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	This page explains how physics compares paths. It uses no mathematics. The		This page explains how physics compares paths. It uses no mathematics. The

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{{Technical|Hamilton's principle/Mathematics/Functionals and the space of paths|Hamilton's principle/Mathematics/The first variation}}

This page explains how physics compares paths. It uses no mathematics. The
technical pages it goes with are named above.

== A number for a whole journey ==

[[File:OPL-HP-F-01.png|thumb|right|400px|A true path and several comparison paths between fixed endpoints.]]

Think about driving from your home to a shop. There is more than one route. Each
route takes a certain amount of fuel.

Fuel use is a number. A route is a whole path. So fuel use is a rule that takes a
whole path and gives back one number.

Physics uses rules like this all the time. The rule might give the time a journey
takes. It might give the length of a curve. It might give a quantity called the
'''action''', which this volume is about.

A rule that turns a whole path into one number has a name. It is called a
'''functional'''. That is the only new word on this page, and it means exactly
what the last paragraph said.

== Why this is different from ordinary maths ==

[[File:OPL-HP-F-02.png|thumb|right|400px|A one-parameter family of paths through the true path.]]

A function takes a number and gives back a number. Put in 3, get out 9.

A functional is different. You do not put in a number. You put in a '''whole
path''' — every point of it, from start to finish.

That makes the usual methods harder to apply. To find the lowest point of a hill
you walk downhill. But a path is not a point on a hill, and there are infinitely
many ways to change one: push it up in the middle, wobble it near an end, stretch
it. Which of those counts as a small change?

Answering that is the first real task.

== The rules of the comparison ==

Physics compares paths under strict rules.

'''All the paths start and end at the same two points.''' If you are asking which
route to the shop is best, every route must actually reach the shop. Routes that
end somewhere else are not in the competition.

'''All the paths are smooth.''' No sudden jumps. No sharp corners, at least to
begin with. Later work relaxes this, but the simple version starts here.

'''Only the true path is real.''' The other paths do not happen. They are
imaginary journeys, invented so that the true one can be compared against them.

That last rule matters more than it looks. When physics compares the real motion
of a ball with a nearby path, that nearby path is not a motion the ball could
make. It breaks the laws of physics. It is a made-up comparison, and it exists
only to be worse than the real thing.

Hamilton said this himself, in 1834. He described the comparison paths as
imaginable in geometry but impossible in dynamics.

== Nudging a path ==

Here is the trick that makes the whole subject work.

Take the true path. Take any smooth nudge — a shape that is zero at both ends and
whatever you like in between. Now add a small amount of the nudge to the path.
How small is set by a single dial.

Turn the dial to zero and you have the true path back. Turn it up a little and you
get a nearby path. Turn it up more and you get a path further away.

The dial is one number. So the problem has changed shape completely. Instead of
asking about infinitely many paths at once, you can ask what happens to your
functional as you turn one dial.

That is an ordinary question, of the kind school mathematics can answer.

== What "stationary" means ==

Turn the dial and watch the number.

For most paths the number changes straight away. Turn the dial a little and the
number goes up, or down.

For the true path something different happens. Turn the dial a little and the
number '''barely moves'''. It changes far less than the size of the nudge would
suggest. To a first approximation it does not change at all.

That is what '''stationary''' means. The number stops responding to small changes
in the path.

You have seen this before. At the top of a hill the ground is flat. Take a step in
any direction and your height barely changes. The same is true at the bottom of a
valley. And it is true at a mountain pass, which is neither a top nor a bottom.

'''Flat is flat.''' The ground being level tells you that you are at a top, a
bottom, or a pass. It does not tell you which.

== The word "least" is wrong ==

This is why this volume does not say the action is '''least'''.

Being stationary means the number stops changing for small nudges. It does not
mean the number is as small as it can be. Sometimes it is. Sometimes it is as
'''large''' as it can be. Sometimes it is neither.

Here is a case anyone can picture. On a globe, the shortest way between London and
Tokyo is the short way round. But you could also fly the long way round, the other
side of the world, along the same great circle. That long route obeys exactly the
same rule at every point of the journey. Locally it looks just as straight. It is
stationary — and it is very clearly not the shortest.

Nothing you could measure while flying would tell you which of the two you were
on. The difference is not in any single place. It is in the journey as a whole.

Working out exactly when stationary does mean shortest took mathematicians most of
the nineteenth century. It is covered later in this volume.

== Where to go next ==

The next step is to work out what actually happens to the number when you turn the
dial. That gives a rule the true path must obey, and it is the most useful single
result in this subject.

It is explained at
[[Hamilton's principle/Explainers/What the Euler-Lagrange equation says]].

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