Talk:Hamilton's optico-mechanical analogy

???

Here are some issues which jump at me at the time of the current article:

• Hamilton is not even mentioned in the article, except the title and lead "Hamilton's optico-mechanical analogy".
• "The wavefronts are two-dimensional curved surfaces. The rays are one-dimensional curved lines." <--- the "rays" are extremal paths that light takes aren't they? which are always straight lines in flat space.
• "Thus, a wave is a foliated set of moving two-dimensional surfaces. It is not part of the definition of a wave that it be distinctly oscillatory." <--- every wave is the propagation of a disturbance, which means oscillatory motion (even if transient)?
• "This is wave–particle duality for a single particle in ordinary three-dimensional physical space or for a wave of some property of a logically dense spatially distributed medium." <--- what does "logically dense spatially distributed medium" mean?
• "Going beyond ordinary three-dimensional physical space, one can imagine a higher dimensional abstract "space", with a dimension a multiple of 3." <--- why a multiple of 3? Why not any integer or even any real number (fractal dimensions)?

But even ignoring these, this seems a duplicate of the Huygens–Fresnel principle anyway. M∧Ŝc²ħεИ_τlk 10:13, 27 January 2015 (UTC)

Perhaps I ought to thank you for these thoughts, even though they make it clear that you do not like to read of the views of Cornelius Lanczos and John Lighton Synge. It seems evident that you are taking an interest in this article for reasons that I do not need to analyse here, but are perhaps improper.Chjoaygame (talk) 12:39, 27 January 2015 (UTC)

• As to "Hamilton is not even mentioned in the article, except the title and lead "Hamilton's optico-mechanical analogy"."

The eponym 'Hamilton's' is standard in the literature, but so far I have focused on more urgent physical aspects. In a history section it would figure prominently.Chjoaygame (talk) 12:39, 27 January 2015 (UTC)

• As to ""The wavefronts are two-dimensional curved surfaces. The rays are one-dimensional curved lines." <--- the "rays" are extremal paths that light takes aren't they? which are always straight lines in flat space."

In a medium with well-behavedly variable refractive index the rays are often curved.Chjoaygame (talk) 12:39, 27 January 2015 (UTC)

• As to ""Thus, a wave is a foliated set of moving two-dimensional surfaces. It is not part of the definition of a wave that it be distinctly oscillatory." <--- every wave is the propagation of a disturbance, which means oscillatory motion (even if transient)?"

John Lighton Synge was a well respected writer on such matters. He points out that what makes a wave not a particle is that it extends over a more or less wide region, unlike a particle which is often regarded as not extending at all. The sentence you attack says "distinctly oscillatory", but you attack it as if it said only 'oscillatory'. You have created a straw man and attacked it. There is a difference between a disturbance and an oscillation, more so for a distinct oscillation. A solitary wave is not distinctly oscillatory but is still a wave. It is possible to make a Fourier analysis of heat propagation that is not distinctly oscillatory. It is not the possibility of Fourier analysis or even of superposition that makes a wave. It is its extension. That is Synge's point.Chjoaygame (talk) 12:39, 27 January 2015 (UTC)

• As to ""This is wave–particle duality for a single particle in ordinary three-dimensional physical space or for a wave of some property of a logically dense spatially distributed medium." <--- what does "logically dense spatially distributed medium" mean?"

Rough speaking it means the medium has a continuous or dense variable refractive index, except for finitely or countably many discontinuities or interfaces.Chjoaygame (talk) 12:39, 27 January 2015 (UTC)

• As to ""Going beyond ordinary three-dimensional physical space, one can imagine a higher dimensional abstract "space", with a dimension a multiple of 3." <--- why a multiple of 3? Why not any integer or even any real number (fractal dimensions)?"

For a classical configuration space, each particle has three position coordinates.Chjoaygame (talk) 12:39, 27 January 2015 (UTC)

• As to "But even ignoring these, this seems a duplicate of the Huygens–Fresnel principle anyway.".

I think that article is not mainly about Hamilton's optico-mechanical analogy. At present it says nothing about it, and doesn't talk about particles.Chjoaygame (talk) 12:39, 27 January 2015 (UTC)

The present article is about a well recognized physical principle that seems not to discussed elsewhere in Wikipedia. The article is a work in progress.Chjoaygame (talk) 12:39, 27 January 2015 (UTC)

I was mistaken about the rays which can be curved in an inhomogeneous medium. I am aware that Cornelius Lanczos and John Lighton Synge were excellent writers (and happen to have Synge and Schild's tensor calculus book) but you can't expect every one to have almost every reference to verify exactly what they wrote, it takes time to access and read sources.

Those points aside, you have vaguely worded the above points which have the ambiguities mentioned, then responded with comments accusing me of screwing up.

When you wrote ""Going beyond ordinary three-dimensional physical space, one can imagine a higher dimensional abstract "space"," that could have meant one particle in d dimensions, the "with a dimension a multiple of 3." is left hanging and means little to a typical reader.

About "Thus, a wave is a foliated set of moving two-dimensional surfaces. It is not part of the definition of a wave that it be distinctly oscillatory." and responded with blabber about me attacking straw men, the soliton wave example is oscillatory in a localized region of space, and so is every other wave at least locally. Yes, I know there is a difference between a disturbance and an oscillation, but in the context of waves they are intertwined. By the way, what is the difference between "distinctly oscillatory" and "oscillatory"? You didn't define the difference, and is it actually standard terminology anyway?

Your final response "I think that article is not mainly about Hamilton's optico-mechanical analogy. At present it says nothing about it, and doesn't talk about particles." is just a contradiction... This article is talking about particles, and should be about the topic in the title.

And you think I came here just to fight you after YohanN7 posted on my talk page? No, it's because I wanted to see what this topic is about, and how it is related to QM. M∧Ŝc²ħεИ_τlk 13:05, 27 January 2015 (UTC)

As to "Your final response "I think that article is not mainly about Hamilton's optico-mechanical analogy. At present it says nothing about it, and doesn't talk about particles." is just a contradiction... This article is talking about particles, and should be about the topic in the title.".

I am guilty of unresolved or ambiguous anaphora. I ought to have written 'I think that article Huygens–Fresnel principle is not mainly about Hamilton's optico-mechanical analogy. At present that article Huygens–Fresnel principle says nothing about Hamilton's optico-mechanical analogy, and doesn't talk about particles.'Chjoaygame (talk) 13:37, 27 January 2015 (UTC)

Removing erroneous claim about Schrodinger.

I'm removing a sentence:

According to Erwin Schrödinger, for micromechanical motions, the Hamiltonian analogy of mechanics to optics is inadequate to treat diffraction, which requires it to be extended to a vibratory wave equation in configuration space.

The sentence implies that Schrödinger claimed the analogy fails.

Thankfully this sentence is fully referenced and thus is easily checked. Here is what Schrödinger's abstract says:

"THE Hamiltonian analogy of mechanics to optics (pp. 13-18) is an analogy to geometrical optics, since to the path of the representative point in configuration space there corresponds on the optical side the light ray, which is only rigorously defined in terms of geometrical optics. The undulatory elaboration of the optical picture (pp. 19-30) leads to the surrender of the idea of the path of the system, as soon as the dimensions of the path are not great in comparison with the wave-length (pp. 25-26). Only when they are so does the idea of the path remain, and with it classical mechanics as an approximation (pp. 20-24, 41-44); whereas for "micro-mechanical" motions the fundamental equations of mechanics are just as useless as geometrical optics is for the treatment of diffraction problems. In analogy with the latter case, a wave equation in configuration space must replace the fundamental equations of mechanics. In the first instance, this equation is stated for purely periodic vibrations sinusoidal with respect to time (p. 27 et seq.); it may also be derived from a "Hamiltonian variation principle" (p. 1 et seq., pp. 11-12). "

(I copied this text from an online preview of a copyrighted book)

So what is inadequate? According to Schrödinger: "for "micro-mechanical" motions the fundamental equations of mechanics are just as useless" So he simply saying that Newtonian mechanics does not work for subatomic particles. He then elaborates with an analogy: "as useless as geometrical optics is for the treatment of diffraction problems." Therefore this sentence is actually a kind of application of the Hamiltonian analogy; Schrödinger is setting up to apply the analogy that connects geometrical optics (mechanical trajectories) to waves (that do work for diffraction). Johnjbarton (talk) 16:31, 9 June 2023 (UTC)

Good removal. XOR'easter (talk) 22:46, 12 June 2023 (UTC)

References