Optico

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Hamiltons optico-mechanical analogy is a conceptual parallel between trajectories in classical mechanics and wavefronts in optics, introduced by William Rowan Hamilton around 1831.[1] It may be viewed as linking Huygens' principle of optics with Maupertuis' principle of mechanics.[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.[7] According to Cornelius Lanczos, the analogy has been important in the development of ideas in quantum physics.[3] Erwin Schrdinger cites the analogy in the very first sentence of his paper introducing his wave mechanics.[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.[9]

The propagation of light can be considered in terms of rays and wavefronts in ordinary physical three-dimensional space. The wavefronts are two-dimensional curved surfaces; the rays are one-dimensional curved lines.[11]Hamilton's analogy amounts to two interpretations of a figure like the one shown here. In the optical interpretation, the green wavefronts are lines of constant phase and the orthogonal red lines are the rays of geometrical optics. In the mechanical interpretation, the green lines denote constant values of action derived by applying Hamilton's principle to mechanical motion and the red lines are the orthogonal object trajectories.[11]

The wavefronts and rays or the equal-action surfaces and trajectories are dual objects linked by orthogonality.[10]On one hand, a ray can be regarded as the orbit of a particle of light. It successively punctures the wave surfaces. The successive punctures can be regarded as defining the trajectory of the particle.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. Then the physical meaning of the rays is less evident.[12]

Going beyond ordinary three-dimensional physical space, one can imagine a higher dimensional abstract configuration "space", with a dimension a multiple of 3. In this space, one can imagine again rays as one-dimensional curved lines. Now the wavefronts are hypersurfaces of dimension one less than the dimension of the space.[6] Such a multi-dimensional space can serve as a configuration space for a multi-particle system.

Albert Messiah considers a classical limit of the Schrdinger equation. He finds there an optical analogy. The trajectories of his particles are orthogonal to the surfaces of equal phase. He writes "In the language of optics, the latter are the wave fronts, and the trajectories of the particles are the rays. Hence the classical approximation is equivalent to the geometric optics approximation: we find once again, as a consequence of the Schrdinger equation, the basic postulate of the theory of matter waves."[13]

Hamilton's optico-mechanical analogy played a critical part[14][11] in the thinking of Schrdinger, one of the originators of quantum mechanics. Section 1 of his paper published in December 1926 is titled "The Hamiltonian analogy between mechanics and optics".[15] Section 1 of the first of his four lectures on wave mechanics delivered in 1928 is titled "Derivation of the fundamental idea of wave mechanics from Hamilton's analogy between ordinary mechanics and geometrical optics".[16]

In a brief paper in 1923, de Broglie wrote : "Dynamics must undergo the same evolution that optics has undergone when undulations took the place of purely geometrical optics."[17] In his 1924 thesis, though Louis de Broglie did not name the optico-mechanical analogy, he wrote in his introduction,[18]

... a single principle, that of Maupertuis, and later in another form as Hamilton's Principle of least action ... Fermat's ... principle ..., which nowadays is usually called the principle of least action. ... 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).

Acquired copper deficiency has recently been recognized as a cause of myeloneuropathy mimicking subacute combined degeneration due to vitamin B-12 deficiency. A remote history of gastric surgery is frequently associated with this syndrome. However, the very limited prevalence of severe copper deficiency in patients with a history of gastric surgery suggests that additional contributing factors are likely to be involved. We describe a patient with copper deficiency and a previous Billroth II partial gastrectomy for gastric carcinoma, presenting with severe myelo-optico-neuropathy, demyelinating lesions of the brain, and subjective hyposmia. An abnormal glucose breath test also revealed small bowel bacterial overgrowth syndrome. Copper replacement therapy associated with antibiotic therapy was effective in preventing further neurological damage and in obtaining mild improvement. We propose that copper status should be evaluated in all patients presenting with unexplained noninflammatory myeloneuropathy. Small bowel bacterial overgrowth syndrome should be investigated as a cause of generalized malabsorption and a possible contributing factor to copper deficiency after gastric surgery, as should occult zinc ingestion.

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