Lightning is the electric breakdown of air by strong electric fields. Heat and light from a lightning produces a plasm motion of air molecules.

In physics and other sciences, energy (from the Greek ενεργός, energos, "active, working")^[1] is a scalar physical quantity that is a property of objects and systems which is conserved by nature. Energy is often defined as the ability to do work. It has been suggested that Energy (Disambiguation) be merged into this article or section. ... Image File history File linksMetadata Download high resolution version (2048x3072, 3589 KB) This is a rotated version of Lightning over Oradea Romania. ... Image File history File linksMetadata Download high resolution version (2048x3072, 3589 KB) This is a rotated version of Lightning over Oradea Romania. ... Not to be confused with lighting. ... A magnet levitating above a high-temperature superconductor demonstrates the Meissner effect. ... A magnet levitating above a high-temperature superconductor demonstrates the Meissner effect. ... See scalar for an account of the broader concept also used in mathematics and computer science. ... A physical quantity is either a quantity within physics that can be measured (e. ... In thermodynamics, work is the quantity of energy transferred from one system to another without an accompanying transfer of entropy. ...

Several different forms of energy, including kinetic, potential, thermal, gravitational,sound energy, Light energy, elastic, electromagnetic, chemical, nuclear, and mass have been defined to explain all known natural phenomena. The cars of a roller coaster reach their maximum kinetic energy when at the bottom of their path. ... Potential energy can be thought of as energy stored within a physical system. ... In thermal physics, thermal energy is the energy portion of a system that increases with its temperature. ... Energy of two or more masses (or other forms of energy-momentum) gravitationally interacting with each other. ... The elastic energy is the energy which causes or is released by the elastic distortion of a solid or a fluid. ... This box:      Electromagnetic (EM) radiation is a self-propagating wave in space with electric and magnetic components. ... In chemistry, a chemical bond is the force which holds together atoms in molecules or crystals. ... This article concerns the energy stored in the nuclei of atoms; for the use of nuclear fission as a power source, see Nuclear power. ... The rest energy of a particle is its energy when it is not moving relative to a given inertial reference frame. ...

Energy is converted from one form to another. This principle, the conservation of energy, was first postulated in the early 19th century, and applies to any isolated system. According to Noether's theorem, the conservation of energy is a consequence of the fact that the laws of physics do not change over time.^[2] Energy Transformation in Energy Systems Language In physics and engineering, poop transformation or energy conversion, is any process of transforming one form of energy to another. ... This article is about the law of conservation of energy in physics. ... In thermodynamics, an isolated system, as contrasted with a closed system, is a physical system that does not interact with its surroundings. ... Noethers theorem is a central result in theoretical physics that shows that a conservation law can be derived from any continuous symmetry. ...

Although the total energy of a system does not change with time, its value may depend on the frame of reference. For example, a seated passenger in a moving airplane has zero kinetic energy relative to the airplane, but nonzero kinetic energy relative to the earth. This article or section is in need of attention from an expert on the subject. ...

[edit] History

Main articles: History of energy, Timeline of thermodynamics, statistical mechanics, and random processes, and History of physics

Thomas Young - the first to use the term "energy" in the modern sense.

The concept of energy emerged out of the idea of vis viva, which Leibniz defined as the product of the mass of an object and its velocity squared; he believed that total vis viva was conserved. To account for slowing due to friction, Leibniz claimed that heat consisted of the random motion of the constituent parts of matter — a view shared by Isaac Newton, although it would be more than a century until this was generally accepted. In 1807, Thomas Young was the first to use the term "energy", instead of vis viva, in its modern sense.^[3] Gustave-Gaspard Coriolis described "kinetic energy" in 1829 in its modern sense, and in 1853, William Rankine coined the term "potential energy." It was argued for some years whether energy was a substance (the caloric) or merely a physical quantity, such as momentum. The word energy seems to appear for the first time in the works of Aristotle. ... A timeline of events related to thermodynamics, statistical mechanics, and random processes. ... Since antiquity, human beings have sought to understand the workings of nature: why unsupported objects drop to the ground, why different materials have different properties, the character of the universe such as the form of the Earth and the behavior of celestial objects such as the Sun and the Moon... Image File history File links Download high resolution version (921x1152, 226 KB) This image is in the public domain because its copyright has expired in the United States and those countries with a copyright term of life of the author plus 100 years or less. ... Image File history File links Download high resolution version (921x1152, 226 KB) This image is in the public domain because its copyright has expired in the United States and those countries with a copyright term of life of the author plus 100 years or less. ... Thomas Young, English scientist Thomas Young (June 13, 1773-May 10, 1829) was an English polymath, contributing to the scientific understanding of vision, light, solid mechanics, energy, physiology, and Egyptology. ... Vis Viva is the principle that the difference between the aggregate work of the accelerating forces of a system and that of the retarding forces is equal to one half the vis viva accumulated or lost in the system while the work is being done. ... Leibniz redirects here. ... Sir Isaac Newton FRS (4 January 1643 – 31 March 1727) [ OS: 25 December 1642 – 20 March 1727][1] was an English physicist, mathematician, astronomer, natural philosopher, and alchemist. ... Thomas Young, English scientist Thomas Young (June 13, 1773-May 10, 1829) was an English polymath, contributing to the scientific understanding of vision, light, solid mechanics, energy, physiology, and Egyptology. ... Vis Viva is the principle that the difference between the aggregate work of the accelerating forces of a system and that of the retarding forces is equal to one half the vis viva accumulated or lost in the system while the work is being done. ... Gaspard-Gustave de Coriolis or Gustave Coriolis (May 21, 1792–September 19, 1843), mathematician, mechanical engineer and scientist born in Paris, France. ... The cars of a roller coaster reach their maximum kinetic energy when at the bottom of their path. ... William John Macquorn Rankine (July 2, 1820 - December 24, 1872) was a Scottish engineer and physicist. ... Potential energy can be thought of as energy stored within a physical system. ... Caloric redirects here. ... This article is about momentum in physics. ...

He^{[citation needed]} amalgamated all of these laws into the laws of thermodynamics, which aided in the rapid development of explanations of chemical processes using the concept of energy by Rudolf Clausius, Josiah Willard Gibbs and Walther Nernst. It also led to a mathematical formulation of the concept of entropy by Clausius, and to the introduction of laws of radiant energy by Jožef Stefan. Thermodynamics (from the Greek θερμη, therme, meaning heat and δυναμις, dynamis, meaning power) is a branch of physics that studies the effects of changes in temperature, pressure, and volume on physical systems at the macroscopic scale by analyzing the collective motion of their particles using statistics. ... Rudolf Clausius - physicist and mathematician Rudolf Julius Emanuel Clausius (January 2, 1822 – August 24, 1888), was a German physicist and mathematician. ... Josiah Willard Gibbs (February 11, 1839 New Haven – April 28, 1903 New Haven) was one of the very first American theoretical physicists and chemists. ... Walther Hermann Nernst (June 25, 1864 – November 18, 1941) was a German physicist who is known for his theories behind the calculation of chemical affinity as embodied in the third law of thermodynamics, for which he won the 1920 Nobel Prize in chemistry. ... For other uses, see: information entropy (in information theory) and entropy (disambiguation). ... Light (a form of radiant energy) observed in a forest Radiant energy is the energy of electromagnetic waves, or sometimes of other forms of radiation. ... Joseph Stefan (Slovene Jožef Stefan) (March 24, 1835 – January 7, 1893) was a Slovene physicist, mathematician and poet. ...

During a 1961 lecture^[4] for undergraduate students at the California Institute of Technology, Richard Feynman, a celebrated physics teacher and Nobel Laureate, said this about the concept of energy: The California Institute of Technology (commonly referred to as Caltech)[1] is a private, coeducational research university located in Pasadena, California, in the United States. ... This article is about the physicist. ... The Nobel Prizes (pronounced no-BELL or no-bell) are awarded annually to people who have done outstanding research, invented groundbreaking techniques or equipment, or made outstanding contributions to society. ...

Since 1918 it has been known that the law of conservation of energy is the direct mathematical consequence of the translational symmetry of the quantity conjugate to energy, namely time. That is, energy is conserved because the laws of physics do not distinguish between different moments of time (see Noether's theorem). This article is about the law of conservation of energy in physics. ... In geometry, a translation slides an object by a vector a: Ta(p) = p + a. ... In physics, especially in quantum mechanics, conjugate variables are pairs of variables that share an uncertainty relation. ... This article is about the concept of time. ... Noethers theorem is a central result in theoretical physics that shows that a conservation law can be derived from any continuous symmetry. ...

[edit] Energy in various contexts since the beginning of the universe

The concept of energy and its transformations is useful in explaining and predicting most natural phenomena. The direction of transformations in energy (what kind of energy is transformed to what other kind) is often described by entropy (equal energy spread among all available degrees of freedom) considerations, since in practice all energy transformations are permitted on a small scale, but certain larger transformations are not permitted because it is statistically unlikely that energy or matter will randomly move into more concentrated forms or smaller spaces. For other uses, see: information entropy (in information theory) and entropy (disambiguation). ... Degrees of freedom is a general term used in explaining dependence on parameters, and implying the possibility of counting the number of those parameters. ...

The concept of energy is used often in all fields of science.

In chemistry, the energy differences between substances determine whether, and to what extent, they can be converted into other substances or react with other substances.

In biology, chemical bonds are broken and made during metabolic processes, and the associated changes in available energy are studied in the subfield of bioenergetics. Energy is often stored by cells in the form of substances such as carbohydrate molecules (including sugars) and lipids, which release energy when reacted with oxygen.

In geology and meteorology, continental drift, mountain ranges, volcanos, and earthquakes are phenomena that can be explained in terms of energy transformations in the Earth's interior. ^[5] While meteorological phenomena like wind, rain, hail, snow, lightning, tornadoes and hurricanes, are all a result of energy transformations brought about by solar energy on the planet Earth.

In cosmology and astronomy the phenomena of stars, nova, supernova, quasars and gamma ray bursts are the universe's highest-output energy transformations of matter. All stellar phenomena (including solar activity) are driven by various kinds of energy transformations. Energy in such transformations is either from gravitational collapse of matter (usually molecular hydrogen) into various classes of astronomical objects (stars, black holes, etc.), or from nuclear fusion (of lighter elements, primarily hydrogen).

Energy transformations in the universe over time are characterized by various kinds of potential energy which has been available since the Big Bang, later being "released" (transformed to more active types of energy such as kinetic or radiant energy), when a triggering mechanism is available. For other uses, see Chemistry (disambiguation). ... Water and steam are two different forms of the same chemical substance A chemical substance is a material with a definite chemical composition. ... For the song by Girls Aloud see Biology (song) Biology studies the variety of life (clockwise from top-left) E. coli, tree fern, gazelle, Goliath beetle Biology (from Greek: Βιολογία - βίος, bio, life; and λόγος, logos, speech lit. ... A chemical bond is the physical process responsible for the attractive interactions between atoms and molecules, and that which confers stability to diatomic and polyatomic chemical compounds. ... Structure of the coenzyme adenosine triphosphate, a central intermediate in energy metabolism. ... Bioenergetics, loosely defined, is the study of energy investment and flow through living systems. ... Drawing of the structure of cork as it appeared under the microscope to Robert Hooke from Micrographia which is the origin of the word cell being used to describe the smallest unit of a living organism Cells in culture, stained for keratin (red) and DNA (green) The cell is the... Lactose is a disaccharide found in milk. ... Some common lipids. ... This article is about the chemical element and its most stable form, or dioxygen. ... Earth science (also known as geoscience, the geosciences or the Earth Sciences), is an all-embracing term for the sciences related to the planet Earth. ... Plates in the crust of the earth, according to the plate tectonics theory Continental drift refers to the movement of the Earths continents relative to each other. ... For other uses, see Mountain (disambiguation). ... Cleveland Volcano in the Aleutian Islands of Alaska photographed from the International Space Station For other uses, see Volcano (disambiguation). ... This article is about the natural seismic phenomenon. ... Energy Transformation in Energy Systems Language In physics and engineering, poop transformation or energy conversion, is any process of transforming one form of energy to another. ... For other uses, see Wind (disambiguation). ... This article is about precipitation. ... This article is about the precipitation. ... For other uses, see Snow (disambiguation). ... Not to be confused with lighting. ... This article is about the weather phenomenon. ... This article is about weather phenomena. ... Ultraviolet image of the Sun. ... This article is about the physics subject. ... This article is about the astronomical object. ... Artists conception of a white dwarf star accreting hydrogen from a larger companion A nova (pl. ... For other uses, see Supernova (disambiguation). ... This article is about the astronomical object. ... The image above shows the optical afterglow of gamma ray burst GRB-990123 taken on January 23, 1999. ... Energy Transformation in Energy Systems Language In physics and engineering, poop transformation or energy conversion, is any process of transforming one form of energy to another. ... For other uses, see Big Bang (disambiguation). ...

Familiar examples of such processes include nuclear decay, in which energy is released which was originally "stored" in heavy isotopes (such as uranium and thorium), by nucleosynthesis, a process which ultimately uses the gravitational potential energy released from the gravitational collapse of supernovae, to store energy in the creation of these heavy elements before they were incorporated into the solar system and the Earth. This energy is triggered and released in nuclear fission bombs. In a slower process, heat from nuclear decay of these atoms in the core of the Earth releases heat, which in turn may lift mountains, via orogenesis. This slow lifting represents a kind of gravitational potential energy storage of the heat energy, which may be released to active kinetic energy in landslides, after a triggering event. Earthquakes also release stored elastic potential energy in rocks, a store which has been produced ultimately from the same radioactive heat sources. Thus, according to present understanding, familiar events such as landslides and earthquakes release energy which has been stored as potential energy in the Earth's gravitational field or elastic strain (mechanical potential energy) in rocks; but prior to this, represents energy that has been stored in heavy atoms since the collapse of long-destroyed stars created these atoms. This article is about the chemical element. ... General Name, Symbol, Number thorium, Th, 90 Chemical series Actinides Group, Period, Block n/a, 7, f Appearance silvery white Standard atomic weight 232. ... Nucleosynthesis is the process of creating new atomic nuclei from preexisting nucleons (protons and neutrons). ... The mushroom cloud of the atomic bombing of Nagasaki, Japan, in 1945 lifted nuclear fallout some 18 km (60,000 feet) above the epicenter. ... Orogeny is the process of mountain building. ...

In another similar chain of transformations beginning at the dawn of the universe, nuclear fusion of hydrogen in the Sun releases another store of potential energy which was created at the time of the Big Bang. At that time, according to theory, space expanded and the universe cooled too rapidly for hydrogen to completely fuse into heavier elements. This meant that hydrogen represents a store of potential energy which can be released by fusion. Such a fusion process is triggered by heat and pressure generated from gravitational collapse of hydrogen clouds when they produce stars, and some of the fusion energy is then transformed into sunlight. Such sunlight from our Sun may again be stored as gravitational potential energy after it strikes the Earth, as (for example) water evaporates from oceans and is deposited upon mountains (where, after being released at a hydroelectric dam, it can be used to drive turbine/generators to produce electricity). Sunlight also drives all weather phenomenon, including such events as those triggered in a hurricane, when large unstable areas of warm ocean, heated over months, give up some of their thermal energy suddenly to power a few days of violent air movement. Sunlight is also is captured by plants as chemical potential energy, when carbon dioxide and water are converted into a combustible combination of carbohydrates, lipids, and oxygen. Release of this energy as heat and light may be triggered suddenly by a spark, in a forest fire; or it may be available more slowly for animal or human metabolism, when these molecules are ingested, and catabolism is triggered by enzyme action. Through all of these transformation chains, potential energy stored at the time of the Big Bang is later released by intermediate events, sometimes being stored in a number of ways over time between releases, as more active energy. In all these events, one kind of energy is converted to other types of energy, including heat. The deuterium-tritium (D-T) fusion reaction is considered the most promising for producing sustainable fusion power. ... For other uses, see Big Bang (disambiguation). ... The deuterium-tritium (D-T) fusion reaction is considered the most promising for producing sustainable fusion power. ... Anabolism is the aspect of metabolism that contributes to growth. ... Ribbon diagram of the enzyme TIM, surrounded by the space-filling model of the protein. ...

[edit] Regarding applications of the concept of energy

Energy is subject to a strict global conservation law; that is, whenever one measures (or calculates) the total energy of a system of particles whose interactions do not depend explicitly on time, it is found that the total energy of the system always remains constant.^[6] In physics, a conservation law states that a particular measurable property of an isolated physical system does not change as the system evolves. ...

• The total energy of a system can be subdivided and classified in various ways. For example, it is sometimes convenient to distinguish potential energy (which is a function of coordinates only) from kinetic energy (which is a function of coordinate time derivatives only). It may also be convenient to distinguish gravitational energy, electric energy, thermal energy, and other forms. These classifications overlap; for instance thermal energy usually consists partly of kinetic and partly of potential energy.
• The transfer of energy can take various forms; familiar examples include work, heat flow, and advection, as discussed below.
• The word "energy" is also used outside of physics in many ways, which can lead to ambiguity and inconsistency. The vernacular terminology is not consistent with technical terminology. For example, the important public-service announcement, "Please conserve energy" uses vernacular notions of "conservation" and "energy" which make sense in their own context but are utterly incompatible with the technical notions of "conservation" and "energy" (such as are used in the law of conservation of energy).^[7]

In classical physics energy is considered a scalar quantity, the canonical conjugate to time. In special relativity energy is also a scalar (although not a Lorentz scalar but a time component of the energy-momentum 4-vector).^[8] In other words, energy is invariant with respect to rotations of space, but not invariant with respect to rotations of space-time (= boosts). For other uses, see System (disambiguation). ... Potential energy can be thought of as energy stored within a physical system. ... The cars of a roller coaster reach their maximum kinetic energy when at the bottom of their path. ... For other uses, see Derivative (disambiguation). ... Classical physics is physics based on principles developed before the rise of quantum theory, usually including the special theory of relativity and general theory of relativity. ... A pair of variables mathematically defined in such a way that they become Fourier transform duals of one-another, or more generally are related through Pontryagin duality. ... This article is about the concept of time. ... For a less technical and generally accessible introduction to the topic, see Introduction to special relativity. ... In physics a Lorentz scalar is a scalar which is invariant under a Lorentz transformation. ... It has been suggested that this article or section be merged with Momentum#Momentum_in_relativistic_mechanics. ... In relativity, a four-vector is a vector in a four-dimensional real vector space, called Minkowski space, whose components transform like the space and time coordinates (t, x, y, z) under spatial rotations and boosts (a change by a constant velocity to another inertial reference frame). ... This article is about the idea of space. ... In special relativity and general relativity, time and three-dimensional space are treated together as a single four-dimensional pseudo-Riemannian manifold called spacetime. ... The Lorentz transformation (LT), named after its discoverer, the Dutch physicist and mathematician Hendrik Antoon Lorentz (1853-1928), forms the basis for the special theory of relativity, which has been introduced to remove contradictions between the theories of electromagnetism and classical mechanics. ...

[edit] Energy transfer

Because energy is strictly conserved and is also locally conserved (wherever it can be defined), it is important to remember that by definition of energy the transfer of energy between the "system" and adjacent regions is work. A familiar example is mechanical work. In simple cases this is written as: In physics, mechanical work is the amount of energy transferred by a force. ...

ΔE = W             (1)

if there are no other energy-transfer processes involved. Here ΔE  is the amount of energy transferred, and W  represents the work done on the system.

More generally, the energy transfer can be split into two categories:

ΔE = W + Q             (2)

where Q  represents the heat flow into the system.

There are other ways in which an open system can gain or lose energy. If mass is counted as energy (as in many relativistic problems) then E must contain a term for mass lost or gained. In chemical systems, energy can be added to a system by means of adding substances with different chemical potentials, which potentials are then extracted (both of these process are illustrated by fueling an auto, a system which gains in energy thereby, without addition of either work or heat). Winding a clock would be adding energy to a mechanical system. These terms may be added to the above equation, or they can generally be subsumed into a quantity called "energy addition term E" which refers to any type of energy carried over the surface of a control volume or system volume. Examples may be seen above, and many others can be imagined (for example, the kinetic energy of a stream of particles entering a system, or energy from a laser beam adds to system energy, without either being either work-done or heat-added, in the classic senses).

ΔE = W + Q + E             (3)

Where E in this general equation represents other additional advected energy terms not covered by work done on a system, or heat added to it.

Energy is also transferred from potential energy (E_p) to kinetic energy (E_k) and then back to potential energy constantly. This is referred to as conservation of energy. In this closed system, energy can not be created or destroyed, so the initial energy and the final energy will be equal to each other. This can be demonstrated by the following:

E_pi + E_ki = E_pF + E_kF'''

The equation can then be simplified further since E_p = mgh (mass times acceleration due to gravity times the height) and (half times mass times velocity squared). Then the total amount of energy can be found by adding E_p + E_k = E_total.

[edit] Energy and the laws of motion

Classical mechanics

Newton's Second Law

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In classical mechanics, energy is a conceptually and mathematically useful property since it is a conserved quantity. Classical mechanics (commonly confused with Newtonian mechanics, which is a subfield thereof) is used for describing the motion of macroscopic objects, from projectiles to parts of machinery, as well as astronomical objects, such as spacecraft, planets, stars, and galaxies. ... Newtons First and Second laws, in Latin, from the original 1687 edition of the Principia Mathematica. ... The Greeks, and Aristotle in particular, were the first to propose that there are abstract principles governing nature. ... This article is about the idea of space. ... This article is about the concept of time. ... For other uses, see Mass (disambiguation). ... For other uses, see Force (disambiguation). ... This article is about momentum in physics. ... Newtons First and Second laws, in Latin, from the original 1687 edition of the Principia Mathematica. ... Lagrangian mechanics is a re-formulation of classical mechanics that combines conservation of momentum with conservation of energy. ... Hamiltonian mechanics is a re-formulation of classical mechanics that was invented in 1833 by William Rowan Hamilton. ... Applied mechanics, also known as theoretical and applied mechanics, is a branch of the physical sciences and the practical application of mechanics. ... Celestial mechanics is a division of astronomy dealing with the motions and gravitational effects of celestial objects. ... Continuum mechanics is a branch of physics (specifically mechanics) that deals with continuous matter, including both solids and fluids (i. ... See also list of optical topics. ... Statistical mechanics is the application of probability theory, which includes mathematical tools for dealing with large populations, to the field of mechanics, which is concerned with the motion of particles or objects when subjected to a force. ... Galileo redirects here. ... Kepler redirects here. ... Sir Isaac Newton FRS (4 January 1643 – 31 March 1727) [ OS: 25 December 1642 – 20 March 1727][1] was an English physicist, mathematician, astronomer, natural philosopher, and alchemist. ... Pierre-Simon, marquis de Laplace (March 23, 1749 - March 5, 1827) was a French mathematician and astronomer whose work was pivotal to the development of mathematical astronomy. ... For other persons named William Hamilton, see William Hamilton (disambiguation). ... Jean le Rond dAlembert, pastel by Maurice Quentin de La Tour Jean le Rond dAlembert (November 16, 1717 – October 29, 1783) was a French mathematician, mechanician, physicist and philosopher. ... Augustin Louis Cauchy (August 21, 1789 – May 23, 1857) was a French mathematician. ... Joseph-Louis, comte de Lagrange (January 25, 1736 Turin, Kingdom of Sardinia - April 10, 1813 Paris) was an Italian-French mathematician and astronomer who made important contributions to all fields of analysis and number theory and to classical and celestial mechanics as arguably the greatest mathematician of the 18th century. ... Euler redirects here. ... Classical mechanics (commonly confused with Newtonian mechanics, which is a subfield thereof) is used for describing the motion of macroscopic objects, from projectiles to parts of machinery, as well as astronomical objects, such as spacecraft, planets, stars, and galaxies. ... This article is about the law of conservation of energy in physics. ...

[edit] The Hamiltonian

The total energy of a system is sometimes called the Hamiltonian, after William Rowan Hamilton. The classical equations of motion can be written in terms of the Hamiltonian, even for highly complex or abstract systems. These classical equations have remarkably direct analogs in nonrelativistic quantum mechanics.^[9] In physics and mathematics, Hamiltons equations is the set of differential equations that arise in Hamiltonian mechanics, but also in many other related and sometimes apparently not related areas of science. ... For other persons named William Hamilton, see William Hamilton (disambiguation). ...

[edit] The Lagrangian

Another energy-related concept is called the Lagrangian, after Joseph Louis Lagrange. This is even more fundamental than the Hamiltonian, and can be used to derive the equations of motion. In non-relativistic physics, the Lagrangian is the kinetic energy minus potential energy. A Lagrangian of a dynamical system, named after Joseph Louis Lagrange, is a function of the dynamical variables and concisely describes the equations of motion of the system. ... Joseph-Louis, comte de Lagrange (January 25, 1736 Turin, Kingdom of Sardinia - April 10, 1813 Paris) was an Italian-French mathematician and astronomer who made important contributions to all fields of analysis and number theory and to classical and celestial mechanics as arguably the greatest mathematician of the 18th century. ...

Usually, the Lagrange formalism is mathematically more convenient than the Hamiltonian for non-conservative systems (like systems with friction).

[edit] Energy and thermodynamics

[edit] Internal energy

Internal energy – the sum of all microscopic forms of energy of a system. It is related to the molecular structure and the degree of molecular activity and may be viewed as the sum of kinetic and potential energies of the molecules; it comprises the following types of energy:^[10] In thermodynamics, the internal energy of a thermodynamic system, or a body with well-defined boundaries, denoted by U, or sometimes E, is the total of the kinetic energy due to the motion of molecules (translational, rotational, vibrational) and the potential energy associated with the vibrational and electric energy of...

In thermodynamics, the internal energy of a thermodynamic system, or a body with well-defined boundaries, denoted by U, or sometimes E, is the total of the kinetic energy due to the motion of molecules (translational, rotational, vibrational) and the potential energy associated with the vibrational and electric energy of... Sensible heat is heat energy that is transported by a body that has a temperature higher than its surroundings via conduction, convection, or both. ... In thermodynamics, the internal energy of a thermodynamic system, or a body with well-defined boundaries, denoted by U, or sometimes E, is the total of the kinetic energy due to the motion of molecules (translational, rotational, vibrational) and the potential energy associated with the vibrational and electric energy of... In thermochemistry, latent heat is the amount of energy in the form of heat released or absorbed by a substance during a change of phase (i. ... In the physical sciences, a phase is a set of states of a macroscopic physical system that have relatively uniform chemical composition and physical properties (i. ... In chemistry, a chemical bond is the force which holds together atoms in molecules or crystals. ... Properties For alternative meanings see atom (disambiguation). ... This article is about matter in physics and chemistry. ... This article concerns the energy stored in the nuclei of atoms; for the use of nuclear fission as a power source, see Nuclear power. ... This article concerns the energy stored in the nuclei of atoms; for the use of nuclear fission as a power source, see Nuclear power. ... A fundamental interaction is a mechanism by which particles interact with each other, and which cannot be explained by another more fundamental interaction. ... In thermal physics, heat transfer is the passage of thermal energy from a hot to a colder body. ... Mass transfer is the phrase commonly used in engineering for physical processes that involve molecular and convective transport of atoms and molecules within physical systems. ... In thermodynamics, work is the quantity of energy transferred from one system to another without an accompanying transfer of entropy. ... Thermodynamics (Greek: thermos = heat and dynamic = change) is the physics of energy, heat, work, entropy and the spontaneity of processes. ... In thermal physics, thermal energy is the energy portion of a system that increases with its temperature. ...

[edit] The laws of thermodynamics

According to the second law of thermodynamics, work can be totally converted into heat, but not vice versa.This is a mathematical consequence of statistical mechanics. The first law of thermodynamics simply asserts that energy is conserved,^[11] and that heat is included as a form of energy transfer. A commonly-used corollary of the first law is that for a "system" subject only to pressure forces and heat transfer (e.g. a cylinder-full of gas), the differential change in energy of the system (with a gain in energy signified by a positive quantity) is given by: The second law of thermodynamics is an expression of the universal law of increasing entropy. ... For other uses, see Heat (disambiguation) In physics, heat, symbolized by Q, is energy transferred from one body or system to another due to a difference in temperature. ... Statistical mechanics is the application of probability theory, which includes mathematical tools for dealing with large populations, to the field of mechanics, which is concerned with the motion of particles or objects when subjected to a force. ... In thermodynamics, the first law of thermodynamics is an expression of the more universal physical law of the conservation of energy. ... This article is about pressure in the physical sciences. ...

,

where the first term on the right is the heat transfer into the system, defined in terms of temperature T and entropy S (in which entropy increases and the change dS is positive when the system is heated); and the last term on the right hand side is identified as "work" done on the system, where pressure is P and volume V (the negative sign results since compression of the system is needed to do work on it, so that the volume change dV is negative when work is done on the system). Although this equation is the standard text-book example of energy conservation in classical thermodynamics, it is highly specific, ignoring all chemical, electric, nuclear, and gravitational forces, effects such as advection of any form of energy other than heat, and because it contains a term that depends on temperature. The most general statement of the first law — i.e. conservation of energy — is valid even in situations in which temperature is undefinable. For other uses, see Temperature (disambiguation). ... For other uses, see: information entropy (in information theory) and entropy (disambiguation). ... Advection is the transport of a conserved scalar quantity that is transported in a vector field. ...

Energy is sometimes expressed as:

,

which is unsatisfactory^[7] because there cannot exist any thermodynamic state functions W or Q that are meaningful on the right hand side of this equation, except perhaps in trivial cases.

[edit] Equipartition of energy

The energy of a mechanical harmonic oscillator (a mass on a spring) is alternatively kinetic and potential. At two points in the oscillation cycle it is entirely kinetic, and alternatively at two other points it is entirely potential. Over the whole cycle, or over many cycles net energy is thus equally split between kinetic and potential. This is called equipartition principle - total energy of a system with many degrees of freedom is equally split among all available degrees of freedom. An undamped spring-mass system is a simple harmonic oscillator. ... The cars of a roller coaster reach their maximum kinetic energy when at the bottom of their path. ... In physics, a potential may refer to the scalar potential or to the vector potential. ... Cycle or Cycles may be: Look up cycle in Wiktionary, the free dictionary. ... In classical statistical mechanics, the equipartition theorem is a general formula that allows average energies of many physical systems to be calculated as a function of temperature. ...

This principle is vitally important to understanding the behavior of a quantity closely related to energy, called entropy. Entropy is a measure of evenness of a distribution of energy between parts of a system. When an isolated system is given more degrees of freedom (= is given new available energy states which are the same as existing states), then total energy spreads over all available degrees equally without distinction between "new" and "old" degrees. This mathematical result is called the second law of thermodynamics. For other uses, see: information entropy (in information theory) and entropy (disambiguation). ... In mathematical analysis, distributions (also known as generalized functions) are objects which generalize functions and probability distributions. ... An energy level is a quantified stable energy, which a physical system can have; the term is most commonly used in reference to the electron configuration of electrons, in atoms or molecules. ... The second law of thermodynamics is an expression of the universal law of increasing entropy. ...

[edit] Oscillators, phonons, and photons

In an ensemble (connected collection) of unsynchronized oscillators, the average energy is spread equally between kinetic and potential types.

In a solid, thermal energy (often referred to loosely as heat content) can be accurately described by an ensemble of thermal phonons that act as mechanical oscillators. In this model, thermal energy is equally kinetic and potential. In thermal physics, thermal energy is the energy portion of a system that increases with its temperature. ...

In an ideal gas, the interaction potential between particles is essentially the delta function which stores no energy: thus, all of the thermal energy is kinetic.

Because an electric oscillator (LC circuit) is analogous to a mechanical oscillator, its energy must be, on average, equally kinetic and potential. It is entirely arbitrary whether the magnetic energy is considered kinetic and the electric energy considered potential, or vice versa. That is, either the inductor is analogous to the mass while the capacitor is analogous to the spring, or vice versa.

1. By extension of the previous line of thought, in free space the electromagnetic field can be considered an ensemble of oscillators, meaning that radiation energy can be considered equally potential and kinetic. This model is useful, for example, when the electromagnetic Lagrangian is of primary interest and is interpreted in terms of potential and kinetic energy.

1. On the other hand, in the key equation m²c⁴ = E² − p²c², the contribution mc² is called the rest energy, and all other contributions to the energy are called kinetic energy. For a particle that has mass, this implies that the kinetic energy is 0.5p² / m at speeds much smaller than c, as can be proved by writing E = mc² √(1 + p²m ^{− 2}c ^{− 2}) and expanding the square root to lowest order. By this line of reasoning, the energy of a photon is entirely kinetic, because the photon is massless and has no rest energy. This expression is useful, for example, when the energy-versus-momentum relationship is of primary interest.

The two analyses are entirely consistent. The electric and magnetic degrees of freedom in item 1 are transverse to the direction of motion, while the speed in item 2 is along the direction of motion. For non-relativistic particles these two notions of potential versus kinetic energy are numerically equal, so the ambiguity is harmless, but not so for relativistic particles. Light (a form of radiant energy) observed in a forest Radiant energy is the energy of electromagnetic waves, or sometimes of other forms of radiation. ...

[edit] Work and virtual work

Main articles: Mechanics, Mechanical work, Thermodynamics, and Quantum mechanics

Work is roughly force times distance. But more precisely, it is For other uses, see Mechanic (disambiguation). ... In physics, mechanical work is the amount of energy transferred by a force. ... Thermodynamics (from the Greek θερμη, therme, meaning heat and δυναμις, dynamis, meaning power) is a branch of physics that studies the effects of changes in temperature, pressure, and volume on physical systems at the macroscopic scale by analyzing the collective motion of their particles using statistics. ... For a generally accessible and less technical introduction to the topic, see Introduction to quantum mechanics. ...

This says that the work (W) is equal to the integral (along a certain path) of the force; for details see the mechanical work article. For other uses, see Force (disambiguation). ... In physics, mechanical work is the amount of energy transferred by a force. ...

Work and thus energy is frame dependent. For example, consider a ball being hit by a bat. In the center-of-mass reference frame, the bat does no work on the ball. But, in the reference frame of the person swinging the bat, considerable work is done on the ball. Refers to reference frame dependance. ...

[edit] Quantum mechanics

In quantum mechanics energy is defined in terms of the energy operator as a time derivative of the wave function. The Schrödinger equation equates the energy operator to the full energy of a particle or a system. It thus can be considered as a definition of measurement of energy in quantum mechanics. The Schrödinger equation describes the space- and time-dependence of slow changing (non-relativistic) wave function of quantum systems. The solution of this equation for bound system is discrete (a set of permitted states, each characterized by an energy level) which results in the concept of quanta. In the solution of the Schrödinger equation for any oscillator (vibrator) and for electromagnetic wave in vacuum, the resulting energy states are related to the frequency by the Planck equation E = hν (where h is the Planck's constant and ν the frequency). In the case of electromagnetic wave these energy states are called quanta of light or photons. The quantum Hamiltonian is the physical state of a system, which may be characterized as a ray in an abstract Hilbert space (or, in the case of ensembles, as a trace class operator with trace 1). ... A wave function is a mathematical tool that quantum mechanics uses to describe any physical system. ... This box:      For a non-technical introduction to the topic, please see Introduction to quantum mechanics. ... A wave function is a mathematical tool that quantum mechanics uses to describe any physical system. ... A quantum mechanical system can only be in certain states, so that only certain energy levels are possible. ... In physics quanta is the plural of quantum. ... This article is about Planck, the German physicist. ... A commemoration plaque for Max Planck on his discovery of Plancks constant, in front of Humboldt University, Berlin. ... For other uses, see Light (disambiguation). ... In modern physics the photon is the elementary particle responsible for electromagnetic phenomena. ...

[edit] Relativity

When calculating kinetic energy (= work to accelerate a mass from zero speed to some finite speed) relativistically - using Lorentz transformations instead of Newtonian mechanics, Einstein discovered unexpected by-product of these calculations to be an energy term which does not vanish at zero speed. He called it rest mass energy - energy which every mass must possess even when being at rest. The amount of energy is directly proportional to the mass of body: In physics, mechanical work is the amount of energy transferred by a force. ... For other uses, see Mass (disambiguation). ... This article does not cite any references or sources. ... The Lorentz transformation (LT), named after its discoverer, the Dutch physicist and mathematician Hendrik Antoon Lorentz (1853-1928), forms the basis for the special theory of relativity, which has been introduced to remove contradictions between the theories of electromagnetism and classical mechanics. ... It has been suggested that this article or section be merged with Classical mechanics. ... Energy E = mc^2 of mass m. ...

E = mc²,

where

m is the mass,

c is the speed of light in vacuum,

E is the rest mass energy.

For example, consider electron-positron annihilation, in which the rest mass of individual particles is destroyed, but the inertia equivalent of the system of the two particles (its invariant mass) remains (since all energy is associated with mass), and this inertia and invariant mass is carried off by photons which individually are massless, but as a system retain their mass. This is a reversible process - the inverse process is called pair creation - in which the rest mass of particles is created from energy of two (or more) annihilating photons. A line showing the speed of light on a scale model of Earth and the Moon, taking about 1â…“ seconds to traverse that distance. ... For other uses, see Electron (disambiguation). ... The first detection of the positron in 1932 by Carl D. Anderson The positron is the antiparticle or the antimatter counterpart of the electron. ... The invariant mass or intrinsic mass or proper mass or just mass is a measurement or calculation of the mass of an object that is the same for all frames of reference. ... Pair production is a nuclear physics process which occurs where a high-energy photon, generally interacting with an atomic nucleus, produces a particle and an antiparticle. ...

In general relativity, the stress-energy tensor serves as the source term for the gravitational field, in rough analogy to the way mass serves as the source term in the non-relativistic Newtonian approximation.^[8]

It is not uncommon to hear that energy is "equivalent" to mass. It would be more accurate to state that every energy has inertia and gravity equivalent, and because mass is a form of energy, then mass too has inertia and gravity associated with it.

[edit] Measurement