翻訳と辞書 Words near each other ・ Fermat curve ・ Fermat number ・ Fermat point ・ Fermat polygonal number theorem ・ Fermat primality test ・ Fermat Prize ・ Fermat pseudoprime ・ Fermat quintic threefold ・ Fermat quotient ・ FermaT Transformation System ・ Fermat's factorization method ・ Fermat's Last Theorem ・ Fermat's Last Theorem (book) ・ Fermat's Last Theorem in fiction ・ Fermat's little theorem ・ Fermat's principle ・ Fermat's right triangle theorem ・ Fermat's Room ・ Fermat's spiral ・ Fermat's theorem ・ Fermat's theorem (stationary points) ・ Fermat's theorem on sums of two squares ・ Fermata ・ Fermata Arts Foundation ・ Fermatta Music Academy ・ Fermat–Catalan conjecture ・ Fermat–Weber problem ・ Fermat’s and energy variation principles in field theory ・ Fermax ・ Ferme castrale of Hermalle-sous-Huy Dictionary Lists mini英和辞書 mini和英辞書 Webster 1913 Latin-English FOLDOC Wikipedia English ウィキペディア

翻訳と辞書　辞書検索 [ 開発暫定版 ] スポンサード リンク Fermat's principle ： ウィキペディア英語版 Fermat's principle In optics, Fermat's principle or the principle of least time is the principle that the path taken between two points by a ray of light is the path that can be traversed in the least time. This principle is sometimes taken as the definition of a ray of light.〔Arthur Schuster, ''An Introduction to the Theory of Optics'', London: Edward Arnold, 1904 (online ).〕 However, this version of the principle is not general; a more modern statement of the principle is that rays of light traverse the path of stationary optical length with respect to variations of the path.〔 〕 In other words, a ray of light prefers the path such that there are other paths, arbitrarily nearby on either side, along which the ray would take almost exactly the same time to traverse. Fermat's principle can be used to describe the properties of light rays reflected off mirrors, refracted through different media, or undergoing total internal reflection. It follows mathematically from Huygens' principle (at the limit of small wavelength). Fermat's text ''Analyse des réfractions'' exploits the technique of adequality to derive Snell's law of refraction and the law of reflection. Fermat's principle has the same form as Hamilton's principle and it is the basis of Hamiltonian optics. ==Modern version== The time T a point of the electromagnetic wave needs to cover a path between the points A and B is given by: :$T=\backslash int\_\}\; \backslash ,\; dt\; =\; \backslash frac\; \backslash int\_\}\; \backslash frac\; \backslash frac\backslash ,\; dt\; =\; \backslash frac\; \backslash int\_\}\; n\backslash ,\; ds\backslash$ ''c'' is the speed of light in vacuum, ''ds'' an infinitesimal displacement along the ray, ''v'' = ''ds''/''dt'' the speed of light in a medium and ''n'' = ''c''/''v'' the refractive index of that medium, $t\_0$ is the starting time (the wave front is in A), $t\_1$ is the arrival time at B. The optical path length of a ray from a point A to a point B is defined by: :$S=\backslash int\_\}\; n\backslash ,\; ds\backslash$ and it is related to the travel time by ''S'' = ''cT''. The optical path length is a purely geometrical quantity since time is not considered in its calculation. An extremum in the light travel time between two points A and B is equivalent to an extremum of the optical path length between those two points. The historical form proposed by French mathematician Pierre de Fermat is incomplete. A complete modern statement of the variational Fermat principle is that In the context of calculus of variations this can be written as :$\backslash delta\; S=\; \backslash delta\backslash int\_\}\; n\; \backslash ,\; ds\; =0$ In general, the refractive index is a scalar field of position in space, that is, $n=n\backslash left(x\_1,x\_2,x\_3\backslash right)\; \backslash$ in 3D euclidean space. Assuming now that light has a component that travels along the ''x''3 axis, the path of a light ray may be parametrized as $s=\backslash left(x\_1\backslash left(x\_3\backslash right),x\_2\backslash left(x\_3\backslash right),x\_3\backslash right)\; \backslash$ and :$nds=n\; \backslash fracdx\_3=n\; \backslash sqrt\_2^2\}\; \backslash \; dx\_3$ where $\backslash dot\_k=dx\_k/dx\_3$. The principle of Fermat can now be written as :$\backslash delta\; S=\; \backslash delta\backslash int\_\}\; n\backslash left(x\_1,x\_2,x\_3\backslash right)\; \backslash sqrt\_2^2\}\backslash ,\; dx\_3$ :$=\; \backslash delta\backslash int\_\}\; L\backslash left(x\_1\backslash left(x\_3\backslash right),x\_2\backslash left(x\_3\backslash right),\backslash dot\_1\backslash left(x\_3\backslash right),\backslash dot\_2\backslash left(x\_3\backslash right),x\_3\backslash right)\backslash ,\; dx\_3=0$ which has the same form as Hamilton's principle but in which ''x''3 takes the role of time in classical mechanics. Function $L\backslash left(x\_1,x\_2,\backslash dot\_1,\backslash dot\_2,x\_3\backslash right)$ is the optical Lagrangian from which the Lagrangian and Hamiltonian (as in Hamiltonian mechanics) formulations of geometrical optics may be derived.〔Julio Chaves, ''Introduction to Nonimaging Optics'', CRC Press, 2008 (ISBN 978-1420054293)〕 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Fermat's principle」の詳細全文を読む スポンサード リンク 翻訳と辞書 : 翻訳のためのインターネットリソース Copyright(C) kotoba.ne.jp 1997-2016. All Rights Reserved.