Fermat's principle

Calculus of variations is a field of mathematics which deals with functions of functions, as opposed to ordinary calculus which deals with functions of numbers. Such functionals can for example be formed as integrals involving an unknown function and its derivatives. The interest is in extremal functions: those making the functional attain a maximum or minimum value. Some classical problems on curves were posed in this form: one example is the brachistochrone, the path along which a particle would descend under gravity i ...

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• Calculus of variations - Reference books

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Light is electromagnetic radiation with a wavelength that is visible to the eye (visible light) or, in a technical or scientific context, electromagnetic radiation of any wavelength. The three basic dimensions of light (i.e., all electromagnetic radiation) are: Intensity (or brilliance or amplitude), which is related to the human perception of brightness of the light, Frequency (or wavelength), perceived by humans as the color of the light, and Polarization (or angle of vibration), which is not perceptible by ...

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• Light - Visible electromagnetic radiation
• Light - Speed of light
• Light - Refraction
• Light - Optics
• Light - Color and wavelengths
• Light - Measurement of light
• Light - Light sources
• Light - Theories about light
• Light - Early Greek ideas
• Light - 10th century optical theory
• Light - The 'plenum'
• Light - Particle theory
• Light - Wave theory
• Light - Electromagnetic theory
• Light - Particle theory revisited
• Light - Quantum theory
• Light - Wave-particle duality
• Light - A light wave

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Light - Early Greek ideas. In 55 BC Lucretius, continuing the ideas of earlier atomists, wrote that light and heat from the Sun were composed of minute particles. Ptolemy also wrote about the refraction of light. Light - 10th century optical theory. The scientist Abu Ali al-Hasan ibn al-Haytham (965-c.1040), also known as Alhazen, developed a broad theory that explained vision, using geometry and anatomy, which stated that each point on an illuminated area or object radi ...

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Light, Light - Visible electromagnetic radiation, Light - Speed of light, Light - Refraction, Light - Optics, Light - Color and wavelengths, Light - Measurement of light, Light - Light sources, Light - Theories about light, Light - Early Greek ideas, Light - 10th century optical theory, Light - The 'plenum', Light - Particle theory, Light - Wave theory, Light - Electromagnetic theory, Light - Particle theory revisited, Light - Quantum theory, Light - Wave-particle duality, Light - A light wave

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Light - Early Greek ideas. In 55 BC Lucretius, continuing the ideas of earlier atomists, wrote that light and heat from the Sun were composed of minute particles. Ptolemy also wrote about the refraction of light. Light - 10th century optical theory. The Persian scientist Abu Ali al-Hasan ibn al-Haytham (965-c.1040), also known as Alhazen, developed a broad theory that explained vision, using geometry and anatomy, which stated that each point on an illuminated area or obj ...

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Light, Light - Visible electromagnetic radiation, Light - Speed of light, Light - Refraction, Light - Optics, Light - Color and wavelengths, Light - Measurement of light, Light - Light sources, Light - Theories about light, Light - Early Greek ideas, Light - 10th century optical theory, Light - The 'plenum', Light - Particle theory, Light - Wave theory, Light - Electromagnetic theory, Light - Particle theory revisited, Light - Quantum theory, Light - Wave-particle duality, Light - A light wave

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There are many sources of light. The most common light sources are thermal: a body at a given temperature emits a characteristic spectrum of black body radiation. Examples include sunlight (the radiation emitted by the chromosphere of the Sun at around 6,000 K peaks in the visible region of the electromagnetic spectrum), incandescent light bulbs (which emit only around 10% of their energy as visible light and the remainder as infrared), and glowing solid particles in flames. The peak of the blackbody spectrum is in the infrared for rela ...

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Light, Light - Visible electromagnetic radiation, Light - Speed of light, Light - Refraction, Light - Optics, Light - Color and wavelengths, Light - Measurement of light, Light - Light sources, Light - Theories about light, Light - Early Greek ideas, Light - 10th century optical theory, Light - The 'plenum', Light - Particle theory, Light - Wave theory, Light - Electromagnetic theory, Light - Particle theory revisited, Light - Quantum theory, Light - Wave-particle duality, Light - A light wave

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A variational principle is a principle in physics which is expressed in terms of the calculus of variations. According to Cornelius Lanczos, any physical law which can be expressed as a variational principle describes an expression which is self-adjoint. These expressions are also called Hermitian. Such an expression describes an invariant under a Hermitian transformation. Felix Klein's Erlangen program attempted to identify such invariants under a group of transformations. In what is referred to in phys ...

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• Variational principle - Examples
• Variational principle - Variational principle in quantum mechanics
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In physics, the action principle is an assertion about the nature of motion, from which the trajectory of an object subject to forces can be determined. The path of an object is the one that yields a stationary value for a quantity called the action. Thus, instead of thinking about an object accelerating in response to applied forces, one might think of them picking out the path with a stationary action. The principle is also called the principle of stationary action and also Hamilton's principle. Other sta ...

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• Action physics - Some applications of the action principle
• Action physics - History
• Action physics - Action principle in classical mechanics
• Action physics - Euler-Lagrange equations for the action integral
• Action physics - Example: Free particle in polar coordinates
• Action physics - Einstein-Hilbert action
• Action physics - Literature

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The different wavelengths are detected by the human eye and then interpreted by the brain as colors, ranging from red at the longest wavelengths of about 700 nm. (lowest frequencies) to violet at the shortest wavelengths of about 400 nm. (highest frequencies). The intervening frequencies are seen as orange, yellow, green, cyan, blue, and, conventionally, indigo. The wavelengths of the electromagnetic spectrum immediately outside the range that the human eye is able to perceive are called ultraviolet (UV) at ...

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Light, Light - Visible electromagnetic radiation, Light - Speed of light, Light - Refraction, Light - Optics, Light - Color and wavelengths, Light - Measurement of light, Light - Light sources, Light - Theories about light, Light - Early Greek ideas, Light - 10th century optical theory, Light - The 'plenum', Light - Particle theory, Light - Wave theory, Light - Electromagnetic theory, Light - Particle theory revisited, Light - Quantum theory, Light - Wave-particle duality, Light - A light wave

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Visible light is the portion of the electromagnetic spectrum between the frequencies of 380 THz (3.8×1014 hertz) and 750 THz (7.5×1014 hertz). The speed (c), frequency (f or ν), and wavelength (λ) of a wave obey the relation: Because the speed of light in a vacuum is fixed, visible light can als ...

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Light, Light - Visible electromagnetic radiation, Light - Speed of light, Light - Refraction, Light - Optics, Light - Color and wavelengths, Light - Measurement of light, Light - Light sources, Light - Theories about light, Light - Early Greek ideas, Light - 10th century optical theory, Light - The 'plenum', Light - Particle theory, Light - Wave theory, Light - Electromagnetic theory, Light - Particle theory revisited, Light - Quantum theory, Light - Wave-particle duality, Light - A light wave

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Although some people speak of the "velocity of light", the word velocity should be reserved for vector quantities, that is, those with both magnitude and direction. The speed of light is a scalar quantity, having only magnitude and no direction, and therefore speed is the correct term. The speed of light has been measured many times, by many physicists. The best early measurement is Ole Rømer's (a Danish physicist), in 1676. By observing the motions of Jupiter and one of its moons, Io, with a telescope, and noting discr ...

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Light, Light - Visible electromagnetic radiation, Light - Speed of light, Light - Refraction, Light - Optics, Light - Color and wavelengths, Light - Measurement of light, Light - Light sources, Light - Theories about light, Light - Early Greek ideas, Light - 10th century optical theory, Light - The 'plenum', Light - Particle theory, Light - Wave theory, Light - Electromagnetic theory, Light - Particle theory revisited, Light - Quantum theory, Light - Wave-particle duality, Light - A light wave

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The following quantities and units are used to measure the quantity or "brightness" of light. edit edit Sometimes confusingly called "intensity". Sometimes confusingly called "intensity". Sometimes confusingly called "intensity". watt per steradian per squ ...

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Light, Light - Visible electromagnetic radiation, Light - Speed of light, Light - Refraction, Light - Optics, Light - Color and wavelengths, Light - Measurement of light, Light - Light sources, Light - Theories about light, Light - Early Greek ideas, Light - 10th century optical theory, Light - The 'plenum', Light - Particle theory, Light - Wave theory, Light - Electromagnetic theory, Light - Particle theory revisited, Light - Quantum theory, Light - Wave-particle duality, Light - A light wave

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All light propagates at a finite speed. Even moving observers always measure the same value of c, the speed of light in vacuum, as c = 299,792,458 metres per second (186,282.397 miles per second). When light passes through a transparent substance, such as air, water or glass, its speed is reduced, and it undergoes refraction. The reduction of the speed of light in a denser material can be indicated by the refractive index, n, which is defined a ...

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Light, Light - Visible electromagnetic radiation, Light - Speed of light, Light - Refraction, Light - Optics, Light - Color and wavelengths, Light - Measurement of light, Light - Light sources, Light - Theories about light, Light - Early Greek ideas, Light - 10th century optical theory, Light - The 'plenum', Light - Particle theory, Light - Wave theory, Light - Electromagnetic theory, Light - Particle theory revisited, Light - Quantum theory, Light - Wave-particle duality, Light - A light wave

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In the diagram on the right, two media of refractive indices n1 (on the left) and n2 (on the right) meet at a surface or interface (vertical line). n2 > n1, and light has a slower phase velocity within the second medium. A light ray PO in the leftmost medium strikes the interface at the point O. From point O, we project a straight line at right angles to the line of the interface; this is known as the normal to the surface (horizontal line). The angle between the normal and the light ray PO is known as the See also:

Snell's law, Snell's law - Overview, Snell's law - Total internal reflection, Snell's law - Vector form, Snell's law - Derivation, Snell's law - History

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Variational principle - Variational principle in quantum mechanics. For a hamiltonian H that describes the studied system and any normalizable function Ψ with arguments appropriate for the unknown wave function of the system, we define the functional The variational principle states that , where E0 is the lowest energy eigenstate (ground state) of the hamiltonian if and only if See also:

Variational principle, Variational principle - Examples, Variational principle - Variational principle in quantum mechanics, Variational principle - Further readings

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The stationary point of an integral along a path is equivalent to a set of differential-equations, called the Euler-Lagrange equations. This can be seen as follows where we restrict ourselves to one coordinate only. The extension to more coordinates is straightforward. Suppose we have an action integral S of an integrand L which depends on coordinates x(t) and dx ...

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Action physics, Action physics - Some applications of the action principle, Action physics - History, Action physics - Action principle in classical mechanics, Action physics - Euler-Lagrange equations for the action integral, Action physics - Example: Free particle in polar coordinates, Action physics - Einstein-Hilbert action, Action physics - Literature

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Before Max Planck suggested that light is quantized, optics consisted mainly of the application of electromagnetism and its high frequency approximations to light. Classical optics divides into two main branches: geometric optics and physical optics. Geometric optics, or ray optics, describes light propagation in terms of "rays". Rays are bent at the interface between two dissimilar media, and may be curved in a medium in which the refractive index is a function of position. The "ray" in geometric optics is an abstract o ...

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Optics, Optics - Classical optics, Optics - Topics related to classical optics, Optics - Modern optics, Optics - Topics related to modern optics, Optics - Other optical fields, Optics - Everyday optics, Optics - Wikibooks modules

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There are several ways to derive Snell's Law, and therefore the Angle of Refraction. The first way it was discovered was by an application of Fermat's principle which states that a light wave must take a path that is an extremum in time subject to the constraints present. Normally this is translated into "Light will always take the quickest path it can." From this principle, and using a bit of differential calculus, Snell’s Law can be derived thus leading to the Angle of Refraction. If one looks into the meaning of Fermat’s principle, ot ...

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Angle of refraction, Angle of refraction - Derivation and Meaning of the Angle of Refraction, Angle of refraction - Results the Angle of Refraction

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Given a normalized ray vector v and a normalized plane normal vector p, one can work out the normalized reflected and refracted rays: (note that the actual angles θ1 and θ2 are not worked out) The cosines may be recycled and used in the Fresnel equations for working out the intensity of the resulting rays. During total internal reflection an evanescent wave is produced, which rapidly decays from the surface into the second medium. Conservation of energy is maintained by the circulation of energy across the boundary, ...

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Snell's law, Snell's law - Overview, Snell's law - Total internal reflection, Snell's law - Vector form, Snell's law - Derivation, Snell's law - History

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Although equivalent in classical mechanics with Newton's laws, the action principle is better suited for generalizations and plays an important role in modern physics. Indeed, this principle is one of the great generalizations in physical science. In particular, it is fully appreciated and best understood within quantum mechanics. Richard Feynman's path integral formulation of quantum mechanics is based on a stationary-action principle, using path integ ...

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Action physics, Action physics - Some applications of the action principle, Action physics - History, Action physics - Action principle in classical mechanics, Action physics - Euler-Lagrange equations for the action integral, Action physics - Example: Free particle in polar coordinates, Action physics - Einstein-Hilbert action, Action physics - Literature

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The principle of least action was first formulated by Maupertuis [1] in 1746 and further developed (from 1748 onwards) by the mathematicians Euler, Lagrange, and Hamilton. Maupertuis arrived at this principle from a feeling that the very perfection of the universe demands a certain economy in nature and is opposed to any needless expenditure of energy. Natural motions must be such as to make some quantity a minimum. It was only necessary to find that quantity, and this he proceeded to do. It was the product of the duration (time) of movement within a system by the "vis viva" or twice what we ...

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Action physics, Action physics - Some applications of the action principle, Action physics - History, Action physics - Action principle in classical mechanics, Action physics - Euler-Lagrange equations for the action integral, Action physics - Example: Free particle in polar coordinates, Action physics - Einstein-Hilbert action, Action physics - Literature

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