|
|
| (One intermediate revision by one other user not shown) |
| Line 1: |
Line 1: |
| [[Image:Snells law.svg|thumb|250px|Fermat's principle leads to [[Snell's law]]; when the sines of the angles in the different media are in the same proportion as the propagation velocities, the time to get from P to Q is minimized.]]
| | Role Playing Games, or RPGs for short, are the most [https://play.google.com/store/apps/details?id=com.animecartoon.games.swing.wizards addicting games] to play, be it on consoles like the Xbox 360, Playstation 3, Nintendo Wii, or using the pc. Most RPGs are mounted in fantasy worlds, but work with games have reached sci-fi or alternative reality settings. The look at these top 10 RPGs recently!<br><br>Dealer's Room - Centrally located in the convention hall, new attendees often get so distracted by games and events the player miss this room the first year or less. This is a shame, since this is one of the highest quality features in the conventions. Hundreds of dealers fill the floor, including within a dozen major game manufacturers. Every major game company holds free demos all weekend long and achievable play lots of new too unreleased games at at no cost. You should find at least four hours during the convention to visit this hotel room.<br><br><br><br>Bubble Trouble 2- provide you . one from the most popular online addicting game with regards to of arcade games. Within the arcade games, the most common played games are ski ball machines, street fighter machines, and Pac-Man. These games are popular associated with its simplicity yet very fun step already started the game and is actually why perfect for the newbie.<br><br>Let's get all apparent and most well-liked sites outside of the way fundamental. Unless you have been living under a rock in the past five years you know online video giant Yt. Youtube does have filters for videos which is often inappropriate much more under the age of 18. Many videos which flagged can always be inappropriate for children. I strongly suggest parental supervision for any video webpage. For adults with a liking for extreme videos then Ebaums World may be your new favorite website. Ebaums world offers videos, prank calls, and animations from gross to hilarious. To buy list of video websites check out Video Choice.<br><br>The game of roulette amongst the of the oldest really addicting games in casinos. Learning the game is easy which means anyone become skilled in the game quickly and start having amusing. If you want start learning the game, you don't have to drive to did find casino inside your neighbourhood because online casinos provide the mechanics in the game. Create to the fun, internet casinos offer free roulette about the. Playing the game for free means simply have to register your name and you given play money may can practice on how to play recreation.<br><br>The subsequent few internet sites had been quite connected with the prior sites so i skipped a couple of pages and found another excellent site. These pages states that you can download the game titles but you do not have to down load them to play them. It has a regarding video games, most of such are based on common cartoons and Disney movies. This website is very worthwhile to young children and kids. It even has classic video games like Clue, Yahtzee, Bejeweled! I any whole regarding enjoyable playing Mystery PI New York, Sweeney Todd, and Wonderful secrets Da Vinci obviously dare say grownups will adore this website also! Back again . the your own domain name just before, you do not possess to download something or sign anywhere up to use this incredible movie site.<br><br>The second screenshot shows the game in action is a passing room. The passing room is played just about the same but don't click with the acorns through to the last ball has been revealed. Players in these games love to get multiple bingos fast and this is why you save the acorns until last ball. |
| | |
| In [[optics]], '''[[Pierre de Fermat|Fermat]]'s principle''' or the '''principle of least time''' is the principle that the path taken between two points by a [[ray (optics)|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.<ref>Arthur Schuster, ''An Introduction to the Theory of Optics'', London: Edward Arnold, 1904 [http://books.google.com/books?vid=OCLC03146755&id=X0AcBd-bcCwC&pg=PA41&lpg=PA41&dq=fermat%27s-principle online].</ref> 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.<ref>{{Citation
| |
| |last = Ghatak
| |
| |first = Ajoy
| |
| |year = 2009
| |
| |title = Optics
| |
| |edition = 4th
| |
| |isbn = 0-07-338048-2
| |
| }}
| |
| </ref> 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.<ref>{{Citation|last=Feynman|first=Richard|title=The Feynman Lectures on Physics, Vol. 1|pages=26–7}}</ref>
| |
| | |
| Fermat's principle can be used to describe the properties of light rays [[Reflection (physics)|reflected]] off mirrors, [[refraction|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]]<ref>{{citation
| |
| | last1 = Katz | first1 = Mikhail G.
| |
| | author1-link = Mikhail Katz
| |
| | last2 = Schaps | first2 = David
| |
| | last3 = Shnider | first3 = Steve
| |
| | author3-link = Steve Shnider
| |
| | arxiv = 1210.7750
| |
| | doi =
| |
| | issue = 3
| |
| | journal = [[Perspectives on Science]]
| |
| | pages = 7750
| |
| | title = Almost Equal: The Method of Adequality from Diophantus to Fermat and Beyond
| |
| | volume = 21
| |
| | year = 2013|bibcode = 2012arXiv1210.7750K
| |
| }}</ref> 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:
| |
| | |
| :<math>T=\int_{\mathbf{A}}^{\mathbf{B}} \, dt = \frac{1}{c} \int_{\mathbf{A}}^{\mathbf{B}} \frac{c}{v} \frac{ds}{dt}\, dt = \frac{1}{c} \int_{\mathbf{A}}^{\mathbf{B}} n\, ds\ </math>
| |
| | |
| ''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. The optical path length of a ray from a point '''A''' to a point '''B''' is defined by:
| |
| | |
| :<math>S=\int_{\mathbf{A}}^{\mathbf{B}} n\, ds\ </math>
| |
| | |
| 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 {{quote|the optical length of the path followed by light between two fixed points, '''A''' and '''B''', is an extremum. The optical length is defined as the physical length multiplied by the refractive index of the material."<ref>R. Marques, F. Martin, and M. Sorolla. Metamaterials with Negative Parameters. Wiley, 2008.</ref>}} In the context of [[calculus of variations]] this can be written as
| |
| | |
| :<math>\delta S= \delta\int_{\mathbf{A}}^{\mathbf{B}} n \, ds =0 </math>
| |
| | |
| In general, the refractive index is a [[scalar field]] of position in space, that is, <math>n=n\left(x_1,x_2,x_3\right) \ </math> in [[Three-dimensional space|3D]] [[euclidean space]]. Assuming now that light has a component that travels along the ''x''<sub>3</sub> axis, the path of a light ray may be parametrized as <math>s=\left(x_1\left(x_3\right),x_2\left(x_3\right),x_3\right) \ </math> and
| |
| | |
| :<math>nds=n \frac{\sqrt{dx_1^2+dx_2^2+dx_3^2}}{dx_3}dx_3=n \sqrt{1+\dot{x}_1^2+\dot{x}_2^2} \ dx_3</math>
| |
| | |
| where <math>\dot{x}_k=dx_k/dx_3</math>. The principle of Fermat can now be written as
| |
| | |
| :<math>\delta S= \delta\int_{x_{3A}}^{x_{3B}} n\left(x_1,x_2,x_3\right) \sqrt{1+\dot{x}_1^2+\dot{x}_2^2}\, dx_3</math>
| |
| :<math>= \delta\int_{x_{3A}}^{x_{3B}} L\left(x_1\left(x_3\right),x_2\left(x_3\right),\dot{x}_1\left(x_3\right),\dot{x}_2\left(x_3\right),x_3\right)\, dx_3=0</math>
| |
| | |
| which has the same form as [[Hamilton's principle]] but in which ''x''<sub>3</sub> takes the role of time in [[classical mechanics]]. Function <math>L\left(x_1,x_2,\dot{x}_1,\dot{x}_2,x_3\right)</math> is the optical [[Lagrangian]] from which the Lagrangian and Hamiltonian (as in [[Hamiltonian mechanics]]) formulations of geometrical optics may be derived.<ref name="IntroductionNIO">Julio Chaves, ''Introduction to Nonimaging Optics'', CRC Press, 2008 (ISBN 978-1420054293)</ref>
| |
| | |
| ==Derivation==
| |
| | |
| Classically, Fermat's principle can be considered as a mathematical consequence of [[Huygens' principle]]. Indeed, of all secondary waves (along all possible paths) the waves with the extrema (stationary) paths contribute most due to constructive interference. Supposing that light waves propagate from A to B by all possible routes AB<sub>j</sub>, unrestricted initially by rules of geometrical or physical optics. The various optical paths AB<sub>j</sub> will vary by amounts greatly in excess of one wavelength, and so the waves arriving at B will have a large range of phases and will tend to interfere destructively. But if there is a shortest route AB<sub>0</sub>, and the optical path varies smoothly through it, then a considerable number of neighboring routes close to AB<sub>0</sub> will have optical paths differing from AB<sub>0</sub> by second-order amounts only and will therefore interfere constructively. Waves along and close to this shortest route will thus dominate and AB<sub>0</sub> will be the route along which the light is seen to travel.<ref>Ariel Lipson, Stephen G. Lipson, Henry Lipson, ''Optical Physics 4th Edition'', Cambridge University Press, ISBN 978-0-521-49345-1.</ref>
| |
| | |
| Fermat's principle is the main principle of [[quantum electrodynamics]] where it states that any particle (e.g. a photon or an electron) propagates over all available (unobstructed) paths and the interference (sum, or superposition) of its wavefunction over all those paths (at the point of observer or detector) gives the correct probability of detection of this particle (at this point). Thus the extremal (shortest, longest or stationary) paths contribute into this interference most as they can not be completely canceled out.
| |
| | |
| In the [[classic mechanics]] of [[waves]], Fermat's principle follows from the [[extremum principle of mechanics]] (see [[variational principle]]).
| |
| | |
| ==History==
| |
| [[Hero of Alexandria]] (Heron) (c. 60) described a principle of reflection, which stated that a ray of light that goes from point A to point B, suffering any number of reflections on flat mirrors, in the same medium, has a smaller path length than any nearby path.<ref>History of Geometric Optics/Richard Fitzpatrick</ref>
| |
| | |
| [[Ibn al-Haytham]] (Alhacen), in his ''[[Book of Optics]]'' (1021), expanded the principle to both reflection and refraction, and expressed an early version of the principle of least time. His experiments were based on earlier works on refraction carried out by the Greek scientist [[Ptolemy]]<ref>Pavlos Mihas (2005). [http://www.ihpst2005.leeds.ac.uk/papers/Mihas.pdf Use of History in Developing ideas of refraction, lenses and rainbow], Demokritus University, Thrace, Greece.</ref>
| |
| [[File:Pierre de Fermat2.png|thumb|140px|Pierre de Fermat]]
| |
| The generalized principle of least time in its modern form was stated by Fermat in a letter dated January 1, 1662, to Cureau de la Chambre.<ref>[[Michael Sean Mahoney]], ''The Mathematical Career of Pierre de Fermat, 1601-1665'', 2nd edition (Princeton University Press, 1994), p. 401</ref> It was met with objections made in May 1662 by Claude Clerselier, an expert in optics and leading spokesman for the Cartesians at that time. Amongst his objections, Clerselier states:
| |
| | |
| <blockquote>
| |
| ''... Fermat's principle can not be the cause, for otherwise we would be attributing knowledge to nature: and here, by nature, we understand only that order and lawfulness in the world, such as it is, which acts without foreknowledge, without choice, but by a necessary determination.''
| |
| </blockquote>
| |
| | |
| The original French, from Mahoney, is as follows: | |
| | |
| <blockquote>
| |
| ''Le principe que vous prenez pour fondement de votre démonstration, à savoir que la nature agit toujours par les voies les plus courtes et les plus simples, n’est qu’un principe moral et non point physique, qui n’est point et qui ne peut être la cause d’aucun effet de la nature.'' | |
| </blockquote>
| |
| | |
| Indeed Fermat's principle does not hold standing alone, we now know it can be derived from earlier principles such as [[Huygens' principle]]. <br />
| |
| Historically, Fermat's principle has served as a guiding principle in the formulation of physical laws with the use of variational calculus (see [[Principle of least action]]).
| |
| | |
| ==See also==
| |
| *[[Adequality]]
| |
| *[[Eikonal equation]]
| |
| *[[Fermat’s and energy variation principles in field theory]]
| |
| *[[Geodesic]]
| |
| *[[Hamilton's principle]]
| |
| *[[Huygens' principle]]
| |
| *[[Path integral formulation]]
| |
| *[[Pierre de Fermat]]
| |
| *[[Principle of least action]]
| |
| | |
| ==Notes==
| |
| <references/>
| |
| | |
| {{DEFAULTSORT:Fermat's Principle}}
| |
| [[Category:Geometrical optics]]
| |
| [[Category:Calculus of variations]]
| |
| [[Category:Principles]]
| |
Role Playing Games, or RPGs for short, are the most addicting games to play, be it on consoles like the Xbox 360, Playstation 3, Nintendo Wii, or using the pc. Most RPGs are mounted in fantasy worlds, but work with games have reached sci-fi or alternative reality settings. The look at these top 10 RPGs recently!
Dealer's Room - Centrally located in the convention hall, new attendees often get so distracted by games and events the player miss this room the first year or less. This is a shame, since this is one of the highest quality features in the conventions. Hundreds of dealers fill the floor, including within a dozen major game manufacturers. Every major game company holds free demos all weekend long and achievable play lots of new too unreleased games at at no cost. You should find at least four hours during the convention to visit this hotel room.
Bubble Trouble 2- provide you . one from the most popular online addicting game with regards to of arcade games. Within the arcade games, the most common played games are ski ball machines, street fighter machines, and Pac-Man. These games are popular associated with its simplicity yet very fun step already started the game and is actually why perfect for the newbie.
Let's get all apparent and most well-liked sites outside of the way fundamental. Unless you have been living under a rock in the past five years you know online video giant Yt. Youtube does have filters for videos which is often inappropriate much more under the age of 18. Many videos which flagged can always be inappropriate for children. I strongly suggest parental supervision for any video webpage. For adults with a liking for extreme videos then Ebaums World may be your new favorite website. Ebaums world offers videos, prank calls, and animations from gross to hilarious. To buy list of video websites check out Video Choice.
The game of roulette amongst the of the oldest really addicting games in casinos. Learning the game is easy which means anyone become skilled in the game quickly and start having amusing. If you want start learning the game, you don't have to drive to did find casino inside your neighbourhood because online casinos provide the mechanics in the game. Create to the fun, internet casinos offer free roulette about the. Playing the game for free means simply have to register your name and you given play money may can practice on how to play recreation.
The subsequent few internet sites had been quite connected with the prior sites so i skipped a couple of pages and found another excellent site. These pages states that you can download the game titles but you do not have to down load them to play them. It has a regarding video games, most of such are based on common cartoons and Disney movies. This website is very worthwhile to young children and kids. It even has classic video games like Clue, Yahtzee, Bejeweled! I any whole regarding enjoyable playing Mystery PI New York, Sweeney Todd, and Wonderful secrets Da Vinci obviously dare say grownups will adore this website also! Back again . the your own domain name just before, you do not possess to download something or sign anywhere up to use this incredible movie site.
The second screenshot shows the game in action is a passing room. The passing room is played just about the same but don't click with the acorns through to the last ball has been revealed. Players in these games love to get multiple bingos fast and this is why you save the acorns until last ball.