Editing Energy (section)

== Scientific use ==

=== Classical mechanics ===
{{Classical mechanics}}
{{Main|Mechanics|Mechanical work|Thermodynamics}}

In classical mechanics, energy is a conceptually and mathematically useful property, as it is a [[conserved quantity]]. Several formulations of mechanics have been developed using energy as a core concept.

[[Work (physics)|Work]], a function of energy, is force times distance.
: <math> W = \int_C \mathbf{F} \cdot \mathrm{d} \mathbf{s}</math>

This says that the work (<math>W</math>) is equal to the [[line integral]] of the [[force]] '''F''' along a path ''C''; for details see the [[mechanical work]] article. 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.

The total energy of a system is sometimes called the [[Hamilton's equations|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 direct analogs in nonrelativistic quantum mechanics.<ref>[https://web.archive.org/web/20071011135413/http://www.sustech.edu/OCWExternal/Akamai/18/18.013a/textbook/HTML/chapter16/section03.html The Hamiltonian] MIT OpenCourseWare website 18.013A Chapter 16.3 Accessed February 2007</ref>

Another energy-related concept is called the [[Lagrangian mechanics|Lagrangian]], after [[Joseph-Louis Lagrange]]. This formalism is as fundamental as the Hamiltonian, and both can be used to derive the equations of motion or be derived from them. It was invented in the context of [[classical mechanics]], but is generally useful in modern physics. The Lagrangian is defined as the kinetic energy ''minus'' the potential energy. Usually, the Lagrange formalism is mathematically more convenient than the Hamiltonian for non-conservative systems (such as systems with friction).

[[Noether's theorem]] (1918) states that any differentiable symmetry of the action of a physical system has a corresponding conservation law. Noether's theorem has become a fundamental tool of modern theoretical physics and the calculus of variations. A generalisation of the seminal formulations on constants of motion in Lagrangian and Hamiltonian mechanics (1788 and 1833, respectively), it does not apply to systems that cannot be modeled with a Lagrangian; for example, dissipative systems with continuous symmetries need not have a corresponding conservation law.

=== Chemistry ===
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In the context of [[Chemistry#Energy|chemistry]], [[Chemical energy|energy]] is an attribute of a substance as a consequence of its atomic, molecular, or aggregate structure. Since a chemical transformation is accompanied by a change in one or more of these kinds of structure, it is usually accompanied by a decrease, and sometimes an increase, of the total energy of the substances involved. Some energy may be transferred between the surroundings and the reactants in the form of heat or light; thus the products of a reaction have sometimes more but usually less energy than the reactants. A reaction is said to be [[Exothermic process|exothermic]] or [[exergonic]] if the final state is lower on the energy scale than the initial state; in the less common case of [[Endothermic process|endothermic]] reactions the situation is the reverse.

[[Chemical reaction]]s are usually not possible unless the reactants surmount an energy barrier known as the [[activation energy]]. The ''speed'' of a chemical reaction (at a given temperature&nbsp;''T'') is related to the activation energy&nbsp;''E'' by the Boltzmann's population factor&nbsp;e<sup>−''E''/''kT''</sup>; that is, the probability of a molecule to have energy greater than or equal to&nbsp;''E'' at a given temperature&nbsp;''T''. This exponential dependence of a reaction rate on temperature is known as the [[Arrhenius equation]]. The activation energy necessary for a chemical reaction can be provided in the form of thermal energy.

=== Biology ===
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{{Main|Bioenergetics|Food energy}}
[[File:Energy and life.svg|thumb|Basic overview of [[Bioenergetics|energy and human life]]]]
{{anchor|Biology}}In [[biology#Energy|biology]], energy is an attribute of all biological systems, from the biosphere to the smallest living organism. Within an organism it is responsible for growth and development of a biological [[Cell (biology)|cell]] or [[organelle]] of a biological organism. Energy used in [[respiration (physiology)|respiration]] is stored in substances such as [[carbohydrate]]s (including sugars), [[lipid]]s, and [[protein]]s stored by [[Cell (biology)|cells]]. In human terms, the [[human equivalent]] (H-e) (Human energy conversion) indicates, for a given amount of energy expenditure, the relative quantity of energy needed for human [[metabolism]], using as a standard an average human energy expenditure of 6,900&nbsp;kJ per day and a [[basal metabolic rate]] of 80 watts.

For example, if our bodies run (on average) at 80 watts, then a light bulb running at 100 watts is running at 1.25 human equivalents (100 ÷ 80) i.e. 1.25 H-e. For a difficult task of only a few seconds' duration, a person can put out thousands of watts, many times the 746 watts in one official horsepower. For tasks lasting a few minutes, a fit human can generate perhaps 1,000 watts. For an activity that must be sustained for an hour, output drops to around 300; for an activity kept up all day, 150 watts is about the maximum.<ref>{{cite web |url=http://www.uic.edu/aa/college/gallery400/notions/human%20energy.htm |title=Retrieved on May-29-09 |publisher=Uic.edu |access-date=2010-12-12 |url-status=live |archive-url=https://web.archive.org/web/20100604191319/http://www.uic.edu/aa/college/gallery400/notions/human%20energy.htm |archive-date=2010-06-04 }}</ref> The human equivalent assists understanding of energy flows in physical and biological systems by expressing energy units in human terms: it provides a "feel" for the use of a given amount of energy.<ref>Bicycle calculator – speed, weight, wattage etc. {{cite web |url=http://bikecalculator.com/ |title=Bike Calculator |access-date=2009-05-29 |url-status=live |archive-url=https://web.archive.org/web/20090513091201/http://bikecalculator.com/ |archive-date=2009-05-13 }}.</ref>

Sunlight's radiant energy is also captured by plants as ''chemical potential energy'' in [[photosynthesis]], when carbon dioxide and water (two low-energy compounds) are converted into carbohydrates, lipids, proteins and oxygen. Release of the energy stored during photosynthesis as heat or light may be triggered suddenly by a spark in a forest fire, or it may be made available more slowly for animal or human metabolism when organic molecules are ingested and [[catabolism]] is triggered by [[enzyme]] action.

All living creatures rely on an external source of energy to be able to grow and reproduce – radiant energy from the Sun in the case of green plants and chemical energy (in some form) in the case of animals. The daily 1500–2000&nbsp;[[kilocalorie|Calories]] (6–8&nbsp;MJ) recommended for a human adult are taken as food molecules, mostly carbohydrates and fats, of which [[glucose]] (C<sub>6</sub>H<sub>12</sub>O<sub>6</sub>) and [[stearin]] (C<sub>57</sub>H<sub>110</sub>O<sub>6</sub>) are convenient examples. The food molecules are oxidized to [[carbon dioxide]] and [[water (molecule)|water]] in the [[Mitochondrion|mitochondria]]
<chem display="block">C6H12O6 + 6O2 -> 6CO2 + 6H2O</chem>
<chem display="block">C57H110O6 + (81 1/2) O2 -> 57CO2 + 55H2O</chem>
and some of the energy is used to convert [[Adenosine diphosphate|ADP]] into [[Adenosine triphosphate|ATP]]:
{{block indent|em=1.6|text=ADP + HPO<sub>4</sub><sup>2−</sup> → ATP + H<sub>2</sub>O}}
The rest of the chemical energy of the carbohydrate or fat are converted into heat: the ATP is used as a sort of "energy currency", and some of the chemical energy it contains is used for other [[metabolism]] when ATP reacts with OH groups and eventually splits into ADP and phosphate (at each stage of a [[metabolic pathway]], some chemical energy is converted into heat). Only a tiny fraction of the original chemical energy is used for [[Work (physics)|work]]:<ref group=note>These examples are solely for illustration, as it is not the energy available for work which limits the performance of the athlete but the [[power (physics)|power]] output (in case of a sprinter) and the [[force (physics)|force]] (in case of a weightlifter).</ref>
: gain in kinetic energy of a sprinter during a 100&nbsp;m race: 4&nbsp;kJ
: gain in gravitational potential energy of a 150&nbsp;kg weight lifted through 2&nbsp;metres: 3&nbsp;kJ
: daily food intake of a normal adult: 6–8&nbsp;MJ

It would appear that living organisms are remarkably [[Energy conversion efficiency|inefficient (in the physical sense)]] in their use of the energy they receive (chemical or radiant energy); most [[machine]]s manage higher efficiencies. In growing organisms the energy that is converted to heat serves a vital purpose, as it allows the organism tissue to be highly ordered with regard to the molecules it is built from. The [[second law of thermodynamics]] states that energy (and matter) tends to become more evenly spread out across the universe: to concentrate energy (or matter) in one specific place, it is necessary to spread out a greater amount of energy (as heat) across the remainder of the universe ("the surroundings").<ref group=note>[[Crystal]]s are another example of highly ordered systems that exist in nature: in this case too, the order is associated with the transfer of a large amount of heat (known as the [[lattice energy]]) to the surroundings.</ref> Simpler organisms can achieve higher energy efficiencies than more complex ones, but the complex organisms can occupy [[ecological niche]]s that are not available to their simpler brethren. The conversion of a portion of the chemical energy to heat at each step in a metabolic pathway is the physical reason behind the pyramid of biomass observed in [[ecology]]. As an example, to take just the first step in the [[food chain]]: of the estimated 124.7&nbsp;Pg/a of carbon that is [[carbon fixation|fixed]] by [[photosynthesis]], 64.3&nbsp;Pg/a (52%) are used for the metabolism of green plants,<ref>Ito, Akihito; Oikawa, Takehisa (2004). "[http://www.terrapub.co.jp/e-library/kawahata/pdf/343.pdf Global Mapping of Terrestrial Primary Productivity and Light-Use Efficiency with a Process-Based Model.] {{webarchive|url=https://web.archive.org/web/20061002083948/http://www.terrapub.co.jp/e-library/kawahata/pdf/343.pdf |date=2006-10-02 }}" in Shiyomi, M. et al. (Eds.) ''Global Environmental Change in the Ocean and on Land.'' pp.&nbsp;343–58.</ref> i.e. reconverted into carbon dioxide and heat.

=== Earth sciences ===
In [[Earth science#earth's energy|geology]], [[continental drift]], [[mountain|mountain ranges]], [[volcano]]es, and [[earthquake]]s are phenomena that can be explained in terms of energy transformations in the Earth's interior,<ref>{{cite web |url=http://okfirst.ocs.ou.edu/train/meteorology/EnergyBudget.html |title=Earth's Energy Budget |publisher=Okfirst.ocs.ou.edu |access-date=2010-12-12 |url-status=live |archive-url=https://web.archive.org/web/20080827194704/http://okfirst.ocs.ou.edu/train/meteorology/EnergyBudget.html |archive-date=2008-08-27 }}</ref> while [[metereology|meteorological]] phenomena like wind, rain, [[hail]], snow, lightning, [[tornado]]es and [[tropical cyclone|hurricanes]] are all a result of energy transformations in our [[atmosphere]] brought about by [[solar energy]].

Sunlight is the main input to [[Earth's energy budget]] which accounts for its temperature and climate stability. Sunlight may be stored as gravitational potential energy after it strikes the Earth, as (for example when) water evaporates from oceans and is deposited upon mountains (where, after being released at a hydroelectric dam, it can be used to drive turbines or generators to produce electricity). Sunlight also drives most weather phenomena, save a few exceptions, like those generated by volcanic events for example. An example of a solar-mediated weather event is a hurricane, which occurs when large unstable areas of warm ocean, heated over months, suddenly give up some of their thermal energy to power a few days of violent air movement.

In a slower process, [[radioactive decay]] of atoms in the core of the Earth releases heat. This thermal energy drives [[plate tectonics]] and may lift mountains, via [[orogenesis]]. This slow lifting represents a kind of gravitational potential [[energy storage]] of the thermal energy, which may later be transformed into active kinetic energy during landslides, after a triggering event. Earthquakes also release stored elastic potential energy in rocks, a store that has been produced ultimately from the same radioactive heat sources. Thus, according to present understanding, familiar events such as landslides and earthquakes release energy that has been stored as potential energy in the Earth's gravitational field or elastic strain (mechanical potential energy) in rocks. Prior to this, they represent release of energy that has been stored in heavy atoms since the collapse of long-destroyed supernova stars (which created these atoms).

=== Cosmology ===
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In [[Physical cosmology#Energy of the cosmos|cosmology and astronomy]] the phenomena of [[star]]s, [[nova]], [[supernova]], [[quasar]]s and [[gamma-ray burst]]s are the universe's highest-output energy transformations of matter. All [[wikt:stellar|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).

The [[nuclear fusion]] of hydrogen in the Sun also 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 that 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.
{{anchor|Physics}}<!-- courtesy note per [[WP:LINK2SECT]]: [[Energy (physics) links here]] -->

=== Quantum mechanics ===
{{Main|Energy operator}}
In [[quantum mechanics]], energy is defined in terms of the [[Hamiltonian (quantum mechanics)|energy operator]]
(Hamiltonian) 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. Its results can be considered as a definition of measurement of energy in quantum mechanics. The Schrödinger equation describes the space- and time-dependence of a slowly changing (non-relativistic) [[wave function]] of quantum systems. The solution of this equation for a bound system is discrete (a set of permitted states, each characterized by an [[energy level]]) which results in the concept of [[quantum|quanta]]. In the solution of the Schrödinger equation for any oscillator (vibrator) and for electromagnetic waves in a vacuum, the resulting energy states are related to the frequency by [[Planck's relation]]: <math>E = h\nu</math> (where <math>h</math> is the [[Planck constant]] and <math>\nu</math> the frequency). In the case of an electromagnetic wave these energy states are called quanta of light or [[photon]]s.

=== Relativity ===
When calculating kinetic energy ([[Mechanical work|work]] to accelerate a [[mass|massive body]] from zero [[speed]] to some finite speed) relativistically – using [[Lorentz transformations]] instead of [[Newtonian mechanics]] – Einstein discovered an unexpected by-product of these calculations to be an energy term which does not vanish at zero speed. He called it [[rest energy]]: energy which every massive body must possess even when being at rest. The amount of energy is directly proportional to the mass of the body:
<math display="block"> E_0 = m_0 c^2 ,</math>
where
* ''m''<sub>0</sub> is the [[Rest Mass|rest mass]] of the body,
* ''c'' is the [[speed of light]] in vacuum,
* <math>E_0</math> is the rest energy.

For example, consider [[electron]]–[[positron]] annihilation, in which the rest energy of these two individual particles (equivalent to their rest mass) is converted to the radiant energy of the photons produced in the process. In this system the [[matter]] and [[antimatter]] (electrons and positrons) are destroyed and changed to non-matter (the photons). However, the total mass and total energy do not change during this interaction. The photons each have no rest mass but nonetheless have radiant energy which exhibits the same inertia as did the two original particles. This is a reversible process – the inverse process is called [[pair creation]] – in which the rest mass of particles is created from the radiant energy of two (or more) annihilating photons.

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.<ref name="MTW"/>

Energy and mass are manifestations of one and the same underlying physical property of a system. This property is responsible for the inertia and strength of gravitational interaction of the system ("mass manifestations"), and is also responsible for the potential ability of the system to perform work or heating ("energy manifestations"), subject to the limitations of other physical laws.

In [[classical physics]], energy is 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]]).<ref name="MTW">{{Cite book |author=Misner |first1=Charles W. |title=Gravitation |last2=Thorne |first2=Kip S. |last3=Wheeler |first3=John Archibald |publisher=W.H. Freeman |year=1973 |isbn=978-0-7167-0344-0 |location=San Francisco}}</ref> In other words, energy is invariant with respect to rotations of [[space]], but not invariant with respect to rotations of [[spacetime]] (= [[Lorentz boost|boosts]]).