || equation is genuinely second order; corners excluded || [[Hamilton's principle/Mathematics/Boundary terms and natural boundary conditions]]
|}

The middle row connects to the Weierstrass-Erdmann corner conditions: a corner
requires <math>P = 0</math>, so the strengthened condition rules corners out. In
several variables <math>P</math> becomes the Hessian matrix
<math>W_{ik} = \partial^2 L / \partial y_i' \partial y_k'</math> of
[[Hamilton's principle/Mathematics/Several dependent variables]], and (1) becomes
positive semidefiniteness of <math>W</math>.

== What it does not buy ==

The strengthened condition <math>P > 0</math> is '''not''' sufficient, and the
counterexample is already on the record. On
[[Hamilton's principle/Mathematics/The second variation]] the integrand
<math>L = \tfrac{1}{2}(y'^2 - k^2y^2)</math> has <math>P = 1 > 0</math>
everywhere, and yet

:<math>\delta^2 J = \frac{b}{2}\left( \frac{\pi^2}{b^2} - k^2 \right)
\qquad\qquad\mathrm{(6)}</math>

is negative for <math>b > \pi/k</math>. Legendre's condition holds, uniformly and
strictly, on an interval where the extremal is not a minimum.

The reason is visible in the derivation. The needle argument is '''local''': it
probes one point at a time and therefore detects only pointwise failures. When
<math>Q < 0</math>, the reward accumulates over the interval and can defeat the
<math>P</math> penalty using a variation spread over the whole domain — a
variation the needle construction never builds.

So the condition is blind in exactly one direction, and it is a direction that
matters. Detecting the global failure requires the accessory equation, which is
[[Hamilton's principle/Mathematics/The Jacobi condition and conjugate points]].

== A historical caution ==

Legendre proposed in 1786 that the strengthened condition ''was'' sufficient.
Lagrange identified the error. The gap stood until Jacobi supplied the missing
condition in 1837, and Weierstrass completed the picture for strong minima later
in the century.

The mistake is instructive rather than embarrassing: it is precisely the
finite-dimensional analogy, in which a positive definite Hessian does settle the
question. The failure of that analogy on a function space is the reason the
sufficiency theory has three named conditions instead of one.

<blockquote>
Provenance: the attributions in this section are editorial summary. No primary
source for Legendre 1786 or Jacobi 1837 has been verified for this volume; the
Jacobi row of the survey on [[Hamilton's principle/Sources]] is UNCOMPUTED. No
equation on this page is attributed to either.
</blockquote>

== Where the three conditions stand ==

{| class="wikitable"
! condition !! type !! status
|-
| Euler-Lagrange || local, first order || necessary
|-
| Legendre, <math>P \geq 0</math> || local, second order || necessary
|-
| Jacobi, no conjugate point in <math>(a,b)</math> || global, second order || necessary
|-
| all three, strengthened || || sufficient for a '''weak''' minimum
|-
| Weierstrass excess function || strong || needed for a '''strong''' minimum
|}

Only the fourth line is a sufficiency statement, and it is confined to weak
minima. The last line is
[[Hamilton's principle/Mathematics/The Weierstrass excess function]].

== Next ==

[[Hamilton's principle/Mathematics/The Jacobi condition and conjugate points]]
supplies the global condition and the geometric counterexample.

== References ==

<references />

← Older revision		Revision as of 04:57, 16 September 2026
Line 53:		Line 53:
	so as <math>h \to 0</math> the derivative term grows two powers of		so as <math>h \to 0</math> the derivative term grows two powers of
	<math>h</math> faster than the amplitude term. In the Legendre form of		<math>h</math> faster than the amplitude term. In the Legendre form of
	[[Hamilton's principle/Mathematics/The second variation]],		[[Hamilton's principle/Mathematics/The second variation\|the second variation]],

	:<math>\delta^2 J[y;\eta_h] = \int_I \left[ P\,\eta_h'^2 + Q\,\eta_h^2 \right] dx		:<math>\delta^2 J[y;\eta_h] = \int_I \left[ P\,\eta_h'^2 + Q\,\eta_h^2 \right] dx
Line 68:		Line 68:

	contradicting the necessary condition (7) of		contradicting the necessary condition (7) of
	[[Hamilton's principle/Mathematics/The second variation]]. Hence		[[Hamilton's principle/Mathematics/The second variation\|the second variation]]. Hence
	<math>P \geq 0</math> on the interior, and at the endpoints by continuity.		<math>P \geq 0</math> on the interior, and at the endpoints by continuity.
	<math>\blacksquare</math>		<math>\blacksquare</math>
Line 90:		Line 90:
	\| <math>P < 0</math> somewhere \|\| no minimum \|\| this page		\| <math>P < 0</math> somewhere \|\| no minimum \|\| this page
	\|-		\|-
	\| <math>P = 0</math> somewhere \|\| Euler-Lagrange degenerates; corners possible \|\| [[Hamilton's principle/Mathematics/The Euler-Lagrange equation]]		\| <math>P = 0</math> somewhere \|\| Euler-Lagrange degenerates; corners possible \|\| [[Hamilton's principle/Mathematics/The Euler-Lagrange equation\|the Euler-Lagrange equation]]
	\|-		\|-
	\| <math>P > 0</math> throughout \|\| equation is genuinely second order; corners excluded \|\| [[Hamilton's principle/Mathematics/Boundary terms and natural boundary conditions]]		\| <math>P > 0</math> throughout \|\| equation is genuinely second order; corners excluded \|\| [[Hamilton's principle/Mathematics/Boundary terms and natural boundary conditions\|Boundary terms and natural boundary conditions]]
	\|}		\|}

Line 99:		Line 99:
	several variables <math>P</math> becomes the Hessian matrix		several variables <math>P</math> becomes the Hessian matrix
	<math>W_{ik} = \partial^2 L / \partial y_i' \partial y_k'</math> of		<math>W_{ik} = \partial^2 L / \partial y_i' \partial y_k'</math> of
	[[Hamilton's principle/Mathematics/Several dependent variables]], and (1) becomes		[[Hamilton's principle/Mathematics/Several dependent variables\|Several dependent variables]], and (1) becomes
	positive semidefiniteness of <math>W</math>.		positive semidefiniteness of <math>W</math>.

Line 106:		Line 106:
	The strengthened condition <math>P > 0</math> is '''not''' sufficient, and the		The strengthened condition <math>P > 0</math> is '''not''' sufficient, and the
	counterexample is already on the record. On		counterexample is already on the record. On
	[[Hamilton's principle/Mathematics/The second variation]] the integrand		[[Hamilton's principle/Mathematics/The second variation\|the second variation]] the integrand
	<math>L = \tfrac{1}{2}(y'^2 - k^2y^2)</math> has <math>P = 1 > 0</math>		<math>L = \tfrac{1}{2}(y'^2 - k^2y^2)</math> has <math>P = 1 > 0</math>
	everywhere, and yet		everywhere, and yet
Line 124:		Line 124:
	So the condition is blind in exactly one direction, and it is a direction that		So the condition is blind in exactly one direction, and it is a direction that
	matters. Detecting the global failure requires the accessory equation, which is		matters. Detecting the global failure requires the accessory equation, which is
	[[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]].

	== A historical caution ==		== A historical caution ==
Line 141:		Line 141:
	Provenance: the attributions in this section are editorial summary. No primary		Provenance: the attributions in this section are editorial summary. No primary
	source for Legendre 1786 or Jacobi 1837 has been verified for this volume; the		source for Legendre 1786 or Jacobi 1837 has been verified for this volume; the
	Jacobi row of the survey on [[Hamilton's principle/Sources]] is UNCOMPUTED. No		Jacobi row of the survey on [[Hamilton's principle/Sources\|Sources]] is UNCOMPUTED. No
	equation on this page is attributed to either.		equation on this page is attributed to either.
	</blockquote>		</blockquote>
Line 163:		Line 163:
	Only the fourth line is a sufficiency statement, and it is confined to weak		Only the fourth line is a sufficiency statement, and it is confined to weak
	minima. The last line is		minima. The last line is
	[[Hamilton's principle/Mathematics/The Weierstrass excess function]].		[[Hamilton's principle/Mathematics/The Weierstrass excess function\|the Weierstrass excess function]].

	== Next ==		== Next ==

	[[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]]
	supplies the global condition and the geometric counterexample.		supplies the global condition and the geometric counterexample.