Hamilton's principle/Explainers/Where the minimum fails

Technical version: 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

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

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.