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	{{Plebeian\|Hamilton's principle/Explainers/Strong and weak}}		{{Plebeian\|Hamilton's principle/Explainers/Strong and weak}}

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	<references />		<references />

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{{Plebeian|Hamilton's principle/Explainers/Strong and weak}}

Every sufficiency result so far has been for '''weak''' minima — minima against
competitors that stay close in both position and slope. Widening the comparison
class to paths that stay close in position only, with slope unrestricted,
requires one further condition. It is Weierstrass's, and it completes the
classical theory.

== The two comparison classes ==

[[File:OPL-HP-F-13.png|thumb|right|400px|A sawtooth competitor close in position and far in slope.]]

From [[Hamilton's principle/Mathematics/Functionals and the space of paths]], the
weak norm controls position and slope, the strong norm controls position only. A
strong neighbourhood is the larger set, so a strong minimum is the stronger
claim, and every strong minimum is a weak minimum.

The gap is real. A competitor may run alongside the extremal in position while
oscillating steeply in slope — a sawtooth of small amplitude and large
gradient. Such a path is far from the extremal in the weak norm and close in the
strong one, so no weak result says anything about it.

The Legendre and Jacobi conditions of the previous two pages are both derived
from the second variation, which is a small-<math>\varepsilon</math> object. They
cannot see a competitor whose slope differs by <math>O(1)</math>.

== The excess function ==

[[File:OPL-HP-F-34.png|thumb|right|400px|The Weierstrass excess function as a gap above the tangent.]]

'''Definition.''' The '''Weierstrass excess function''' is

:<math>E(x, y, p, q) = L(x, y, q) - L(x, y, p) - (q - p)\, L_{y'}(x, y, p)
\qquad\qquad\mathrm{(1)}</math>

Read (1) as a remainder. <math>L(x,y,p) + (q-p)L_{y'}(x,y,p)</math> is the
tangent line to <math>L</math> in its third slot at <math>p</math>, evaluated at
<math>q</math>. So <math>E</math> is the amount by which <math>L</math> exceeds
its own tangent line.

<math>E \geq 0</math> for all <math>q</math> is therefore exactly the statement
that <math>L</math> is '''convex''' in its slope argument — not merely at
<math>p</math>, which is Legendre's local condition, but across the whole range
of slopes a competitor might use.

The relation between the two is worth setting down. Taylor expansion of (1) in
<math>q</math> about <math>p</math> gives

:<math>E = \tfrac{1}{2}(q - p)^2\, L_{y'y'}(x, y, \bar p)
\qquad\qquad\mathrm{(2)}</math>

for some <math>\bar p</math> between <math>p</math> and <math>q</math>. So
<math>E \geq 0</math> near <math>q = p</math> is Legendre's condition, and
<math>E \geq 0</math> for all <math>q</math> is a genuinely stronger demand:
<math>L_{y'y'} \geq 0</math> must hold not only along the extremal but at every
slope, at each point of a neighbourhood.

== The conditions assembled ==

[[File:OPL-HP-F-42.png|thumb|right|400px|A corner in an extremal.]]

'''Weierstrass's necessary condition.''' If <math>y</math> is a strong local
minimum then

:<math>E\bigl(x,\, y(x),\, y'(x),\, q\bigr) \geq 0
\qquad\qquad\mathrm{(3)}</math>

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

'''Sufficiency.''' If <math>y</math> satisfies the Euler-Lagrange equation, can
be embedded in a field of extremals — which the Jacobi condition of
[[Hamilton's principle/Mathematics/The Jacobi condition and conjugate points]]
guarantees — and <math>E > 0</math> for all <math>q \neq p</math> at every point
of a neighbourhood of the extremal, then <math>y</math> is a strong local minimum.

The necessary form is proved by a '''needle''' variation of a different kind from
Legendre's: the competitor departs from the extremal at one point with a slope
<math>q</math> quite unlike <math>y'</math>, runs briefly, and rejoins. Such a
path is strongly close and weakly far. Computing <math>J</math> along it to first
order in the departure length gives exactly <math>E</math> as the leading change,
so <math>E < 0</math> anywhere produces a cheaper competitor.

== The full hierarchy ==

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

{| class="wikitable"
! condition !! scope !! necessary for !! sufficient with the others for
|-
| Euler-Lagrange || pointwise, 1st order || both || —
|-
| Legendre <math>P \geq 0</math> || pointwise, 2nd order || both || —
|-
| Jacobi, no conjugate point || interval, 2nd order || both || weak minimum
|-
| Weierstrass <math>E \geq 0</math> || pointwise, all slopes || strong only || strong minimum
|}

Four conditions, and only the strengthened combinations in the last column are
sufficient for anything. That is the honest summary of the classical theory, and
it is the reason this volume has taken seven pages to reach a sufficiency
statement.

== Worked example: where the strong claim fails ==

Take

:<math>J[y] = \int_0^1 \bigl( y'^2 - y'^4 \bigr) dx
\qquad\qquad\mathrm{(4)}</math>

with <math>y(0) = y(1) = 0</math>. The extremal <math>y \equiv 0</math> satisfies
the Euler-Lagrange equation, and

:<math>L_{y'y'} = 2 - 12y'^2
\qquad\qquad\mathrm{(5)}</math>

which at <math>y' = 0</math> equals <math>2 > 0</math>. Legendre's strengthened
condition holds. <math>Q = 0</math>, so the accessory equation is
<math>\eta'' = 0</math>, whose solution vanishing at the origin is
<math>\eta = cx</math>, with no second zero. No conjugate point. Jacobi's
condition holds. The extremal is therefore a '''weak''' minimum, and
<math>J[0] = 0</math>.

Now compute the excess function at <math>p = 0</math>:

:<math>E(x, y, 0, q) = (q^2 - q^4) - 0 - q\cdot 0 = q^2(1 - q^2)
\qquad\qquad\mathrm{(6)}</math>

which is '''negative''' for <math>\vert q \vert > 1</math>. Weierstrass's
condition fails.

The failure is realised by an explicit competitor. Let <math>y_n</math> be a
sawtooth of slope <math>\pm 2</math> with <math>n</math> teeth on
<math>[0,1]</math>. Its amplitude is <math>1/n</math>, so
<math>\Vert y_n \Vert_0 \to 0</math> and it is strongly close to the extremal.
But <math>y_n'^2 = 4</math> and <math>y_n'^4 = 16</math> everywhere, so

:<math>J[y_n] = \int_0^1 (4 - 16)\, dx = -12
\qquad\qquad\mathrm{(7)}</math>

independent of <math>n</math>. Every one of these paths beats
<math>J[0] = 0</math>, and they crowd arbitrarily close to the extremal in
position.

So <math>y \equiv 0</math> is a weak minimum and is not a strong minimum. Nothing
in the second-variation theory could have detected this: the sawtooth is not a
small variation in the weak norm, and the whole of
[[Hamilton's principle/Mathematics/The second variation]] is about small weak
variations.

== Corners ==

The strong class admits paths with corners, so the Weierstrass-Erdmann conditions
of
[[Hamilton's principle/Mathematics/Boundary terms and natural boundary conditions]]
apply. The relation to (1) is direct: <math>E \equiv 0</math> along a segment
means <math>L</math> is affine in the slope there, which is the degeneracy that
permits a corner. Strict convexity, <math>E > 0</math>, forbids corners and
recovers the smooth theory.

== What this track has and has not settled ==

O6 closes here. The track has established what a minimum requires and, under
strengthened hypotheses, what guarantees one. It has also produced two explicit
failures — the conjugate-point failure on
[[Hamilton's principle/Mathematics/The Jacobi condition and conjugate points]]
and the strong-class failure above — in each case with an exhibited competitor
rather than an assertion.

Two limits are worth stating plainly. All of this is '''local''': it compares an
extremal with nearby paths and says nothing about a distant competitor. And
existence has been assumed throughout: every theorem here has the form ''if an
extremal is a minimum then'', or ''if these conditions hold then it is a local
minimum''. Whether a minimiser exists at all is the direct method, which this
volume does not treat and which is named as further reading on
[[Hamilton's principle/Sources]].

== Next ==

[[Hamilton's principle/Mathematics/The Legendre transform]] opens the third
mathematics track, on structure.

== References ==

<references />

← Older revision		Revision as of 04:57, 16 September 2026
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	[[File:OPL-HP-F-13.png\|thumb\|right\|400px\|A sawtooth competitor close in position and far in slope.]]		[[File:OPL-HP-F-13.png\|thumb\|right\|400px\|A sawtooth competitor close in position and far in slope.]]

	From [[Hamilton's principle/Mathematics/Functionals and the space of paths]], the		From [[Hamilton's principle/Mathematics/Functionals and the space of paths\|Functionals and the space of paths]], the
	weak norm controls position and slope, the strong norm controls position only. A		weak norm controls position and slope, the strong norm controls position only. A
	strong neighbourhood is the larger set, so a strong minimum is the stronger		strong neighbourhood is the larger set, so a strong minimum is the stronger
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	'''Sufficiency.''' If <math>y</math> satisfies the Euler-Lagrange equation, can		'''Sufficiency.''' If <math>y</math> satisfies the Euler-Lagrange equation, can
	be embedded in a field of extremals — which the Jacobi condition of		be embedded in a field of extremals — which the Jacobi condition of
	[[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]]
	guarantees — and <math>E > 0</math> for all <math>q \neq p</math> at every point		guarantees — and <math>E > 0</math> for all <math>q \neq p</math> at every point
	of a neighbourhood of the extremal, then <math>y</math> is a strong local minimum.		of a neighbourhood of the extremal, then <math>y</math> is a strong local minimum.
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	in the second-variation theory could have detected this: the sawtooth is not a		in the second-variation theory could have detected this: the sawtooth is not a
	small variation in the weak norm, and the whole of		small variation in the weak norm, and the whole of
	[[Hamilton's principle/Mathematics/The second variation]] is about small weak		[[Hamilton's principle/Mathematics/The second variation\|the second variation]] is about small weak
	variations.		variations.

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	The strong class admits paths with corners, so the Weierstrass-Erdmann conditions		The strong class admits paths with corners, so the Weierstrass-Erdmann conditions
	of		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]]
	apply. The relation to (1) is direct: <math>E \equiv 0</math> along a segment		apply. The relation to (1) is direct: <math>E \equiv 0</math> along a segment
	means <math>L</math> is affine in the slope there, which is the degeneracy that		means <math>L</math> is affine in the slope there, which is the degeneracy that
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	strengthened hypotheses, what guarantees one. It has also produced two explicit		strengthened hypotheses, what guarantees one. It has also produced two explicit
	failures — the conjugate-point failure on		failures — the conjugate-point failure 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]]
	and the strong-class failure above — in each case with an exhibited competitor		and the strong-class failure above — in each case with an exhibited competitor
	rather than an assertion.		rather than an assertion.
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	minimum''. Whether a minimiser exists at all is the direct method, which this		minimum''. Whether a minimiser exists at all is the direct method, which this
	volume does not treat and which is named as further reading on		volume does not treat and which is named as further reading on
	[[Hamilton's principle/Sources]].		[[Hamilton's principle/Sources\|Sources]].

	== Next ==		== Next ==

	[[Hamilton's principle/Mathematics/The Legendre transform]] opens the third		[[Hamilton's principle/Mathematics/The Legendre transform\|the Legendre transform]] opens the third
	mathematics track, on structure.		mathematics track, on structure.