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	{{Technical\|Hamilton's principle/Mathematics/Boundary terms and natural boundary conditions\|Hamilton's principle/Mathematics/The Beltrami identity}}		{{Technical\|Hamilton's principle/Mathematics/Boundary terms and natural boundary conditions\|Hamilton's principle/Mathematics/The Beltrami identity}}
			{{Hamilton's principle map}}

	This page explains what happens when a path is not pinned down at both ends. It		This page explains what happens when a path is not pinned down at both ends. It

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	Real systems have many moving parts, not one. That is at		Real systems have many moving parts, not one. That is at
	[[Hamilton's principle/Explainers/More than one variable]].		[[Hamilton's principle/Explainers/More than one variable\|More than one variable]].

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{{Technical|Hamilton's principle/Mathematics/Boundary terms and natural boundary conditions|Hamilton's principle/Mathematics/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 ==

[[File:OPL-HP-F-04.png|thumb|right|400px|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 ==

[[File:OPL-HP-F-23.png|thumb|right|400px|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 ==

[[File:OPL-HP-F-28.png|thumb|right|400px|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 ==

[[File:OPL-HP-F-40.png|thumb|right|400px|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 ==

[[File:OPL-HP-F-43.png|thumb|right|400px|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
[[Hamilton's principle/Explainers/More than one variable]].

Hamilton's principle/Explainers/Free ends - Revision history

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