Hamilton's optico-mechanical analogy

[2][3][4][5][6] While Hamilton discovered the analogy in 1831, it was not applied practically until Hans Busch used it to explain electron beam focusing in 1925.

[8] Later in the body of his paper he says: Unfortunately this powerful and momentous conception of Hamilton is deprived, in most modern reproductions, of its beautiful raiment as a superfluous accessory, in favour of a more colourless representation of the analytical correspondence.

[10] The propagation of light can be considered in terms of rays and wavefronts in ordinary physical three-dimensional space.

On the other hand, a wave-front can be regarded as a level surface of displacement of some quantity, such as electric field intensity, hydrostatic pressure, particle number density, oscillatory phase, or probability amplitude.

"[13] Hamilton's optico-mechanical analogy played a critical part[14][11] in the thinking of Schrödinger, one of the originators of quantum mechanics.

... Huygens propounded an undulatory theory of light, while Newton, calling on an analogy with the material point dynamics that he created, developed a corpuscular theory, the so-called "emission theory", which enabled him even to explain, albeit with a contrived hypothesis, effects nowadays considered wave effects, (i.e., Newton's rings).In the opinion of Léon Rosenfeld, a close colleague of Niels Bohr, "... Schrödinger [was] inspired by Hamilton's beautiful comparison of classical mechanics and geometrical optics ..."[19] The first textbook in English on wave mechanics[20] devotes the second of its two chapters to "Wave mechanics in relation to ordinary mechanics".

It opines "... de Broglie and Schrödinger have turned this false analogy into a true one by using the natural Unit or Measure of Action, h, .... ... We must now go into Hamilton's theory in more detail, for when once its true meaning is grasped the step to wave mechanics is but a short one—indeed now, after the event, almost seems to suggest itself.

"[21] According to one textbook, "The first part of our problem, namely, the establishment of a system of first-order equations satisfying the spacetime symmetry condition, can be solved in a very simple way, with the help of the analogy between mechanics and optics, which was the starting point for the development of wave mechanics and which can still be used—with reservations—as a source of inspiration.