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	{{Technical\|Hamilton's principle/Mathematics/The Jacobi condition and conjugate points}}		{{Technical\|Hamilton's principle/Mathematics/The Jacobi condition and conjugate points}}
			{{Hamilton's principle map}}

	This page explains where a stationary path stops being the best one. It uses no		This page explains where a stationary path stops being the best one. It uses no

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	Jacobi found this in 1837, and it is the missing test that		Jacobi found this in 1837, and it is the missing test that
	[[Hamilton's principle/Explainers/Is it really a minimum]] was looking for. It is		[[Hamilton's principle/Explainers/Is it really a minimum\|Is it really a minimum]] was looking for. It is
	not a local test. It asks whether the path has run far enough for neighbouring		not a local test. It asks whether the path has run far enough for neighbouring
	paths to refocus, and that is a question about the whole journey.		paths to refocus, and that is a question about the whole journey.
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	[[Hamilton's principle/Explainers/Strong and weak]].		[[Hamilton's principle/Explainers/Strong and weak\|Strong and weak]].

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{{Technical|Hamilton's principle/Mathematics/The Jacobi condition and conjugate points}}

This page explains where a stationary path stops being the best one. It uses no
mathematics. The technical page it goes with is named above.

== Paths that meet again ==

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

Stand at the North Pole and set off south, in a straight line, along the surface.
Have a friend do the same, in a different direction.

You start together. You separate. You keep separating for a while.

Then you start converging, and at the South Pole you meet again.

Neither of you turned. Both of you went as straight as the surface allows. The
curvature of the Earth brought you back together.

That is the whole idea of this page. On a curved surface, paths that set off in
slightly different directions can '''refocus'''. The place where they meet again
has a name: a '''conjugate point'''.

== What happens at the meeting point ==

[[File:OPL-HP-F-14.png|thumb|right|400px|Two stationary light paths around a mass.]]

Something breaks there, and it is easy to see what.

From the North Pole to the South Pole, '''every''' meridian is a valid straight
path, and they all have the same length.

So there is no unique shortest route. There are infinitely many, all tied.

That is not a normal situation. Usually the shortest route between two places is
one specific route. Here the answer has become ambiguous, and the ambiguity
arrives exactly at the refocusing point.

== Past the meeting point ==

Now go further. Set off from the North Pole, pass the South Pole, and keep going
until you are nearly back where you started.

This route is still perfectly straight. At every single point it obeys the same
rule as any other straight path on the sphere. Nothing you could measure along the
way would reveal anything unusual.

And it is obviously '''not''' the shortest way to get where you ended up. The
shortest way was to take a few steps in the other direction.

'''That is a stationary path which is not a minimum, and it needs no mathematics
to see.'''

== The rule ==

Putting it together:

'''Before the first conjugate point''' the path is the unique shortest route.

'''At the conjugate point''' the shortest route stops being unique.

'''After it''' the path is still stationary and is no longer shortest.

Jacobi found this in 1837, and it is the missing test that
[[Hamilton's principle/Explainers/Is it really a minimum]] was looking for. It is
not a local test. It asks whether the path has run far enough for neighbouring
paths to refocus, and that is a question about the whole journey.

== Which surfaces do this ==

Refocusing needs curvature of a particular sign.

'''On a flat surface''' paths that diverge keep diverging. There is no conjugate
point, and straight lines are shortest however far you extend them. A ruler on a
table is always right.

'''On a sphere''' paths converge. Conjugate points exist, at the antipode, and
beyond them minimality fails.

'''On a saddle-shaped surface''' paths diverge even faster than on a flat one.
Again no conjugate point, and again straightest is always shortest.

So refocusing is a symptom of curving '''towards''' yourself. A surface that
curves away never does it.

== The same thing in a lens ==

The idea is not confined to geometry. It appears wherever paths can be bent, and
the most familiar case is optical.

A lens takes light spreading out from a point and brings it back together at
another point. That is refocusing, and the meeting place is called a '''focus'''.

A focus and a conjugate point are the same thing. The vocabulary differs because
the subjects grew up separately.

This makes the idea physical rather than abstract. A conjugate point is not a
technical condition inside a proof. It is the place where the image forms.

== Astronomers photograph it ==

There is a case where the whole phenomenon has been photographed.

A galaxy is massive enough to bend light passing near it. Light from a more
distant object behind it can therefore reach us by '''more than one route''',
bending round different sides.

We see several images of the same object.

Each route is a valid stationary path. Only one of them is the shortest — the
first to arrive. The others are stationary and are not shortest, and they are
visible in the photograph.

This is the volume's central point, taken with a telescope. The extra images exist
because light past a conjugate point still travels a legitimate path, even though
that path is not the quickest.

== What the meeting point costs you ==

One more consequence, which matters for the later parts of the volume.

Before the conjugate point, one path connects your start and finish. Afterwards,
several may.

Anything that assumed a single unique path stops working there. Descriptions built
on "the" path become ambiguous, and calculations that divide by the spreading of
nearby paths break down, because the spreading has gone to zero.

In optics this is why a focus is bright — a great deal of light arrives at one
place — and why the simple ray picture is worst exactly there.

Conjugate points are therefore the trouble spots of the entire subject. They are
where minimality fails, where uniqueness fails, where the approximations fail, and
where the interesting physics happens.

== Where to go next ==

There is one more way a stationary path can fail to be the best, and it is
different in kind from this one. It is at
[[Hamilton's principle/Explainers/Strong and weak]].

Hamilton's principle/Explainers/Where the minimum fails - Revision history

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