Hamilton's principle/Explainers/Free ends

Technical version: Boundary terms and natural boundary conditions· The Beltrami identity

This page explains what happens when a path is not pinned down at both ends. It uses no mathematics. The technical pages it goes with are named above.

What was assumed before

A free endpoint and its natural boundary condition.

Everything so far fixed both ends. You knew where the journey started and where it finished, and you compared the routes between.

That assumption was used once, at a particular moment in the argument. Because the nudges had to be zero at both ends, a term that appears during the working dropped out, and the argument went through cleanly.

Now suppose one end is free. A rope tied to a post at one end and hanging loose at the other. A slide where you know where the child gets on but not where they get off.

The dropped term comes back.

The free end sets its own rule

A catenary and the directrix from which its tension is measured.

Here is the pleasant surprise.

You might expect a free end to make the problem vaguer. Less is known, so less can be said.

The opposite happens. The problem produces an extra rule, at the free end, that it did not need before.

The reason is that you now have more nudges available. Before, every nudge had to be zero at both ends. Now nudges can move the free end too. More ways to test the path means more conditions the path has to satisfy.

The rule at the free end has a name: a natural boundary condition. Natural, because nobody imposed it. It came out of the problem.

What the extra rule says

Fundamental modes with both ends fixed and with one end free.

In words: nothing is being pushed at a free end.

For a rope, a free end carries no sideways pull. For a beam, an unloaded end carries no stress. For a moving object, a free end carries no momentum transfer.

These are the same statement about different things, and they all come out of the same piece of working.

The counting works out too. Rules of this kind need two pieces of information to pin down an answer. Pin both ends and you have supplied two. Pin one end and the free end supplies the other itself. Either way you end up with two.

You can hear it

The boundary term left by integration by parts.

There is a case where this is audible.

Take a guitar string, held at both ends. Pluck it and it sounds a note.

Now take an organ pipe. In a pipe closed at one end and open at the other, the air inside is held at the closed end and free at the open one — the same situation as a string free at one end.

The closed pipe sounds a note an octave lower than an open pipe of the same length. It also sounds thinner, because it produces only some of the overtones an open pipe produces.

Both differences come from the free end and the extra rule it imposes. A derivation that carelessly threw that term away would predict the same note for both pipes, and be wrong by an octave.

Meeting a line at a right angle

A quantity conserved along the whole path.

A different case: one end is not pinned to a point but has to land somewhere on a line. Think of walking to a river. You must reach the river, but any spot on the bank will do.

The problem then tells you where to meet it. For the shortest walk, you should cross the bank at a right angle.

Nobody put that in. It came out.

A word of caution: the right angle is special to distance. If you were minimising time rather than distance, and the ground were muddy in places, the best crossing angle would not be ninety degrees. The general rule is not about angles at all. Right angles are what it happens to say for this particular question.

A shortcut worth knowing

The second technical page under this one is about a labour-saving device, and it is easy to state.

If the cost rule does not mention time, one quantity stays the same for the whole journey.

That is enormously helpful. Problems of this kind normally need you to solve a hard equation. If something stays constant, you get one stage of the answer free and the remaining work is much easier.

For a moving object, the constant quantity is the energy. Energy is conserved because the laws of physics do not depend on what time it is.

A chain hanging between two posts

Here is the shortcut in action.

Hang a chain between two posts. It settles into a curve. Which curve?

The chain settles where its energy is lowest. The rule for that energy does not mention sideways position — the chain does not care whereabouts along the gap you are. So the shortcut applies, and one quantity stays the same all along the chain.

That single fact turns a hard problem into an easy one. The answer is a curve called a catenary.

Galileo thought it was a parabola. He was wrong, but understandably so: the two curves look nearly identical near the bottom and only separate towards the ends. The correct answer was found in 1691.

The constant quantity has a meaning, too. It is the sideways pull in the chain, and it is the same at every point — which makes sense, because nothing is pulling the chain sideways anywhere along its length.

That is the pattern worth taking away. The shortcut is not just an algebra trick. The thing that stays constant is usually something you can point at.

Where to go next

Real systems have many moving parts, not one. That is at More than one variable.