The Number That Does Not Move

Here is the whole of a swinging pendulum, written out. There are two numbers:

how far it is from hanging straight down, and how fast that angle is changing.

And there are two rules, one for each number, saying what it is a moment later.

That is not a simplification of a pendulum. Given the two starting numbers,

those two rules decide where it is in ten seconds, in ten minutes, and at every

moment in between. Nothing else enters. Mechanics is full of systems like this,

and the striking thing about them is how little you can say about where they

will be. There is no formula for the angle of a pendulum swinging wide at the

four hundredth step. You have to walk it forward.

What you can say is that a particular combination of the two numbers will not

have changed.

This is a strange kind of statement and it is worth being clear about how

strong it is. It does not say the pendulum is slow, or smooth, or nearly a

circle. It says that if you stop the motion at any moment whatsoever, measure

the two numbers, and put them into that expression, you get the same answer you

would have got at any other moment. One number, checkable anywhere, standing in

for an infinite number of claims about the motion.

That is the only handle there is on this system, so everything in this course

is built on finding one.

Below, the two rules are applied two hundred times, with the energy worked out

after every single application. The pendulum is started a long way out, at

about seventy degrees, where none of the convenient small-angle approximations

hold.

The strip under the picture is doing the work. That line is the energy, and the

figure was only produced because the energy stayed put. Had the two rules been

written in a slightly different way, with one of them reading the other's value

from before the step rather than after, the pendulum above would still swing,

the picture would still look like a pendulum, and the energy would have been

gone by about a third of itself by the right-hand edge. That is not a hard

mistake to make. It is one character.

Against the step count, a conserved quantity is a flat line and that is all it

looks like. Plot the two numbers against each other instead and it becomes a

shape.

The loop closes. That is the conserved quantity, seen from the side. The

pendulum could in principle be at any point of this frame, with any angle and

any speed, and it is not: it is on one curve, and the curve is the set of

states with the right energy. After two hundred steps it comes back to where it

started, because there is nowhere else for it to go.

This is why mechanics is drawn this way so often. In the history picture you

have to compare the amplitude of one swing with the amplitude of a swing a

hundred steps later and judge by eye. In the shape picture, a system that is

losing energy spirals visibly inward, and a system that is gaining it spirals

out.

One honest complication. A calculation that steps forward in finite jumps never

holds a conserved quantity perfectly still, and it is important not to read that

as failure.

Each step lands slightly off the true motion, so the energy is slightly wrong

at the end of the step. In a well-ordered calculation the next step's error

leans the other way, and the energy ends up breathing in and out around a fixed

centre for as long as you care to run it. In a badly ordered one the errors all

lean the same way, and the energy walks steadily off.

The distinction matters because the two look identical over a few steps and

tell you completely different things over a few hundred. A wobble says the step

is coarse. A drift says the rule is wrong.

A moving system is a list of numbers and a rule for advancing them. A conserved

quantity is one number built out of that list which the rule is not allowed to

change, and it is the only claim about such a system that can be checked at a

single moment rather than over the whole motion.

The next lesson takes the pendulum's energy apart into the two pieces named

above and asks what each of them is doing, because the trade between them is the

same trade in every system in this course.