PMA Publisher: Published from the Principia Machina source

2026-10-15T19:30:19Z

Published from the Principia Machina source

← Older revision		Revision as of 19:30, 15 October 2026
Line 1:		Line 1:
	{{DISPLAYTITLE:Light and particles}}		{{DISPLAYTITLE:Light and particles}}
	{{Technical\|Hamilton's principle/Applications/The optical-mechanical analogy}}		{{Technical\|Hamilton's principle/Applications/The optical-mechanical analogy}}
			{{Hamilton's principle map}}

	This page explains why light and moving objects obey the same rule. It uses no		This page explains why light and moving objects obey the same rule. It uses no

BFP Rauch: Pipe internal links so cross-references read as titles, not paths

2026-09-16T04:59:01Z

Pipe internal links so cross-references read as titles, not paths

← Older revision		Revision as of 04:59, 16 September 2026
Line 120:		Line 120:

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

Line 133:		Line 133:

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

BFP Rauch: Add DISPLAYTITLE: title the page by its section rather than its path

2026-09-16T04:47:01Z

Add DISPLAYTITLE: title the page by its section rather than its path

← Older revision		Revision as of 04:47, 16 September 2026
Line 1:		Line 1:
			{{DISPLAYTITLE:Light and particles}}
	{{Technical\|Hamilton's principle/Applications/The optical-mechanical analogy}}		{{Technical\|Hamilton's principle/Applications/The optical-mechanical analogy}}

Line 133:		Line 134:
	The volume now turns to systems where every point can move, at		The volume now turns to systems where every point can move, at
	[[Hamilton's principle/Explainers/Fields]].		[[Hamilton's principle/Explainers/Fields]].

BFP Rauch: 1 revision imported

2026-09-09T05:05:17Z

1 revision imported

← Older revision	Revision as of 05:05, 9 September 2026
(No difference)

hamiltons-principle>HPVBuild: Initial import of B-18

2026-09-08T12:00:00Z

Initial import of B-18

New page

{{Technical|Hamilton's principle/Applications/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 ==

[[File:OPL-HP-F-08.png|thumb|right|400px|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.

{| class="wikitable"
! 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 ==

[[File:OPL-HP-F-38.png|thumb|right|400px|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 ==

[[File:OPL-HP-F-48.png|thumb|right|400px|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]].

Hamilton's principle/Explainers/Light and particles - Revision history

PMA Publisher: Published from the Principia Machina source

BFP Rauch: Pipe internal links so cross-references read as titles, not paths

BFP Rauch: Add DISPLAYTITLE: title the page by its section rather than its path

BFP Rauch: 1 revision imported

hamiltons-principle>HPVBuild: Initial import of B-18