Hamilton's principle/Explainers/Is it really a minimum
This page asks whether a stationary path is actually the best one. It uses no mathematics. The technical pages it goes with are named above.
The question that has been waiting
Every method in this volume so far finds paths where the total stops changing under small nudges.
Not one of them shows the total is smallest.
That gap was flagged on the first page and has been left open ever since. This page begins to close it, and the answer is more interesting than a simple yes.
Flat ground tells you less than you think
Stand somewhere and find the ground perfectly level. Where are you?
At the bottom of a valley. Or at the top of a hill. Or at a mountain pass, which is a low point along the ridge and a high point along the path across it.
Level ground is level. It does not say which.
To tell the difference you have to look at the curvature — how the ground bends away as you step off. Curving up in every direction means a valley. Curving down in every direction means a summit. Curving up one way and down another means a pass.
The same applies to paths. Finding a stationary path is finding level ground. To learn what kind, you have to look at how the total bends.
Two things that push against each other
Look at the bending, and it turns out to have two parts pulling in opposite directions.
A penalty for wobbling. Deviating from the path costs something, and it costs more the more sharply you wobble.
A reward for straying. Being away from the path may, in some problems, be worth something.
If the penalty always wins, the path is a minimum. If the reward can win, it is not.
And these two behave completely differently in an important respect, which is what makes the rest of the subject necessary.
The local test
The wobble penalty can be checked one point at a time.
Here is the reason. You can make a nudge that is tiny in size but very sharp — a narrow, steep wiggle in a short stretch of the path, leaving the rest untouched. Such a wiggle barely moves the path but wobbles it violently.
For that nudge, the wobble penalty is overwhelming and the straying reward is negligible. So the wiggle tests the penalty alone.
Do this everywhere. If the penalty is ever negative — if wobbling somewhere actually pays — then a wiggle there makes the total smaller, and the path is not a minimum.
That gives a necessary condition, and it is easy to check: wobbling must never pay, anywhere along the path.
This test is named after Legendre, who found it in 1786.
Legendre's mistake
Legendre thought his test was enough. If wobbling never pays anywhere, he reasoned, the path must be a minimum.
It is a very natural mistake. For an ordinary hill it would be right: curving upwards in every direction really does mean a valley.
Lagrange found the error. It took another half century for Jacobi to supply what was missing.
The reason for the failure is worth understanding, because it is the reason this part of the subject exists at all.
Why local checks are not enough
The wiggle test only ever looks at one small stretch. That is what makes it convenient, and it is also what makes it blind.
The straying reward does not act in one place. It accumulates along the path. Over a short journey there is not enough of it to matter. Over a long enough journey it adds up, and it can eventually beat a wobble penalty that is perfectly healthy at every single point.
A nudge that exploits this is not a sharp wiggle in one place. It is a gentle bulge spread over the whole path. The wiggle test never builds such a nudge, and so never sees the failure.
The length of the journey is part of the answer. Two paths can satisfy every local condition identically and differ only in how far they run — and one can be a minimum while the other is not.
Nothing about the path at any single point records this. That is the strangeness, and it is why a second, non-local test is needed.
A case you can check
There is a simple setup where a hand calculation shows exactly this.
Take a system where staying near the middle is rewarded, and follow it for a certain time.
Over a short time, the answer is a clean minimum. The wobble penalty dominates and everything behaves.
Extend the time and the reward accumulates. At a particular length the two balance exactly. Beyond it, the reward wins, and the path is no longer a minimum.
Nothing about the system changed. The rules are the same, the path satisfies the same equation at every point, and the local test passes throughout. Only the duration is different.
That case is worked in full on the technical pages, and its answer comes out as an exact length at which the switch happens.
The missing test
What is needed is a test that looks at the journey as a whole rather than point by point.
Jacobi supplied it in 1837, and it has a picture attached that makes it easy to hold on to: paths that set off in slightly different directions can come back together.
Where they meet again, minimality fails.
That is the next page: Hamilton's principle/Explainers/Where the minimum fails.