Hamilton's principle/Explainers/Fixed length problems

Technical version: Isoperimetric problems

Hamilton's principle: map
Reading the volume How to read this volume Notation Glossary Equation index Figure gallery
Reading by level Start here: Comparing paths The mathematics: Functionals and the space of paths Stationary is not least: The Jacobi condition and conjugate points The applications: The brachistochrone The history: Hamilton 1834 and 1835
Sources Sources Chronology
Related Noether's theorems On Physical Lines of Force

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

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

A two-parameter family for an isoperimetric problem.

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

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 Is it really a minimum.