Fermat least time principle

Fundamental to the understanding of optics is the Principle of Least Time, developed by Fermat ( 1650), which says that light will take the path that... 

Use Lagrange s method of multipliers to derive the law of refraction of light from Fermat s principle of least time between two fixed points. 

To develop a system of mechanics from here without the introduction of any other concepts, apart from energy, some general principle that predicts the course of a mechanical change is required. This could be like the Maupertuis principle of least action or Fermat s principle of least time. It means that the actual path of the change will have an extreme value e.g. minimum) of either action or time, compared to all other possible paths. Based on considerations like these Hamilton formulated the principle that the action integral... 

The law of geometrical optics anticipated by Hero of Alexandria was formulated by Fermat (1601-1655) as a principle of least time, consistent with Snell s law of refraction (1621). The time for phase transmission from point P to point Q along a path x(r) is given by... 

The above discussion implicitly obeys Huygens 97 principle, that each point on a spherical wavefront can be regarded as the source of a secondary wavelet (another spherical wave), as well as Fermat s98 principle of least time. 

Not only mechanics but all of physics can be derived from the principle of least action. There are appropriate Lagrangian functions for electrodynamics, quantum mechanics, hydrodynamics, etc., which all allow us to derive the basic equations of the respective discipline from the principle of least action. In this sense, the principle of least action is the most powerful economy principle known in physics since it is sufficient to know the principle of least action, and the rest can be derived. Nature as a whole seems to be organized according to this principle. The principle of least action can be found under various names in nearly every branch of science. For instance the principle of least cost in economy or Fermat s principle of least time in optics. 

Not long after the formulation of Snell s law, the French mathematician Pierre de Fermat unified the laws of reflection and refraction by showing that both could be deduced from the hypotheses that light travels a path of least time. In other words, given two points A and B in a region with mirrors or with different media, the path of a ray of light from point A to point B will be that for which the time of travel is least. The implication of Fermat s principle is that light travels at a finite speed. ... 

Fermat principle The path taken by a ray of light between any two points in a system is always the path that takes the least time. This principle leads to the law of the rectilinear propagation of light and the laws of reflection and refraction. It was discovered by the French mathematician, Pierre de Fermat (1601-65). 

Fermat De Pierre (1601—1665) Fr. math., devised principle of least time (action) and Fermat s small and big (last) theorems, gravity reciprocal attraction, father of modem theory of numbers, probabilities Feynman Richard Phillips (1918-1988) US. phys., quantum electrodynamics, devised Feynman diagrams as means for accounting possible particle transformations ( Theory of Fundamental Processes 1961)... 

Fermat s principle The path followed by light (or other waves) passing through any collection of media from one specified point to another, is that path for which the time of travel is least. 

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