Hamilton's principle/Explainers/Strong and weak
This page explains a second way a stationary path can fail to be the best. It uses no mathematics. The technical page it goes with is named above.
A different kind of failure
Hamilton's principle/Explainers/Where the minimum fails described one way things go wrong: run a path far enough and nearby paths refocus, and minimality is lost.
There is a second way, and it is not a variation on the first. It fails for a different reason and needs a different test.
The clue is that there are two different ways for paths to be close, and they are not the same.
Two kinds of nearby
Two paths are close in position if they never get far apart.
Two paths are close in direction if they also point the same way all along.
A zigzag that criss-crosses a straight road every few centimetres is close in position and far in direction. It never leaves the road's vicinity. It is constantly heading somewhere else.
That gap is where this page lives.
The blind spot
Everything on Hamilton's principle/Explainers/Is it really a minimum and Hamilton's principle/Explainers/Where the minimum fails tested against competitors that are close in both senses. Gentle nudges. Paths that stay near and keep roughly the same heading.
So those tests have a blind spot. They say nothing about zigzag competitors.
A path can therefore beat everything gently nearby and still lose badly to something that stays close in position while wobbling furiously in direction.
Beating only the gentle competitors is called a weak minimum. Beating the zigzags too is called a strong minimum.
The names are unfortunate. A weak minimum is the weaker claim about a stronger restriction on competitors. Reading the two words the wrong way round is a common slip.
A case where it happens
There is a setup where a path passes every earlier test and still loses.
Arrange a cost rule that charges a little for gentle movement but pays out for violent movement. Nothing forbids such a rule; it is a perfectly ordinary formula.
Now consider staying still. Staying still costs nothing.
Check it against gentle competitors and it wins every time — small movements cost a little, so doing nothing is best.
Now try a fine sawtooth: move up and down rapidly, with a very small amplitude, hundreds of times over. In position this path barely leaves the original. In direction it is nothing like it.
Because it moves violently, the rule pays out. The total comes out negative — better than staying still.
And you can make the teeth as small as you like. There are competitors as close in position as you please, all beating the original.
So staying still is a weak minimum and is not a strong one. Every earlier test passed. The failure was invisible to all of them.
The test that catches it
The missing test comes from Weierstrass, later in the nineteenth century, and it is simple to state.
Ask what the rule charges for travelling in a completely different direction, not just a slightly different one.
Earlier tests only ever asked about small changes of heading. Weierstrass's test asks about any heading — including a wild one.
If setting off in some wild direction is ever cheaper than the rule's own prediction, a sawtooth exploiting that direction will beat the path.
That is the whole idea: check all directions, not just nearby ones.
Why this is not academic
Rules that reward violent movement sound artificial. They are not rare.
Anything with a cost that grows and then falls off can behave this way. Some materials get easier to deform once past a threshold; some engineering problems have costs that drop when you go fast enough.
When a problem is like that, the smooth answer you compute may be a weak minimum that a rapidly oscillating solution beats. Physically, the material does not sit smoothly. It forms fine layers, alternating between two states, at a scale limited only by whatever the model left out.
This is real. Certain alloys form fine twinned structures for exactly this reason, and the fineness is set by physics the simple model ignores. The mathematics predicting sawtooths is describing something you can photograph under a microscope.
What the whole sufficiency story amounts to
Four conditions have now appeared, and it is worth seeing them together.
The main equation finds stationary paths. Necessary, never sufficient.
The local test checks that wobbling never pays anywhere. Necessary, not sufficient.
The refocusing test checks the journey is not long enough for nearby paths to meet. Together with the first two, enough for a weak minimum.
The all-directions test checks that no wild heading is cheap. Needed for a strong minimum.
Four tests, and only the last two combinations conclude anything positive. That is why this volume says stationary and not least: getting from one to the other took a century and four conditions, and even then only locally.
One thing still missing
All of these assume a best path exists. They compare a candidate against its neighbours. None of them shows there is anything to find.
Sometimes there is not. Stretch a soap film between two rings and pull the rings apart far enough, and the film snaps into two flat discs. No smooth surface connects them any more. The answer did not become worse; it stopped existing.
Proving that a best path exists is a separate subject, and this volume does not treat it.
Where to go next
The volume now turns to reorganising what it has, starting at Hamilton's principle/Explainers/Reading a curve two ways.