# Fermat's principle

In optics, Fermat's principle or the principle of least time, named after French mathematician Pierre de Fermat, 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.[1] 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.[2] 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[3] and the law of reflection.

Fermat's principle has the same form as Hamilton's principle and it is the basis of Hamiltonian optics.

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.

## 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:

${\displaystyle T=\int _{\mathbf {t_{0}} }^{\mathbf {t_{1}} }\,dt={\frac {1}{c}}\int _{\mathbf {t_{0}} }^{\mathbf {t_{1}} }{\frac {c}{v}}{\frac {ds}{dt}}\,dt={\frac {1}{c}}\int _{\mathbf {A} }^{\mathbf {B} }n\,ds}$

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, ${\displaystyle t_{0}}$ is the starting time (the wave front is in A), ${\displaystyle t_{1}}$ is the arrival time at B. The optical path length of a ray from a point A to a point B is defined by:

${\displaystyle S=\int _{\mathbf {A} }^{\mathbf {B} }n\,ds}$

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 Fermat is incomplete. A complete modern statement of the variational Fermat principle is that

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."[4]

In the context of calculus of variations this can be written as

${\displaystyle \delta S=\delta \int _{\mathbf {A} }^{\mathbf {B} }n\,ds=0}$

In general, the refractive index is a scalar field of position in space, that is, ${\displaystyle n=n\left(x_{1},x_{2},x_{3}\right)}$ in 3D euclidean space. Assuming now that light has a component that travels along the x3 axis, the path of a light ray may be parametrized as ${\displaystyle s=\left(x_{1}\left(x_{3}\right),x_{2}\left(x_{3}\right),x_{3}\right)}$ and

${\displaystyle nds=n{\frac {\sqrt {dx_{1}^{2}+dx_{2}^{2}+dx_{3}^{2}}}{dx_{3}}}dx_{3}=n{\sqrt {1+{\dot {x}}_{1}^{2}+{\dot {x}}_{2}^{2}}}\ dx_{3}}$

where ${\displaystyle {\dot {x}}_{k}=dx_{k}/dx_{3}}$. The principle of Fermat can now be written as

${\displaystyle \delta S=\delta \int _{x_{3A}}^{x_{3B}}n\left(x_{1},x_{2},x_{3}\right){\sqrt {1+{\dot {x}}_{1}^{2}+{\dot {x}}_{2}^{2}}}\,dx_{3}}$
${\displaystyle =\delta \int _{x_{3A}}^{x_{3B}}L\left(x_{1}\left(x_{3}\right),x_{2}\left(x_{3}\right),{\dot {x}}_{1}\left(x_{3}\right),{\dot {x}}_{2}\left(x_{3}\right),x_{3}\right)\,dx_{3}=0}$

which has the same form as Hamilton's principle but in which x3 takes the role of time in classical mechanics. Function ${\displaystyle L\left(x_{1},x_{2},{\dot {x}}_{1},{\dot {x}}_{2},x_{3}\right)}$ is the optical Lagrangian from which the Lagrangian and Hamiltonian (as in Hamiltonian mechanics) formulations of geometrical optics may be derived.[5]

## 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 extremal (stationary) paths contribute most due to constructive interference. Suppose that light waves propagate from A to B by all possible routes ABj, unrestricted initially by rules of geometrical or physical optics. The various optical paths ABj 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 AB0, and the optical path varies smoothly through it, then a considerable number of neighboring routes close to AB0 will have optical paths differing from AB0 by second-order amounts only and will therefore interfere constructively. Waves along and close to this shortest route will thus dominate and AB0 will be the route along which the light is seen to travel.[6]

Fermat's principle is the main principle of quantum electrodynamics which states that any particle (e.g. a photon or an electron) propagates over all available, unobstructed paths and that the interference, or superposition, of its wavefunction over all those paths at the point of observation gives the probability of detecting the particle at this point. Thus, because the extremal paths (shortest, longest, or stationary) cannot be completely canceled out, they contribute most to this interference. In humans, for example, Fermat's principle can be demonstrated in a situation when a lifeguard has to find the fastest way to traverse both beach and water in order to reach a drowning swimmer.[7] The principle has been tested in studies with ants, in which the ants' nest is on one end of a container and food is on the opposite end, but the ants choose to follow the path of least time, rather than the most direct path.[8]

In the classic mechanics of waves, Fermat's principle follows from the extremum principle of mechanics (see variational principle).

## History

Euclid, c. 320 BCE in his Catoptrics (on mirrors, including spherical mirrors) and Optics, laid the foundations for reflection, which was repeated by Ptolemy, and then in his more detailed books that have surfaced, Hero of Alexandria (Heron) (c. 60) described the 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.[9]

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.[10]

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.[11] It was met with objections by Claude Clerselier in May 1662, an expert in optics and leading spokesman for the Cartesians at the time. Amongst his objections, Clerselier states:

... The principle which you take as the basis for your proof, namely that Nature always acts by using the simplest and shortest paths, is merely a moral, and not a physical one. It is not, and cannot be, the cause of any effect in Nature.

The original French, from Mahoney, is as follows:

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.

Although Fermat's principle does not hold standing alone, we now know it can be derived from earlier principles such as Huygens' principle.

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).

## Notes

1. ^ Arthur Schuster, An Introduction to the Theory of Optics, London: Edward Arnold, 1904 online.
2. ^ Ghatak, Ajoy (2009), Optics (4th ed.), ISBN 0-07-338048-2
3. ^ Katz, Mikhail G.; Schaps, David; Shnider, Steve (2013), "Almost Equal: The Method of Adequality from Diophantus to Fermat and Beyond", Perspectives on Science, 21 (3): 7750, arXiv:1210.7750, Bibcode:2012arXiv1210.7750K
4. ^ R. Marques, F. Martin, and M. Sorolla. Metamaterials with Negative Parameters. Wiley, 2008.
5. ^ Chaves, Julio (2015). Introduction to Nonimaging Optics, Second Edition. CRC Press. ISBN 978-1482206739.
6. ^ Ariel Lipson, Stephen G. Lipson, Henry Lipson, Optical Physics 4th Edition, Cambridge University Press, ISBN 978-0-521-49345-1.
7. ^ Aatish Bhatia (24 March 2014). "To Save Drowning People, Ask Yourself "What Would Light Do?"". Nautilus. Retrieved 11 July 2016.
8. ^ Lisa Zyga (1 April 2013). "Ants follow Fermat's principle of least time". Phys.org. Retrieved 11 July 2016.
9. ^ History of Geometric Optics/Richard Fitzpatrick
10. ^ Pavlos Mihas (2005). Use of History in Developing ideas of refraction, lenses and rainbow Archived 2007-09-27 at the Wayback Machine, Demokritus University, Thrace, Greece.
11. ^ Michael Sean Mahoney, The Mathematical Career of Pierre de Fermat, 1601-1665, 2nd edition (Princeton University Press, 1994), p. 401
Abraham–Minkowski controversy

The Abraham–Minkowski controversy is a physics debate concerning electromagnetic momentum within dielectric media. Traditionally, it is argued that in the presence of matter the electromagnetic stress-energy tensor by itself is not conserved (divergenceless). Only the total stress-energy tensor carries unambiguous physical significance, and how one apportions it between an "electromagnetic" part and a "matter" part depends on context and convenience. In other words, the electromagnetic part and the matter part in the total momentum can be arbitrarily distributed as long as the total momentum is kept the same. There are two incompatible equations to describe momentum transfer between matter and electromagnetic fields. These two equations were first suggested by Hermann Minkowski (1908) and Max Abraham (1909), from which the controversy's name derives. Both were claimed to be supported by experimental data. Theoretically, it is usually argued that Abraham's version of momentum "does indeed represent the true momentum density of electromagnetic fields" for electromagnetic waves,

while Minkowski's version of momentum is "pseudomomentum" or "wave momentum".Several papers have now claimed to have resolved this controversy; e.g., a team from the Aalto University argues that the photon EM field induces a dipole in the medium, where the dipole moment causes the medium atoms to bunch, creating a mass density wave. The EM field carries the Abraham momentum and the combined EM field and mass density wave carries momentum equal to the Minkowski momentum. However, a recent study

argues that the physical model set up by the team is not consistent with Einstein's special relativity; and the study further argues that (i) momentum-energy conservation law is consistent with but not included in Maxwell equations, and as a result, the momentum and energy of light in a medium cannot be uniquely defined within the Maxwell-equations frame; (ii) the momentum and energy of a non-radiation field is not measurable experimentally, because the non-radiation field cannot exist independently of the materials which support it, just like one cannot experimentally determine the momentum and energy of the EM field carried by a free electron in free space. In other words, the non-radiation field is the component of the material subsystem, instead of the EM subsystem. This conclusion is apparently supported by Compton photon-electron scattering experiment.The Abraham-Minkowski controversy also has inspired various theories proposing the existence of reactionless drives.

Calculus of variations

Calculus of variations is a field of mathematical analysis that uses variations, which are small changes in functions

and functionals, to find maxima and minima of functionals: mappings from a set of functions to the real numbers. Functionals are often expressed as definite integrals involving functions and their derivatives. Functions that maximize or minimize functionals may be found using the Euler–Lagrange equation of the calculus of variations.

A simple example of such a problem is to find the curve of shortest length connecting two points. If there are no constraints, the solution is obviously a straight line between the points. However, if the curve is constrained to lie on a surface in space, then the solution is less obvious, and possibly many solutions may exist. Such solutions are known as geodesics. A related problem is posed by Fermat's principle: light follows the path of shortest optical length connecting two points, where the optical length depends upon the material of the medium. One corresponding concept in mechanics is the principle of least action.

Many important problems involve functions of several variables. Solutions of boundary value problems for the Laplace equation satisfy the Dirichlet principle. Plateau's problem requires finding a surface of minimal area that spans a given contour in space: a solution can often be found by dipping a frame in a solution of soap suds. Although such experiments are relatively easy to perform, their mathematical interpretation is far from simple: there may be more than one locally minimizing surface, and they may have non-trivial topology.

Fermat's theorem

The works of the 17th-century mathematician Pierre de Fermat engendered many theorems. Fermat's theorem may refer to one of the following theorems:

Fermat's Last Theorem, about integer solutions to an + bn = cn

Fermat's little theorem, a property of prime numbers

Fermat's theorem on sums of two squares, about primes expressible as a sum of squares

Fermat's theorem (stationary points), about local maxima and minima of differentiable functions

Fermat's principle, about the path taken by a ray of light

Fermat polygonal number theorem, about expressing integers as a sum of polygonal numbers

Fermat’s and energy variation principles in field theory

In general relativity the light is assumed to propagate in the vacuum along null geodesic in a pseudo-Riemannian manifold. Besides the geodesics principle in a classical field theory there exists the Fermat's principle for stationary gravity fields.

Gradient-index (GRIN) optics is the branch of optics covering optical effects produced by a gradient of the refractive index of a material. Such gradual variation can be used to produce lenses with flat surfaces, or lenses that do not have the aberrations typical of traditional spherical lenses. Gradient-index lenses may have a refraction gradient that is spherical, axial, or radial.

Hamilton's optico-mechanical analogy

Hamilton's optico-mechanical analogy is a concept of classical physics enunciated by William Rowan Hamilton. It may be viewed as linking Huygens' principle of optics with Jacobi's Principle of mechanics.According to Cornelius Lanczos, the analogy has been important in the development of ideas in quantum physics. According to Erwin Schrödinger, for micromechanical motions, the Hamiltonian analogy of mechanics to optics is inadequate to treat diffraction, which requires it to be extended to a vibratory wave equation in configuration space.

Hamiltonian optics

Hamiltonian optics and Lagrangian optics are two formulations of geometrical optics which share much of the mathematical formalism with Hamiltonian mechanics and Lagrangian mechanics.

List of scientific laws named after people

This is a list of scientific laws named after people (eponymous laws). For other lists of eponyms, see eponym.

List of things named after Pierre de Fermat

This is a list of things named after Pierre de Fermat, a French amateur mathematician.

Fermat–Apollonius circle

Fermat–Catalan conjecture

Fermat cubic

Fermat curve

Fermat's frog

Fermat number

Fermat point

Fermat polygonal number theorem

Fermat polynomial

Fermat primality test

Fermat Prize

Fermat pseudoprime

Fermat quintic threefold

Fermat quotient

Fermat's difference quotient

Fermat's factorization method

Fermat's last theorem

Fermat's little theorem

Fermat's principle

Fermat's spiral

Fermat's theorem (stationary points)

Fermat's theorem on sums of two squares

List of variational topics

This is a list of variational topics in from mathematics and physics. See calculus of variations for a general introduction.

Action (physics)

Averaged Lagrangian

Brachistochrone curve

Calculus of variations

Catenoid

Cycloid

Dirichlet principle

Euler–Lagrange equation cf. Action (physics)

Fermat's principle

Functional (mathematics)

Functional derivative

Functional integral

Geodesic

Isoperimetry

Lagrangian

Lagrangian mechanics

Legendre transformation

Luke's variational principle

Minimal surface

Morse theory

Noether's theorem

Path integral formulation

Plateau's problem

Prime geodesic

Principle of least action

Soap bubble

Soap film

Tautochrone curve

Maupertuis's principle

In classical mechanics, Maupertuis's principle (named after Pierre Louis Maupertuis), states that the path followed by a physical system is the one of least length (with a suitable interpretation of path and length). It is a special case of the more generally stated principle of least action. Using the calculus of variations, it results in an integral equation formulation of the equations of motion for the system.

Optical path length

In optics, optical path length (OPL) or optical distance is the product of the geometric length of the path followed by light through a given system, and the index of refraction of the medium through which it propagates. In many textbooks, it is symbolically written as Λ. A difference in optical path length between two paths is often called the optical path difference (OPD). OPL and OPD are important because they determine the phase of the light and governs interference and diffraction of light as it propagates.

Pierre de Fermat

Pierre de Fermat (French: [pjɛːʁ də fɛʁma]) (between 31 October and 6 December 1607 – 12 January 1665) was a French lawyer at the Parlement of Toulouse, France, and a mathematician who is given credit for early developments that led to infinitesimal calculus, including his technique of adequality. In particular, he is recognized for his discovery of an original method of finding the greatest and the smallest ordinates of curved lines, which is analogous to that of differential calculus, then unknown, and his research into number theory. He made notable contributions to analytic geometry, probability, and optics. He is best known for his Fermat's principle for light propagation and his Fermat's Last Theorem in number theory, which he described in a note at the margin of a copy of Diophantus' Arithmetica.

Principle of least action

This article discusses the history of the principle of least action. For the application, please refer to action (physics).The principle of least action – or, more accurately, the principle of stationary action – is a variational principle that, when applied to the action of a mechanical system, can be used to obtain the equations of motion for that system. In relativity, a different action must be minimized or maximized. The principle can be used to derive Newtonian, Lagrangian and Hamiltonian equations of motion, and even general relativity (see Einstein–Hilbert action). The physicist Paul Dirac, and after him Julian Schwinger and Richard Feynman, demonstrated how this principle can also be used in quantum calculations.

It was historically called "least" because its solution requires finding the path that has the least value. Its classical mechanics and electromagnetic expressions are a consequence of quantum mechanics, but the stationary action method helped in the development of quantum mechanics.The principle remains central in modern physics and mathematics, being applied in thermodynamics, fluid mechanics, the theory of relativity, quantum mechanics, particle physics, and string theory and is a focus of modern mathematical investigation in Morse theory. Maupertuis' principle and Hamilton's principle exemplify the principle of stationary action.

The action principle is preceded by earlier ideas in optics. In ancient Greece, Euclid wrote in his Catoptrica that, for the path of light reflecting from a mirror, the angle of incidence equals the angle of reflection. Hero of Alexandria later showed that this path was the shortest length and least time.Scholars often credit Pierre Louis Maupertuis for formulating the principle of least action because he wrote about it in 1744 and 1746. However, Leonhard Euler discussed the principle in 1744, and evidence shows that Gottfried Leibniz preceded both by 39 years.In 1933, Paul Dirac discerned the quantum mechanical underpinning of the principle in the quantum interference of amplitudes.

Principle of least effort

The principle of least effort is a broad theory that covers diverse fields from evolutionary biology to webpage design. It postulates that animals, people, even well-designed machines will naturally choose the path of least resistance or "effort". It is closely related to many other similar principles: see Principle of least action or other articles listed below. This is perhaps best known or at least documented among researchers in the field of library and information science. Their principle states that an information-seeking client will tend to use the most convenient search method, in the least exacting mode available. Information seeking behavior stops as soon as minimally acceptable results are found. This theory holds true regardless of the user's proficiency as a searcher, or their level of subject expertise. Also, this theory takes into account the user’s previous information-seeking experience. The user will use the tools that are most familiar and easy to use that find results. The principle of least effort is known as a “deterministic description of human behavior”. The principle of least effort applies not only in the library context, but also to any information-seeking activity. For example, one might consult a generalist co-worker down the hall rather than a specialist in another building, so long as the generalist's answers were within the threshold of acceptability.

The principle of least effort is analogous to the path of least resistance.

Snell's law

Snell's law (also known as Snell–Descartes law and the law of refraction) is a formula used to describe the relationship between the angles of incidence and refraction, when referring to light or other waves passing through a boundary between two different isotropic media, such as water, glass, or air.

In optics, the law is used in ray tracing to compute the angles of incidence or refraction, and in experimental optics to find the refractive index of a material. The law is also satisfied in metamaterials, which allow light to be bent "backward" at a negative angle of refraction with a negative refractive index.

Snell's law states that the ratio of the sines of the angles of incidence and refraction is equivalent to the ratio of phase velocities in the two media, or equivalent to the reciprocal of the ratio of the indices of refraction:

${\displaystyle {\frac {\sin \theta _{2}}{\sin \theta _{1}}}={\frac {v_{2}}{v_{1}}}={\frac {n_{1}}{n_{2}}}}$

with each ${\displaystyle \theta }$ as the angle measured from the normal of the boundary, ${\displaystyle v}$ as the velocity of light in the respective medium (SI units are meters per second, or m/s), ${\displaystyle \lambda }$ as the wavelength of light in the respective medium and ${\displaystyle n}$ as the refractive index (which is unitless) of the respective medium.

The law follows from Fermat's principle of least time, which in turn follows from the propagation of light as waves.

"Story of Your Life" is a science fiction novella by American writer Ted Chiang, first published in Starlight 2 in 1998, and in 2002 in Chiang's collection of short stories, Stories of Your Life and Others. Its major themes are language and determinism.

"Story of Your Life" won the 2000 Nebula Award for Best Novella, as well as the 1999 Theodore Sturgeon Award. It was nominated for the 1999 Hugo Award for Best Novella. The novella has been translated into Italian, French and German.A film adaptation of the story by Eric Heisserer, titled Arrival and directed by Denis Villeneuve, was released in 2016. It stars Amy Adams, Jeremy Renner, and Forest Whitaker and was nominated for eight Academy Awards, including Best Picture; it won the award for Best Sound Editing. The film also won the 2017 Ray Bradbury Award for Outstanding Dramatic Presentation and the Hugo Award for Best Dramatic Presentation.

University of Orléans

The University of Orléans (French: Université d'Orléans) is a French university, in the Academy of Orléans and Tours. As of July 2015 it is a member of the regional university association Leonardo da Vinci consolidated University.

Variational principle

A variational principle is a scientific principle used within the calculus of variations, which develops general methods for finding functions which extremize the value of quantities that depend upon those functions. For example, to answer this question: "What is the shape of a chain suspended at both ends?" we can use the variational principle that the shape must minimize the gravitational potential energy.

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