View source for Hamilton's principle/Mathematics/The Jacobi condition and conjugate points

{{DISPLAYTITLE:The Jacobi condition and conjugate points}}
{{Plebeian|Hamilton's principle/Explainers/Where the minimum fails}}

The '''Jacobi condition''' is the global second-order condition that Legendre's
local condition cannot supply. It says that an extremal ceases to be a minimum
once it extends past the first point '''conjugate''' to its start. This page
carries the volume's worked counterexample to the identification of stationary
with least, and it is the reason this volume is not titled with the word
''least''.

== The accessory equation ==

[[File:OPL-HP-F-05.png|thumb|right|400px|Geodesics from a pole refocusing at the antipode.]]

From [[Hamilton's principle/Mathematics/The second variation|the second variation]], the second
variation in Legendre form is
<math>\int_a^b (P\eta'^2 + Q\eta^2)\,dx</math>, and its own Euler-Lagrange
equation is the '''Jacobi accessory equation'''

:<math>\frac{d}{dx}\bigl( P(x)\,\eta' \bigr) - Q(x)\,\eta = 0
\qquad\qquad\mathrm{(1)}</math>

a linear, homogeneous, second-order ordinary differential equation whose
coefficients are fixed by the extremal under test. Assume throughout that the
strengthened Legendre condition <math>P > 0</math> holds, so (1) is genuinely
second order.

'''Definition.''' A point <math>c \in (a,b]</math> is '''conjugate''' to
<math>a</math> if some solution <math>u</math> of (1), not identically zero,
satisfies <math>u(a) = 0</math> and <math>u(c) = 0</math>.

Since (1) is linear and second order, the solution with <math>u(a) = 0</math> is
unique up to scale, so conjugacy is a property of the extremal and not of a
choice.

== Why zeros of the accessory equation matter ==

[[File:OPL-HP-F-22.png|thumb|right|400px|Geodesic behaviour on flat, spherical and saddle surfaces.]]

Suppose <math>c</math> is conjugate to <math>a</math> with
<math>a < c < b</math>, and let <math>u</math> be the corresponding solution.
Define

:<math>\eta^*(x) = \begin{cases}
u(x), & a \leq x \leq c \\
0, & c < x \leq b
\end{cases} \qquad\qquad\mathrm{(2)}</math>

<math>\eta^*</math> is continuous, vanishes at both ends, and is admissible in
the piecewise-smooth class. Evaluate the second variation on it. On
<math>[c,b]</math> it contributes nothing. On <math>[a,c]</math>, integrate by
parts:

:<math>\int_a^c \bigl( Pu'^2 + Qu^2 \bigr) dx
= \Bigl[ P u u' \Bigr]_a^c
- \int_a^c u\left[ \frac{d}{dx}(Pu') - Qu \right] dx
\qquad\qquad\mathrm{(3)}</math>

The integral on the right vanishes because <math>u</math> solves (1). The
boundary term vanishes because <math>u(a) = u(c) = 0</math>. Therefore

:<math>\delta^2 J[y;\eta^*] = 0
\qquad\qquad\mathrm{(4)}</math>

The second variation is '''degenerate''': a nonzero admissible variation makes it
vanish. The extremal sits on the boundary between the positive and negative
cases, and it can be shown that the corner at <math>c</math> can be rounded off
to give a smooth variation with <math>\delta^2 J < 0</math> strictly. So beyond
the conjugate point the second variation takes negative values and the extremal
is not a minimum.

'''Theorem (Jacobi's necessary condition).''' If <math>y</math> is a weak local
minimum and <math>P > 0</math>, then no point of the open interval
<math>(a,b)</math> is conjugate to <math>a</math>.

'''Theorem (sufficiency).''' If <math>y</math> satisfies the Euler-Lagrange
equation, <math>P > 0</math> on <math>[a,b]</math>, and no point of
<math>(a,b]</math> is conjugate to <math>a</math>, then <math>y</math> is a weak
local minimum.

The two differ only at the closed endpoint, and that half-open gap at
<math>b</math> itself is genuinely undecided by these conditions alone.

== The analytic example, revisited ==

[[File:OPL-HP-F-50.png|thumb|right|400px|Rays crossing a focus.]]

Take <math>L = \tfrac{1}{2}(y'^2 - k^2y^2)</math> and the extremal
<math>y \equiv 0</math>, as on
[[Hamilton's principle/Mathematics/The second variation|the second variation]]. Then
<math>P = 1</math>, <math>Q = -k^2</math>, and (1) becomes

:<math>\eta'' + k^2\eta = 0
\qquad\qquad\mathrm{(5)}</math>

with solution <math>u = \sin(kx)</math> for <math>u(0) = 0</math>. Its next zero
is at <math>x = \pi/k</math>. So the first conjugate point is
<math>c = \pi/k</math>, which is '''exactly''' the interval length at which the
direct computation gave <math>\delta^2 J = 0</math>.

The two calculations agree, and they were done by different routes: one by
testing a chosen function, the other by solving a differential equation. The
agreement is the check that the accessory machinery describes the phenomenon
rather than merely accompanying it.

== The geometric counterexample: geodesics on a sphere ==

Take the unit sphere. Its geodesics are the great circles, and every great-circle
arc satisfies the geodesic equation — every one is stationary for arc length. The
question is which of them are shortest.

Let <math>\theta</math> be arc length along a great circle from a starting point
<math>A</math>. The accessory equation for arc length on a unit sphere is the
Jacobi equation of a space of curvature <math>K = 1</math>,

:<math>u'' + K u = 0
\qquad \Longrightarrow \qquad
u'' + u = 0
\qquad\qquad\mathrm{(6)}</math>

so <math>u = \sin\theta</math> for <math>u(0) = 0</math>, and the first conjugate
point is at <math>\theta = \pi</math> — the '''antipode''' of <math>A</math>.

Read off the three regimes:

{| class="wikitable"
! arc length !! conjugate point passed !! status of the arc
|-
| <math>\theta < \pi</math> || no || stationary '''and''' the unique shortest path
|-
| <math>\theta = \pi</math> || at the endpoint || stationary; infinitely many arcs of equal length
|-
| <math>\theta > \pi</math> || yes || stationary and '''not''' shortest
|}

The middle row is where the degeneracy is visible without any calculation. From
the north pole to the south pole every meridian has length <math>\pi</math>. All
of them are stationary, all have the same length, and the solution is not unique.
That non-uniqueness is the geometric face of equation (4).

The bottom row is the counterexample proper. Take an arc running from
<math>A</math> the long way round to a point <math>B</math> just short of a full
revolution, of length <math>\theta = 2\pi - \varepsilon</math>. It satisfies the
geodesic equation at every point. It is stationary. And it is obviously not the
shortest route from <math>A</math> to <math>B</math>: the short arc, of length
<math>\varepsilon</math>, joins the same two points along the same great circle.

'''Nothing local distinguishes the two arcs.''' They satisfy the same equation,
they have the same <math>P = 1 > 0</math>, and no pointwise test can tell them
apart. Only the interval length differs, and only the Jacobi condition sees it.

This is the worked counterexample that
[[Hamilton's principle/Mathematics/The second variation|the second variation]] promised and that the
volume's claim rests on. Stationary means the first variation vanishes. It does
not mean least, and the sphere shows the difference without any appeal to exotic
integrands.

== Geometric reading ==

The accessory equation <math>u'' + Ku = 0</math> is the equation of geodesic
deviation: <math>u</math> measures the separation of two neighbouring geodesics
leaving <math>A</math> in slightly different directions. Positive curvature makes
them converge, and a conjugate point is where they meet again.

That gives the general picture. Conjugate points are '''refocusing''' points of
the family of extremals through <math>A</math>. Before the first one, the
extremal is embedded in a field of non-crossing neighbouring extremals and is
minimising. At it, the field degenerates. Past it, extremals cross and
minimality fails.

Three consequences follow, and each recurs later in the volume:

* '''Flat and negatively curved spaces have no conjugate points.''' <math>u'' - \vert K\vert u = 0</math> has no second zero, so geodesics minimise indefinitely. Straight lines in the plane are the case worked on [[Hamilton's principle/Mathematics/The Euler-Lagrange equation|the Euler-Lagrange equation]], and the minimality claim deferred there is now discharged.
* '''In optics, conjugate points are foci.''' [[Hamilton's principle/Applications/The optical-mechanical analogy|The optical-mechanical analogy]] makes this literal rather than metaphorical: a lens image point is a conjugate point of the object point.
* '''In mechanics, past a conjugate point the action is stationary but not minimal.''' This is why the classical action in [[Hamilton's principle/Applications/The Feynman path integral|the Feynman path integral]] carries a phase correction counting the conjugate points crossed.

== Relation to the field of extremals ==

The sufficiency proof works by embedding the extremal in a field: a family of
extremals covering a neighbourhood, exactly one through each point. Such a field
exists precisely up to the first conjugate point, which is the analytic content
of the geometric picture above. The field construction is also what
[[Hamilton's principle/Mathematics/The Weierstrass excess function|the Weierstrass excess function]] needs, and
what connects this material to [[Hamilton's principle/Mathematics/The Hamilton-Jacobi equation|the Hamilton-Jacobi equation]], where the
action as a function of endpoints becomes singular at conjugate points.

== Next ==

[[Hamilton's principle/Mathematics/The Weierstrass excess function|the Weierstrass excess function]] closes the
track by treating strong minima, where the comparison class admits paths with
wildly different slopes.

== References ==

<references />