Hamilton's optico-mechanical analogy

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. == Huygens' principle; Fermat's principle ==

The Hamilton optico-mechanical analogy is closely related to Fermat's principle and thus to the Huygens–Fresnel principle. Fermat's principle states that the rays between wavefronts will take the path least time; the concept of successive wavefronts derives from Huygens principle. == Extended Huygens' principle ==

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. Such a multi-dimensional space can serve as a configuration space for a multi-particle system. == Classical limit of the Schrödinger equation ==

Albert Messiah considers a classical limit of the Schrödinger 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 Schrödinger equation, the basic postulate of the theory of matter waves." == History ==

Hamilton's optico-mechanical analogy played a critical part in the thinking of Schrödinger, 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". 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". 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." In his 1924 thesis, though Louis de Broglie did not name the optico-mechanical analogy, he wrote in his introduction, 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 ..." The first textbook in English on wave mechanics 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, , .... ... 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." 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." Recently the concept has been extended to wavelength dependent regime. ==Optics, oceanology and QM==

The analogy between the three fields of optics, oceanology and quantum mechanics can be summarized in the following table. The first row shows the full wave equations from optics, oceanology and QM, namely the Helmholtz equation, the PDE governing the sea surface elevation \eta and the Schrödinger equation respectively. The second row is the first order WKB approximation, i.e. the slowly varying amplitude variation and the equations obtained are the Eikonal equation of optics and oceanology and the Hamilton-Jacobi equation. The third row is the second order WKB approximation and the equations obtained are the transport of intensity equation in optics, the wave energy equation in oceanology and the Liouville equation. == References ==