Hamilton's principle/Explainers/Fixed length problems - Revision history

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	{{DISPLAYTITLE:Fixed length problems}}		{{DISPLAYTITLE:Fixed length problems}}
	{{Technical\|Hamilton's principle/Mathematics/Isoperimetric problems}}		{{Technical\|Hamilton's principle/Mathematics/Isoperimetric problems}}
			{{Hamilton's principle map}}

	This page explains problems where a total is fixed in advance. It uses no		This page explains problems where a total is fixed in advance. It uses no

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	[[File:OPL-HP-F-35.png\|thumb\|right\|400px\|A two-parameter family for an isoperimetric problem.]]		[[File:OPL-HP-F-35.png\|thumb\|right\|400px\|A two-parameter family for an isoperimetric problem.]]

	[[Hamilton's principle/Explainers/Multipliers]] dealt with constraints that apply		[[Hamilton's principle/Explainers/Multipliers\|Multipliers]] dealt with constraints that apply
	at every moment. A rod has a fixed length now, and in a second, and always.		at every moment. A rod has a fixed length now, and in a second, and always.

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	== The hanging chain ==		== The hanging chain ==

	[[Hamilton's principle/Explainers/Free ends]] met a chain hanging between two		[[Hamilton's principle/Explainers/Free ends\|Free ends]] met a chain hanging between two
	posts and found the curve it settles into. That account left the length out.		posts and found the curve it settles into. That account left the length out.

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	path is actually the '''best''' one is a separate question, and the answer is more		path is actually the '''best''' one is a separate question, and the answer is more
	interesting than expected. It is at		interesting than expected. It is at
	[[Hamilton's principle/Explainers/Is it really a minimum]].		[[Hamilton's principle/Explainers/Is it really a minimum\|Is it really a minimum]].

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{{Technical|Hamilton's principle/Mathematics/Isoperimetric problems}}

This page explains problems where a total is fixed in advance. It uses no
mathematics. The technical page it goes with is named above.

== The oldest problem of this kind ==

[[File:OPL-HP-F-20.png|thumb|right|400px|Fixed perimeter, varying area.]]

You have a rope of a given length. Lay it on the ground in a loop. Which shape
encloses the most land?

The answer is a circle, and people have believed it for a very long time. Proving
it took much longer than believing it.

Problems of this shape have a name that records the original: '''isoperimetric''',
from the Greek for equal perimeter.

The pattern is: make one thing as large or small as possible, while some '''other'''
total is held fixed.

== How this differs from the last page ==

[[File:OPL-HP-F-35.png|thumb|right|400px|A two-parameter family for an isoperimetric problem.]]

[[Hamilton's principle/Explainers/Multipliers]] dealt with constraints that apply
at every moment. A rod has a fixed length now, and in a second, and always.

This is different. The rope has one length, for the whole loop. It is a single
fact about the entire shape, and it says nothing about any particular point.

That difference changes what the price has to be.

A constraint at every moment needs a price that can vary — cheap here, expensive
there. A constraint on a total needs a '''single fixed number''' for the whole
problem.

'''One rule per moment gives a varying price. One rule for the whole thing gives
one price.''' That is the whole distinction, and getting it backwards is the usual
error.

== Why you need two dials ==

There is a difficulty in setting these problems up, and seeing it makes the method
make sense.

Earlier pages nudged a path with one dial. Turn it and you move to a nearby path.

Try that here and it fails at once. Nudge the rope's shape and you almost
certainly change its length. The nudged shape is no longer a rope of the right
length, so comparing it to the original is meaningless.

The fix is to use '''two''' dials. One nudges the shape. The other adjusts it back
so the length comes out right.

Two dials, working together: one explores, one corrects. Now every path you
compare is a legitimate competitor.

The single price then falls out of the fact that it is one correction for the
whole path, not one per point.

== The hanging chain ==

[[Hamilton's principle/Explainers/Free ends]] met a chain hanging between two
posts and found the curve it settles into. That account left the length out.

Put it back. The chain has a definite length, fixed by how much chain you have.
That is a single fact about the whole chain — an isoperimetric constraint.

The answer is the same family of curves, moved up or down by an amount the price
sets.

And the price has a meaning again. It tells you the '''height of an imaginary
line''' below the chain from which the tension is measured. At any point of the
chain, the pull is the weight of a length of chain equal to the height above that
line.

That is a real and useful fact. It tells you the chain is under least strain at
the bottom and most at the supports. Anyone hanging a cable needs to know that,
and the version that ignored the length constraint could not supply it.

== Why a slack cable and a taut one differ so much ==

The same result explains something you can feel.

Hang a chain loosely and it sags deeply. The tension is small.

Pull the same chain nearly straight and the tension becomes enormous. Not a little
larger — dramatically larger.

The reason is that a nearly straight chain has almost no downward slope for its
weight to press against. To hold the weight up, the sideways pull has to be huge.

This is why a washing line always sags a little, and why you cannot pull one truly
horizontal no matter how hard you haul. The tension needed goes to infinity as the
sag goes to zero. Something breaks first — usually the post.

== The case the method cannot touch ==

There is one situation where the method fails outright, and it is worth knowing
because it looks like the easiest case.

Suppose the rope is '''exactly''' long enough to reach between the two posts and
no longer. Then there is only one shape available: straight, taut, no slack.

There is nothing to compare it against. Every alternative shape needs more rope
than you have. The two dials have nothing to work with — you cannot explore, and
you cannot correct.

The method needs alternatives. If the constraint pins things down so tightly that
only one candidate exists, there is no comparison to make, and the answer is
whatever the constraint allows.

This is not a flaw. It is the method being honest about the fact that it is a
comparison tool, and comparison needs at least two things.

== Working in reverse ==

One pleasant feature of these problems: they come in pairs.

Ask for the largest area with a given perimeter. Or ask for the shortest perimeter
around a given area. Both give the circle.

That is not a coincidence, and it holds generally. Swapping the quantity you
optimise with the quantity you hold fixed gives the same answers.

So a soap bubble, which minimises its surface for the air it holds, is solving the
same problem as the rope on the ground. It arrives at a sphere for the reason the
rope arrives at a circle.

== Where to go next ==

Every result so far has found paths where the total stops changing. Whether such a
path is actually the '''best''' one is a separate question, and the answer is more
interesting than expected. It is at
[[Hamilton's principle/Explainers/Is it really a minimum]].