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

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	<references />		<references />

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

Every result so far has been necessary and none has been sufficient. The
Euler-Lagrange equation says the first variation vanishes; it says nothing about
whether the extremal is a minimum, a maximum, or neither. The '''second
variation''' is the tool that begins to answer that, and this page is where the
volume's central distinction — stationary against least — acquires machinery
rather than assertion.

== Definition ==

[[File:OPL-HP-F-07.png|thumb|right|400px|The second variation changing sign as the interval lengthens.]]

With <math>\Phi(\varepsilon) = J[y + \varepsilon\eta]</math> as on
[[Hamilton's principle/Mathematics/The first variation]], the '''second
variation''' is

:<math>\delta^2 J[y;\eta] = \Phi''(0)
\qquad\qquad\mathrm{(1)}</math>

Taylor's theorem with the first variation vanishing gives

:<math>J[y + \varepsilon\eta] - J[y]
= \tfrac{1}{2}\varepsilon^2\, \delta^2 J[y;\eta] + o(\varepsilon^2)
\qquad\qquad\mathrm{(2)}</math>

So the sign of <math>\delta^2 J</math> controls the comparison, subject to the
error term — and that qualification is where the difficulty lives.

== Computing it ==

[[File:OPL-HP-F-17.png|thumb|right|400px|Local penalty against accumulated reward.]]

Differentiate under the integral twice and set <math>\varepsilon = 0</math>:

:<math>\delta^2 J[y;\eta] = \int_a^b \left[
L_{yy}\,\eta^2
+ 2L_{yy'}\,\eta\eta'
+ L_{y'y'}\,\eta'^2 \right] dx
\qquad\qquad\mathrm{(3)}</math>

with all second partials evaluated along the extremal. This is a quadratic
functional of <math>\eta</math> — the analogue of the Hessian in finite
dimensions.

The cross term can be removed. Since <math>2\eta\eta' = (\eta^2)'</math>,
integrate the middle term by parts; the boundary term vanishes because
<math>\eta</math> does, and

:<math>\int_a^b 2L_{yy'}\,\eta\eta'\, dx
= -\int_a^b \frac{d}{dx}\bigl( L_{yy'} \bigr)\,\eta^2\, dx
\qquad\qquad\mathrm{(4)}</math>

Substituting into (3) gives the '''Legendre form'''

:<math>\delta^2 J[y;\eta] = \int_a^b \left[ P\,\eta'^2 + Q\,\eta^2 \right] dx
\qquad\qquad\mathrm{(5)}</math>

with

:<math>P = L_{y'y'},
\qquad
Q = L_{yy} - \frac{d}{dx} L_{yy'}
\qquad\qquad\mathrm{(6)}</math>

Two terms, two coefficients, both determined by the extremal.

== Necessary conditions, and the trap ==

If <math>y</math> is a weak local minimum then <math>\Phi</math> has a minimum at
<math>\varepsilon = 0</math> for every <math>\eta</math>, so

:<math>\delta^2 J[y;\eta] \geq 0 \quad \text{for all } \eta \in \mathcal{V}
\qquad\qquad\mathrm{(7)}</math>

This is necessary. The temptation is to reverse it and conclude that
<math>\delta^2 J > 0</math> for all <math>\eta \neq 0</math> makes
<math>y</math> a minimum, by analogy with the finite-dimensional second
derivative test.

'''That inference is false in infinite dimensions.''' The reason is the error
term in (2). In finite dimensions a positive definite Hessian is bounded below by
<math>c\Vert v\Vert^2</math> with <math>c > 0</math>, which dominates the
remainder. On a function space, positivity of a quadratic functional does not
give such a bound: the infimum of <math>\delta^2 J[y;\eta]</math> over
<math>\Vert\eta\Vert = 1</math> can be zero without being attained. The
remainder is then not dominated and no conclusion follows.

Repairing this is the whole content of the sufficiency theory. It needs a
'''strengthened''' positivity, and that is what the Legendre and Jacobi
conditions supply.

== What the two coefficients contribute ==

<math>P</math> and <math>Q</math> enter (5) very differently, and seeing why
motivates the next two pages.

'''<math>P</math> is local and dominant.''' <math>\eta'</math> can be made
enormous while <math>\eta</math> stays small — a rapid, low-amplitude wiggle. Any
such <math>\eta</math> makes the <math>P\eta'^2</math> term overwhelm the
<math>Q\eta^2</math> term. So if <math>P < 0</math> anywhere, a wiggle localised
there drives <math>\delta^2 J</math> negative and (7) fails. This yields
<math>P \geq 0</math> as a necessary condition, derived properly on
[[Hamilton's principle/Mathematics/The Legendre condition]].

'''<math>Q</math> is global and accumulative.''' With <math>P > 0</math> the
first term is a penalty on bending and the second may be a reward, if
<math>Q < 0</math>. Over a short interval the penalty wins. Over a long enough
interval the accumulated reward can win, and <math>\delta^2 J</math> turns
negative even though <math>P > 0</math> throughout. The length at which this
happens is set by conjugate points, on
[[Hamilton's principle/Mathematics/The Jacobi condition and conjugate points]].

'''Interval length is therefore part of the answer.''' The same extremal can be a
minimum on <math>[a,b]</math> and fail to be one on <math>[a,b']</math> with
<math>b' > b</math>, with no change to <math>L</math> and no change to the
equation it satisfies. Nothing in the Euler-Lagrange equation sees this.

== The accessory problem ==

Equation (5) is itself a variational problem, in the unknown <math>\eta</math>.
Its Euler-Lagrange equation is

:<math>\frac{d}{dx}\bigl( P\,\eta' \bigr) - Q\,\eta = 0
\qquad\qquad\mathrm{(8)}</math>

called the '''Jacobi accessory equation'''. Analysing the second variation thus
means solving a linear second-order equation derived from the original extremal —
a striking reduction, and the reason the sufficiency theory is tractable at all.

Solutions of (8) are exactly the directions in which
<math>\delta^2 J</math> is stationary, and its zeros are the conjugate points.

== A first sight of the failure ==

Take <math>L = \tfrac{1}{2}(y'^2 - k^2 y^2)</math> on <math>[0,b]</math> with
<math>y \equiv 0</math> as the extremal, endpoints zero. Then
<math>P = 1</math> and <math>Q = -k^2</math>, so

:<math>\delta^2 J[\,0\,;\eta] = \int_0^b \bigl( \eta'^2 - k^2\eta^2 \bigr) dx
\qquad\qquad\mathrm{(9)}</math>

Test with <math>\eta = \sin(\pi x/b)</math>, which vanishes at both ends. Using
<math>\int_0^b \sin^2(\pi x/b)\,dx = \int_0^b \cos^2(\pi x/b)\,dx = b/2</math>,

:<math>\delta^2 J = \frac{b}{2}\left( \frac{\pi^2}{b^2} - k^2 \right)
\qquad\qquad\mathrm{(10)}</math>

This is positive for <math>b < \pi/k</math> and '''negative''' for
<math>b > \pi/k</math>. The extremal, the integrand and the Legendre coefficient
<math>P = 1 > 0</math> are identical in both cases. Only the interval length
changed.

At <math>b = \pi/k</math> exactly, (10) vanishes for this <math>\eta</math>: the
second variation is degenerate, and <math>x = \pi/k</math> is the point conjugate
to the origin. Beyond it the zero solution is stationary and is not a minimum.

Equation (10) is the volume's first exhibited demonstration that stationary does
not imply least. It is a computation, not an assertion, and the geometric version
of the same phenomenon — a great-circle arc past its antipode — is worked on
[[Hamilton's principle/Mathematics/The Jacobi condition and conjugate points]].

== Second variation with constraints ==

For constrained problems the second variation is that of the augmented integrand
of [[Hamilton's principle/Mathematics/Lagrange multipliers]], evaluated with the
multipliers at their solution values, and it is tested only against variations
satisfying the linearised constraint. Both restrictions matter; dropping either
gives the wrong sign test.

== Next ==

[[Hamilton's principle/Mathematics/The Legendre condition]] derives the local
condition <math>P \geq 0</math> and shows exactly how much it does and does not
buy.

== References ==

<references />

← Older revision		Revision as of 04:57, 16 September 2026
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	With <math>\Phi(\varepsilon) = J[y + \varepsilon\eta]</math> as on		With <math>\Phi(\varepsilon) = J[y + \varepsilon\eta]</math> as on
	[[Hamilton's principle/Mathematics/The first variation]], the '''second		[[Hamilton's principle/Mathematics/The first variation\|the first variation]], the '''second
	variation''' is		variation''' is

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	there drives <math>\delta^2 J</math> negative and (7) fails. This yields		there drives <math>\delta^2 J</math> negative and (7) fails. This yields
	<math>P \geq 0</math> as a necessary condition, derived properly on		<math>P \geq 0</math> as a necessary condition, derived properly on
	[[Hamilton's principle/Mathematics/The Legendre condition]].		[[Hamilton's principle/Mathematics/The Legendre condition\|the Legendre condition]].

	'''<math>Q</math> is global and accumulative.''' With <math>P > 0</math> the		'''<math>Q</math> is global and accumulative.''' With <math>P > 0</math> the
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	negative even though <math>P > 0</math> throughout. The length at which this		negative even though <math>P > 0</math> throughout. The length at which this
	happens is set by conjugate points, on		happens is set by conjugate points, 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]].

	'''Interval length is therefore part of the answer.''' The same extremal can be a		'''Interval length is therefore part of the answer.''' The same extremal can be a
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	not imply least. It is a computation, not an assertion, and the geometric version		not imply least. It is a computation, not an assertion, and the geometric version
	of the same phenomenon — a great-circle arc past its antipode — is worked on		of the same phenomenon — a great-circle arc past its antipode — is worked 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]].

	== Second variation with constraints ==		== Second variation with constraints ==

	For constrained problems the second variation is that of the augmented integrand		For constrained problems the second variation is that of the augmented integrand
	of [[Hamilton's principle/Mathematics/Lagrange multipliers]], evaluated with the		of [[Hamilton's principle/Mathematics/Lagrange multipliers\|Lagrange multipliers]], evaluated with the
	multipliers at their solution values, and it is tested only against variations		multipliers at their solution values, and it is tested only against variations
	satisfying the linearised constraint. Both restrictions matter; dropping either		satisfying the linearised constraint. Both restrictions matter; dropping either
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	== Next ==		== Next ==

	[[Hamilton's principle/Mathematics/The Legendre condition]] derives the local		[[Hamilton's principle/Mathematics/The Legendre condition\|the Legendre condition]] derives the local
	condition <math>P \geq 0</math> and shows exactly how much it does and does not		condition <math>P \geq 0</math> and shows exactly how much it does and does not
	buy.		buy.