This derivation makes use of the calculus of maximumminimum determination and the definitions of the triangle trig functions. Fermats principle article about fermats principle by. Suppose fermats last theorem is false, so there are a. This principle states in its simplest form that light waves of a given frequency traverse the path between two points which takes the least time. We show that fermats principle holds, and we derive an extended snells law. Fermats principle metadata this file contains additional information such as exif metadata which may have been added by the digital camera, scanner, or software program used to create or digitize it. Of course the straight line from a to b is the shortest time, but suppose it has a single reflection. Fermats principle of least time university of virginia. In fermats principle the optical path length between points a and b can be calculated in either direction, a to b or b to a, the result is the same. If a point source of light is switched on, the wavefront is an expanding sphere centered at the source. Fermats principle with calculus of variations physics forums. It is also described as the principle of least time in many books. Since light travels at different speeds through different media, the path of least time may not be a straight line.
It is usually presented erroneously as a least time principle. Fermats principle synonyms, fermats principle pronunciation, fermats principle translation, english dictionary definition of fermats principle. And, of course, in a homogeneous medium the leastdistance paths of the greek school were leasttime paths too, so all paths are leasttime paths. Fermat s principle says that the path taken between two points by a ray of light is the path that can be traversed in the least time it occurred to me today that maybe the path is actually the one that covers the smallest distance through spacetime, and that in flattish places the distance would be measured using the minkowski metric. Fermats principle definition of fermats principle by the. Wavesfermats principle wikibooks, open books for an. Fermats principle definition, the law that the path taken by a ray of light in going from one point to another point will be the path that requires the least time. Applied to reflection a b b since it is straight, the blue path is the shortest path from a to b. Fermats principle definition of fermats principle by. In order to be true in all cases, this statement must be weakened by replacing the least time with a time that is.
Light travels through the path in which it can reach the destination in least time. Let the mirror lie in the xz plane, and let p 1lie on the y axis at 0, yi, 0 and p2 in the xy plane at x2, y2, 0. Background and history of fermats little theorem fermats little theorem is stated as follows. In geometrical optics, we assume that the wavelike behaviour of light is. The principle of fermat or of maupertuis contains implicitly the entirety of geometrical optics. In fermat s principle the optical path length between points a and b can be calculated in either direction, a to b or b to a, the result is the same.
The leading thought throughout the derivation is illustrated in fig. We make use of the index of refraction, defined as ncv. Fermats method of infinite descent is an offshoot of mathematical induction that can be used to disprove statements. Mar 22, 2008 proof of snell s law using fermat s principle and the eulerlagrange equation duration. The results confirm the validity of fermats principle when the optical media and the boundaries between them are moving at relativistic speeds. How to get differential equation of fermat principle. Associated to elliptic curves and to certain modular forms are galois repre sentations. One of the most basic analogy is between the fermats principle in optics and the hamiltons principle of least action in classical mechanics. Any actual ray of light in an optical system, if reversed in direction, will retrace the same path backward.
Application of fermats principle to magnetic spectrometers csnsm. The optical length of a path rs taken by a ray of light in passing from point a to. However, some people state fermats little theorem as. Fermats principle states that light travels between two points along the path that requires the least time, as compared to other nearby paths. But much more important for the future of mathematics is the substantial progress wiles made toward the shimurataniyama conjecture. Fermat s principle definition, the law that the path taken by a ray of light in going from one point to another point will be the path that requires the least time. An alternate approach to geometrical optics can be developed from fermats principle. Fermats principle, also known as the principle of least time, is the link between ray optics and wave optics. An outline to the strategy of the proof has been given. Fermats principle and the laws of reflection and refraction scipp. Another way of stating this principle is that the path taken by a ray of light in traveling between two points requires either a minimum or a maximum time. A real optical path should not change even there is a virtual displacement. The path of the ray is such that the opticalpath length.
Radio waves travel at the speed of light, 300,000 km s. In the metaphor of climbing down a ladder, if a higher rung cannot be reached without first reaching a lower rung coupled with the notion that no lowest rung exists, then no rung can ever be reached. Fermat s principle is the main principle of quantum electrodynamics which states that any particle e. If p is a prime number and a is any other natural number not divisible by p, then the number is divisible by p. See the derivation section of the wikipedia article. Fermats principle of stationary ray paths is dual to huygens principle 1678 of. Radio waves travel at the speed of light, 300,000 kms. But it can be derived from a deeper theory, and thats worthwhile. The straight line ac would not take the least time, because the ray, in this case, would be spending a larger part of its course moving more slowly in the denser medium. Fermat s principle synonyms, fermat s principle pronunciation, fermat s principle translation, english dictionary definition of fermat s principle. When one supercube made up of unit cubes is subtracted from a. Fermats principle and snells law wolfram demonstrations. Fermats principle, also known as the principle of least time states that. Fermat s principle, also known as the principle of least time, is the link between ray optics and wave optics.
Snells law can be derived from this by setting the derivative of the time 0. Fermats principle of least time galileo and einstein. Fermat s principle metadata this file contains additional information such as exif metadata which may have been added by the digital camera, scanner, or software program used to create or digitize it. So fermat was able to work out that descartes new law could indeed be worked out from the least total time principle.
Fermats principle states that light travels between two points along the path. Fermats principle of least time was an elegant formulation of the laws of motion of this stream it reduced all observed deflections, etc. The optical path is defined as the actual path followed by the light ray in vacuum or in air medium. Obviously, it means that light may come from p to p1 along a. It is a special case of eulers theorem, and is important in applications of elementary number theory, including primality testing and publickey cryptography. Fermat s principle of least time was an elegant formulation of the laws of motion of this stream it reduced all observed deflections, etc. Fermats principle, in optics, statement that light traveling between two points seeks a path such that the number of waves the optical length between the points is equal, in the first approximation, to that in neighbouring paths. In each case, a detailed analysis of the obtained refraction formula is provided, and in the latter case, an intriguing backward refraction of light is noticed and thoroughly discussed. Aug 31, 2009 permission is granted to copy, distribute andor modify this document under the terms of the gnu free documentation license, version 1. Fermat s method of infinite descent is an offshoot of mathematical induction that can be used to disprove statements. This file is licensed under the creative commons attributionshare alike 4. Do we have a deeper understanding of fermat s principle.
In its original strong form, fermat s principle states that the path taken by a ray between two given points is the path that can be traversed in the least time. Fermats little theorem is a fundamental theorem in elementary number theory, which helps compute powers of integers modulo prime numbers. Fermat showed that the path of such a beam is given by the principle of least time. Pdf fermats principle and the formal equivalence of local lightray. Fermats principle is that the path traveled by light is the path of least travel time, an explicit description of time optimization. The result is called fermats little theorem in order to distinguish it from fermats last theorem. Given an objectpoint a in the air, and an observation point b in the water, the refraction point p is that which minimizes the time taken by the light to travel the path apb.
There are deep and subtle connections between number theory. Jul 25, 2014 show that fermat s principle implies that, on the actual path followed, q lies in the same vertical plane as p 1and p2 and obeys the law of reflection, that 0 1 02. Another way of stating this principle is that the path taken by a. The shimurataniyama conjecture is part of a more general philosophy. The modern formulation of fermats principle states that a light in going between two fixed points traverses an optical path length that is. Pdf fermats principle is considered as a unifying concept. Nov 29, 2010 applications of fermat s principlelet s imagine the situation where all the light emitted in point p is collected in another point p1.
Opticsfermats principle wikibooks, open books for an open. That is, within an y medium where the index of refraction is constan t, ligh t. Pdf software for fermats principle and lenses researchgate. Fermat s principle states that light travels between two points along the path that requires the least time, as compared to other nearby paths. Fermats principle says that the path taken between two points by a ray of light is the path that can be traversed in the least time it occurred to me today that maybe the path is actually the one that covers the smallest distance through spacetime, and that in flattish places the distance would be measured using the minkowski metric. The only case of fermats last theorem for which fermat actually wrote down a proof is for the case n 4. Fermat around 1660, fermats principle was taken as the most general law of the propagation of light. This leads to the principle of geometrical reversibility. The proof of the fermat s last theorem will be derived utilizing such a geometrical representation of integer numbers raised to an integer power. Aug 29, 2011 fermat s principle states that light travels between two points in such a way that the total time traveled is a minimum. Nigel boston university of wisconsin madison the proof.
Proof of snells law using fermats principle and the eulerlagrange equation duration. In particular, he is recognized for his discovery of an original method of finding the. Fermats principle establishes that the path taken by a light ray between 2 given points is such that the time that the light takes is the smallest possible. Wavesfermats principle wikibooks, open books for an open. Fermat s principle, in optics, statement that light traveling between two points seeks a path such that the number of waves the optical length between the points is equal, in the first approximation, to that in neighbouring paths. Fermats principle definition of fermats principle at. It is a fundamental law of optics from which the other laws of geometrical optics can be derived. The law of reflection can be derivedusing fermatsprinciple by solving the equation or. Fermats principle in the case of refraction of light at a flat surface between say air and water. Of course mark harrison is right, physical laws arent really proved. The proof of the fermats last theorem will be derived utilizing such a geometrical representation of integer numbers raised to an integer power. Calculus of variations, fermats principle physics forums. Apr 06, 2010 fermat s principle establishes that the path taken by a light ray between 2 given points is such that the time that the light takes is the smallest possible.
Fermat s principle, also known as the principle of least time states that. Fermats principle a fundamental principle of geometrical optics. Fermats principle of least time predicts refraction of. The wavelength of a radio wave received at 100 megahertz is. Fermat principle article about fermat principle by the free. Pdf we derive a formal description of local lightray rotation in terms of complex. Opticsfermats principle wikibooks, open books for an.
If correct, freys idea shows that no such a, b, c, and n can exist, and hence fermats last. Ray optics, fermats principle, and applications to. Mathematics stack exchange is a question and answer site for people studying math at any level and professionals in related fields. Fermats principle and the laws of reflection and refraction. A third top problem, flight time between two points, with auxiliary travel constraints, will also be considered in this paper. In particular, light travels a longer distance in the medium in which it has a higher speed. Fermats principle of least time states that light rays passing through different media follow the fastest and not the most direct path between two points, leading to refraction at medium borders. From fermats principle, one can derive a the law of reflection the angle of incidence is equal to the angle of reflection and b the law of refraction snells law. In a homogenous medium, ligh tra ys are rectilinear. Many of you are probably familiar with snells law from an optics unit in another course. In the metaphor of climbing down a ladder, if a higher rung cannot be reached without first reaching a lower rung coupled with the notion that. It may therefore be said that fermats principle is the principle of the minimum optical path.
Derivation of snells law using fermats principle youtube. The results confirm the validity of fermat s principle when the optical media and the boundaries between them are moving at relativistic speeds. Fermats principle of least time fermats principle of minimum time argues that light will travel from one point to another along a path that requires the minimum time. Fermat s principle is that the path traveled by light is the path of least travel time, an explicit description of time optimization. From fermat s principle, one can derive a the law of reflection the angle of incidence is equal to the angle of reflection and b the law of refraction snells law. An alternate approach to geometrical optics can be developed from fermat s principle. The solving of fermats last theorem uci mathematics. Fermats principle rests on observation, not on deeper theory. Applications of fermats principlelets imagine the situation where all the light emitted in point p is collected in another point p1.
The reader is assumed to have some basic knowledge of general relativity and some familiarity with differential geometry. In its original strong form, fermats principle states that the path taken by a ray between two given points is the path that can be traversed in the least time. As in classical mechanics, we can use either the lagrangian or the hamiltonian formulations to further study properties of light ray propagation. Fermats principle to the problem of focusing in lenses. The faces on the boundaries of a supercube, not in contact with the other faces, will be referred to as the outer faces. Fermats principle of least time predicts refraction of ant.
Fermat principle article about fermat principle by the. A counterexample to fermat s last theorem would yield an elliptic curve freys curve with remarkable proper ties. In its original strong form, fermats principle states. Fermats principle states that light travels between two points in such a way that the total time traveled is a minimum. Fermats principle of least time in the presence of uniformly moving. It even included phenomena caused by a variable refractive index, and consequent curved paths for light rays, such as mirages, reflections of distant mountains in the middle.
1347 1214 1491 166 402 948 199 384 1522 419 508 496 528 793 1424 778 1653 39 436 20 1618 392 994 179 320 593 1573 1159 863 1551 251 1646 881 1441 108 256 685 389 848 977