Hamilton's principle/Explainers/Light and particles

Technical version: The optical-mechanical analogy

This page explains why light and moving objects obey the same rule. It uses no mathematics. The technical page it goes with is named above.

Two subjects, one rule

Snell's law from stationary time.

Light travelling through glass and a ball rolling through a landscape look like different problems. They are not.

Both are described by the same rule. Substitute one quantity for another and the equations for light become the equations for the ball.

for light	for a moving object
how much the material slows light	how much speed the object has to spare
a ray	a trajectory
a wavefront	a surface of constant total cost
bending at a boundary	a change in speed
a focus	a place where paths refocus

This is not an analogy in the loose sense of two things that resemble each other. It is the same mathematics with the labels changed.

Which came first

The optical-mechanical dictionary.

Hamilton did the optics first.

He built the method for optical systems, in his work on rays, and then transposed it to mechanics. He says so in the opening paragraph of his 1834 essay: the dynamical principle is another form of an idea already applied to optics.

So the famous mechanics is a translation of the less famous optics, and not the other way round.

He also wrote something that reads oddly now. The method, he suggested, is probably not confined to those two sciences, and might one day become a separate branch of analysis. He had no particular candidate in mind.

The hole in the table

The empty corner of the optical-mechanical analogy.

Look at that table again. It matches ray optics to mechanics.

But rays are not all of optics. They are the approximation that works when the wavelength of light is tiny compared with everything else — lenses, apertures, the room.

When the wavelength is not tiny, rays fail. Light spreads through a narrow slit. It bends round edges. Two beams overlap and produce a striped pattern. None of this is in the ray picture.

So the table has an empty cell. Ray optics corresponds to mechanics. Wave optics corresponds to nothing.

If light has a wave theory underlying its ray theory, and mechanics is the same as ray theory, then mechanics ought to have a wave theory underlying it too.

Ninety-two years of nothing

Nobody filled the cell.

The reason is not stupidity. There was no evidence. No experiment on moving objects failed. No wavelength had been observed for anything with mass. Nothing suggested there was anything to be waving.

An empty cell in a table is only a clue if you have a reason to think something belongs there. From 1834 to 1926 there was none.

Filling it

Then evidence arrived. Experiments on light and on atoms showed behaviour that ray thinking could not explain, and the idea that matter might have a wave aspect became worth taking seriously.

Schrödinger took the analogy and worked backwards. Suppose the missing wave theory exists. Suppose ordinary mechanics is what it looks like when the wavelength is negligible. What equation must the wave obey?

That requirement has essentially one answer, and it is the equation at the foundation of quantum mechanics.

It was constructed, not guessed. The construction consisted of insisting that Hamilton's table be completed.

Why the wavelength had to be small

The wavelength associated with an object depends on its momentum. Heavy and fast means short.

For anything you can see, it is unimaginably smaller than the object itself — smaller by a factor with more zeros than are worth writing. Waves that fine are undetectable, and the ray picture is exact for all practical purposes.

For an electron, the wavelength is comparable to the size of an atom. There the wave behaviour is not a correction. It is the whole story, and treating an electron as a small ball on a trajectory fails completely.

So classical mechanics is not wrong. It is the short-wavelength limit of something larger, in the same way ray optics is.

Where both pictures break

One place is worth singling out, because this volume keeps arriving at it.

At a focus, rays come back together and the ray description fails. It predicts infinite brightness, which is not what happens.

That is exactly the conjugate point of Hamilton's principle/Explainers/Where the minimum fails — the place where neighbouring paths refocus and minimality is lost.

The optical version, the geometrical version, the mechanical version and the quantum version are one phenomenon with four names.

That is the practical value of the correspondence. Understand it once, in whichever subject you find easiest, and you understand it in all four. A result proved in optics transfers to mechanics without reproof.

Where to go next

The volume now turns to systems where every point can move, at Hamilton's principle/Explainers/Fields.