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{{Technical|Hamilton's principle/Mathematics/The second variation|Hamilton's principle/Mathematics/The Legendre condition}}

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 ==

[[File:OPL-HP-F-07.png|thumb|right|400px|The second variation changing sign as the interval lengthens.]]

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 ==

[[File:OPL-HP-F-16.png|thumb|right|400px|A needle variation.]]

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 ==

[[File:OPL-HP-F-17.png|thumb|right|400px|Local penalty against accumulated reward.]]

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]].

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