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Fermat's principle is most familiar, however, in the case of visible light: it is the link between geometrical optics, which describes certain optical phenomena in terms of rays, and the wave theory of light, which explains the same phenomena on the hypothesis that light consists of waves.
In the generalized Fermat’s principle [6] the time is used as a functional and together as a variable. It is applied Pontryagin’s minimum principle of the optimal control theory and obtained an effective Hamiltonian for the light-like particle motion in a curved spacetime. It is shown that obtained curves are null geodesics.
Fermat supported the opposing assumptions, i.e., the speed of light is finite, and his derivation depended upon the speed of light being slower in a denser medium. [12] [13] Fermat's derivation also utilized his invention of adequality, a mathematical procedure equivalent to differential calculus, for finding maxima, minima, and tangents. [14] [15]
The general results presented above for Hamilton's principle can be applied to optics using the Lagrangian defined in Fermat's principle.The Euler-Lagrange equations with parameter σ =x 3 and N=2 applied to Fermat's principle result in ˙ = with k = 1, 2 and where L is the optical Lagrangian and ˙ = /.
Pierre de Fermat (French: [pjɛʁ də fɛʁma]; [a] 17 August 1601 – 12 January 1665) was a French mathematician who is given credit for early developments that led to infinitesimal calculus, including his technique of adequality.
He used Fresnel's theory to predict that a bright spot ought to appear in the center of the shadow of a small disc, and deduced from this that the theory was incorrect. However, Arago, another member of the committee, performed the experiment and showed that the prediction was correct. [3]
Number theory: Leonhard Euler: Faraday's law of induction Faraday's law of electrolysis: Electromagnetism Chemistry: Michael Faraday: Faxén's law: Fluid dynamics: Hilding Faxén: Fermat's principle Fermat's Last Theorem Fermat's little theorem: Optics Number theory Number theory: Pierre de Fermat: Fermi paradox Fermi's golden rule Fermi ...
For example, incoming and outgoing light can be considered as reversals of each other, [1] without affecting the bidirectional reflectance distribution function (BRDF) [2] outcome. If light was measured with a sensor and that light reflected on a material with a BRDF that obeys the Helmholtz reciprocity principle one would be able to swap the ...