Energy

{{Short description|Physical quantity}} {{hatnote group| {{About|the scalar physical quantity|an overview of and topical guide|Outline of energy|other uses}} {{redirect2|Energetic|Physical energy|other uses|Energetic (disambiguation)}} }} {{pp-semi-indef}} {{pp-move}} {{Use British English|date=March 2013}} {{CS1 config|mode=cs1}} {{Infobox physical quantity | name = Energy | image = Plasma globe 60th.jpg | caption = A [[plasma globe]] using [[electrical energy]] to create [[plasma (physics)|plasma]], [[light]], [[heat]], [[kinetic energy|movement]], and a faint [[sound]] | unit = [[joule]]<ref name=Soman_2010/> (J) | otherunits = [[Kilowatt-hour|kWh]], [[British thermal unit|BTU]], [[calorie]], [[Electronvolt|eV]], [[erg]], [[foot-pound (energy)|foot-pound]]<ref name=Dougherty_Schissler_2020/> | symbols = ''E'' | baseunits = J = kg⋅m2⋅s−2<ref name=ISU_1964/> | dimension = '''M''' '''L'''2 '''T'''−2<ref name=Fuller_2014/> | extensive = yes<ref name=Chandra_2016>{{cite book | title=Energy, Entropy and Engines: An Introduction to Thermodynamics | first=Sanjeev | last=Chandra | publisher=John Wiley & Sons | year=2016 | isbn=978-1-119-01318-1 | page=34 | url=https://books.google.com/books?id=AKfQCwAAQBAJ&pg=PA34 }}</ref> | conserved = [[Conservation of energy|yes]] | derivations = | image_upright = 1.15 }}

'''Energy''' ({{etymology|grc|''{{Wikt-lang|grc|ἐνέργεια}}'' ({{grc-transl|ἐνέργεια}})|activity}}) is the [[physical quantity|quantitative]] [[physical property|property]] that is transferred to a [[physical body|body]] or to a [[physical system]], recognizable in the capacity to do [[Work (thermodynamics)|work]] and in the form of [[heat]] and [[light]]. Energy is a [[Conservation law|conserved quantity]]—the law of [[conservation of energy]] states that energy can be [[Energy transformation|converted]] in form, but not created or destroyed. The unit of measurement for energy in the [[International System of Units]] (SI) is the [[joule]] (J).

Forms of energy include the [[kinetic energy]] of a moving object, the [[potential energy]] stored by an object (for instance due to its position in a [[Classical field theory|field]]), the [[elastic energy]] stored in a solid object, [[chemical energy]] associated with [[chemical reaction]]s, the [[radiant energy]] carried by [[electromagnetic radiation]], the [[internal energy]] contained within a [[thermodynamic system]], and [[rest energy]] associated with an object's [[rest mass]]. These are not mutually exclusive.

All [[life|living organisms]] constantly take in and release energy. The Earth's [[climate]] and [[ecosystem]]s processes are driven primarily by [[Solar irradiance|radiant energy from the Sun]].<ref>{{cite web |url=https://energyeducation.ca/encyclopedia/Earth%27s_energy_flow |title=Earth's energy flow |website=Energy Education |access-date=2024-08-28}}</ref>

== Forms == [[File:Lightning over Oradea Romania zoom.jpg|thumb|In a typical [[lightning]] strike, 500 [[megajoule]]s of [[electric potential energy]] is converted into the same amount of energy in other forms, mostly [[light energy]], [[sound energy]], and [[thermal energy]].]] [[File:Hot metalwork.jpg|thumb|[[Thermal energy]] is energy of microscopic constituents of matter, which may include both [[kinetic energy|kinetic]] and [[potential energy]].]]

The total energy of a [[system]] can be subdivided and classified into [[potential energy]], [[kinetic energy]], or combinations of the two in various ways. Kinetic energy is determined by the [[motion (physics)|movement]] of an object – or the [[statistical mechanics|composite motion]] of the object's components – while [[potential energy]] reflects the potential of an object to have motion, generally being based upon the object's position within a [[Field (physics)|field]] or what is stored within the field itself.<ref>{{Cite journal |last=Bobrowsky |first=Matt |title=SCIENCE 101: Q: What Is Energy? |url=https://www.jstor.org/stable/27133353 |access-date=February 5, 2024 |journal=[[Science and Children]] |date=2021 |volume=59 |issue=1 |pages=61–65 |language=en |doi=10.1080/19434812.2021.12291716 |jstor=27133353 |s2cid=266084433 |issn=0036-8148|url-access=subscription }}</ref>

While these two categories are sufficient to describe all forms of energy, it is often convenient to refer to particular combinations of potential and kinetic energy as its own form. For example, the sum of translational and [[rotational energy|rotational]] kinetic and potential energy within a system is referred to as [[mechanical energy]], whereas nuclear energy refers to the combined potentials within an atomic nucleus from either the [[nuclear force]] or the [[weak force]], among other examples.<ref>{{Cite web |title=Nuclear Energy {{!}} Definition, Formula & Examples {{!}} nuclear-power.com |url=https://www.nuclear-power.com/nuclear-power/nuclear-energy/ |access-date=2022-07-06 |website=Nuclear Power |language=en-us |archive-date=2022-07-06 |archive-url=https://web.archive.org/web/20220706153815/https://www.nuclear-power.com/nuclear-power/nuclear-energy/ |url-status=live }}</ref>

{| class="wikitable plainrowheaders" |+Some forms of energy (that an object or system can have as a measurable property)<ref name=Rosen_Dincer_2007>{{cite book | title=Exergy: Energy, Environment and Sustainable Development | first1=Marc A. | last1=Rosen | first2=Ibrahim | last2=Dincer | publisher=Elsevier | year=2007 | page=3 | isbn=978-0-08-053135-9 | url=https://books.google.com/books?id=ruR7U3IjrR0C&pg=PA3 }}</ref><ref name=Goel_2021>{{cite book | title=Systems in Mechanical Engineering: Fundamentals and Applications | first=Anup | last=Goel | publisher=Technical Publications | year=2021 | isbn=978-93-332-2183-2 | url=https://books.google.com/books?id=XjcfEAAAQBAJ&pg=SA1-PA20 }}</ref> !scope="col"|Type of energy !scope="col"|Description |- !scope="row"|[[Chemical energy|Chemical]] |potential energy due to chemical bonds |- !scope="row"|[[Quantum chromodynamics binding energy|Chromodynamic]] |potential energy that [[Binding energy|binds]] [[quark]]s to form [[hadron]]s |- !scope="row"|[[Elastic energy|Elastic]] |potential energy due to the deformation of a material (or its container) exhibiting a restorative force as it returns to its original shape |- !scope="row"|[[Electrical energy|Electric]] |potential energy due to or stored in electric fields |- !scope="row"|[[Gravitational energy|Gravitational]] |potential energy due to or stored in gravitational fields |- !scope="row"|[[Ionization energy|Ionization]] |potential energy that [[Binding energy|binds]] an electron to its atom or molecule |- !scope="row"|[[Magnetic energy|Magnetic]] |potential energy due to or stored in magnetic fields |- !scope="row"|[[Mechanical energy|Mechanical]] |the sum of [[macroscopic]] translational and rotational kinetic and potential energies |- !scope="row"|[[Mechanical wave]] |kinetic and potential energy in an elastic material due to a propagating [[oscillation]] of matter |- !scope="row"|[[Nuclear binding energy|Nuclear]] |potential energy that [[Binding energy|binds]] [[nucleons]] to form the [[atomic nucleus]] (and nuclear reactions) |- !scope="row"|[[Radiant energy|Radiant]] |[[Photon energy|potential energy]] stored in the fields of waves propagated by [[electromagnetic radiation]], including [[light]] |- !scope="row"|[[Rest energy|Rest]] |potential energy [[Mass–energy equivalence|due to]] an object's [[Intrinsic mass|rest mass]] |- !scope="row"|[[Rotational energy|Rotational]] |kinetic energy due to the rotation of an object |- !scope="row"|[[Sound energy|Sound wave]] |kinetic and potential energy in a material due to a sound propagated wave (a particular type of mechanical wave) |- !scope="row"|[[Heat|Thermal]] |kinetic energy of the [[microscopic]] motion of particles, a kind of disordered equivalent of mechanical energy |}

== History == {{Main|History of energy|timeline of thermodynamics, statistical mechanics, and random processes|}} [[File:Portrait of Thomas Young with printed autograph.jpg|thumb|upright|[[Thomas Young (scientist)|Thomas Young]], the first person to use the term "energy" in the modern sense]] The word ''energy'' derives from the {{langx|grc|ἐνέργεια|[[energeia]]|activity, operation}},<ref>{{cite web |url=http://www.etymonline.com/index.php?term=energy |title=Energy |work=Online Etymology Dictionary |last=Harper |first=Douglas |access-date=May 1, 2007 |url-status=live |archive-url=https://web.archive.org/web/20071011122441/http://etymonline.com/index.php?term=energy |archive-date=October 11, 2007 }}</ref> which possibly appears for the first time in the work of [[Aristotle]] in the 4th century BC. In contrast to the modern definition, energeia was a qualitative philosophical concept, broad enough to include ideas such as happiness and pleasure.<ref>{{cite journal | title=Different meanings of the term Energeia in the philosophy of Aristotle | first=Chung-Hwan | last=Chen | journal=Philosophy and Phenomenological Research | volume=17 | issue=1 | date=September 1956 | pages=56–65 | doi=10.2307/2104687 | jstor=2104687 }}</ref>

In the late 17th century, [[Gottfried Leibniz]] proposed the idea of the {{langx|la|[[vis viva]]}}, or living force, which 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 theorized that thermal energy consisted of the motions of the constituent parts of matter, although it would be more than a century until this was generally accepted. The modern analog of this property, [[kinetic energy]], differs from ''vis viva'' only by a factor of two.<ref name="McDonough 2021">{{cite encyclopedia | last=McDonough | first=Jeffrey K. | title=Leibniz's Philosophy of Physics | year=2021 | url=https://plato.stanford.edu/archives/fall2021/entries/leibniz-physics/ | encyclopedia=The Stanford Encyclopedia of Philosophy | editor-last=Zalta | editor-first=Edward N. | edition=Fall 2021 | publisher=Metaphysics Research Lab, Stanford University | access-date=2025-07-25 }}</ref> Writing in the early 18th century, [[Émilie du Châtelet]] proposed the concept of [[conservation of energy]] in the marginalia of her French language translation of Newton's ''[[Philosophiæ Naturalis Principia Mathematica|Principia Mathematica]]'', which represented the first formulation of a conserved measurable quantity that was distinct from [[momentum]], and which would later be called "energy".<ref name=Châtelet_2008>{{cite web | title=December 1706: Birth of Émilie du Châtelet | date=December 1, 2008 | series=This month in physics history | work=APS News | publisher=American Physical Society | url=https://www.aps.org/apsnews/2008/12/emilie-du-chatelet | access-date=2025-07-26 }}</ref>

In 1807, [[Thomas Young (scientist)|Thomas Young]] was possibly the first to use the term "energy" instead of ''vis viva'', in its modern sense.<ref>{{Cite book| last = Smith | first = Crosbie | title = The Science of Energy – a Cultural History of Energy Physics in Victorian Britain | publisher = The University of Chicago Press | year = 1998 | isbn = 978-0-226-76420-7}}</ref> [[Gustave-Gaspard Coriolis]] described "[[kinetic energy]]" in 1829 in its modern sense,<ref name=Coriolis>{{cite web | title=Gustave-Gaspard Coriolis | work=Oxford Reference | url=https://www.oxfordreference.com/display/10.1093/oi/authority.20110803095639380 | access-date=2025-07-26 }}</ref> and in 1853, [[William John Macquorn Rankine|William Rankine]] coined the term "[[potential energy]]".<ref name=Young_2015>{{cite journal | title=Heat, work and subtle fluids: a commentary on Joule (1850) 'On the mechanical equivalent of heat' | first=John | last=Young | journal=Philosophical Transactions A | date=April 13, 2015 | volume=373 | issue=2039 | at=20140348 | doi=10.1098/rsta.2014.0348 | pmid=25750152 | pmc=4360093 | bibcode=2015RSPTA.37340348Y }}</ref> The law of [[conservation of energy]] was also first postulated in the early 19th century, and applies to any [[isolated system]].<ref name=Sarton_et_al_1929>{{cite journal | title=The Discovery of the Law of Conservation of Energy | display-authors=1 | first1=G. | last1=Sarton | first2=J. R. | last2=Mayer | first3=J. P. | last3=Joule | first4=Sadi | last4=Carnot | journal=Isis | volume=13 | issue=1 | date=September 1929 | pages=18–44 | jstor=224595 | doi=10.1086/346430 }}</ref> It was argued for some years whether heat was a physical substance, dubbed the [[caloric theory|caloric]], or merely a physical quantity, such as [[momentum]]. In 1845 [[James Prescott Joule]] discovered the link between mechanical work and the generation of heat.<ref name=Williams_2015>{{cite web | title=June 1849: James Prescott Joule and the Mechanical Equivalent of Heat | first=Richard | last=Williams | date=June 1, 2015 | series=This month in physics history | work=APS News | publisher=American Physical Society | url=https://www.aps.org/apsnews/2015/06/joule-mechanical-equivalent-heat | access-date=2025-07-26 }}</ref>

These developments led to the theory of conservation of energy, formalized largely by William Thomson ([[Lord Kelvin]]) as the field of [[thermodynamics]].<ref name=McEvoy_2002>{{cite book | title=Classical Theory | volume=2 | series=Theory of interacting systems | first=Paul | last=McEvoy | publisher=Microanalytix | year=2002 | isbn=978-1-930832-02-2 | page=151 | url=https://books.google.com/books?id=dj0wFIxn-PoC&pg=PA151 }}</ref> Thermodynamics aided the rapid development of explanations of chemical processes by [[Rudolf Clausius]], [[Josiah Willard Gibbs]], [[Walther Nernst]], and others.<ref name=Fegley_Osborne_2013>{{cite book | title=Practical Chemical Thermodynamics for Geoscientists | first1=Bruce | last1=Fegley | first2=Rose | last2=Osborne | publisher=Academic Press | year=2013 | isbn=978-0-12-251100-4 | page=1 | url=https://books.google.com/books?id=Z8VNZNLNsTcC&pg=PA1 }}</ref> It also led to a mathematical formulation of the concept of [[entropy]] by Clausius<ref name=Grossinger_2012>{{cite book | title=Embryos, Galaxies, and Sentient Beings: How the Universe Makes Life | first=Richard | last=Grossinger | publisher=North Atlantic Books | year=2012 | isbn=978-1-58394-698-5 | page=22 | url=https://books.google.com/books?id=NThvbCJiqFUC&pg=PA22 }}</ref> and to the introduction of laws of [[radiant energy]] by [[Jožef Stefan]].<ref name=Stefan>{{cite web | title=Josef Stefan | work=Catholic scientists of the past | publisher=Society of Catholic Scientists | url=https://catholicscientists.org/scientists-of-the-past/josef-stefan/ | access-date=2025-07-26 }}</ref> 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.<ref name="jphysics">{{Cite book |last1=Lofts |first1=G. |title=Jacaranda Physics 1 |last2=O'Keeffe |first2=D. |publisher=John Wiley & Sons Australia Limited |year=2004 |isbn=978-0-7016-3777-4 |edition=2 |location=Milton, Queensland, Australia |page=286 |chapter=11 – Mechanical Interactions |display-authors=etal}}</ref> Thus, since 1918, theorists have understood that the law of [[conservation of energy]] is the direct mathematical consequence of the [[translational symmetry]] of the quantity [[conjugate variables|conjugate]] to energy, namely time.<ref name=Foot_2002>{{cite book | title=Shadowlands: Quest for Mirror Matter in the Universe | first=Robert | last=Foot | publisher=Universal-Publishers | year=2002 | isbn=978-1-58112-645-7 | page=114 | url=https://books.google.com/books?id=3evE2K-ylVIC&pg=PA114 }}</ref>

Albert Einstein's 1905 theory of [[special relativity]] showed that [[Invariant mass|rest mass]] corresponds to an [[Mass–energy equivalence|equivalent amount]] of ''[[rest energy]]''. This means that ''rest mass'' can be converted to or from equivalent amounts of (non-material) forms of energy, for example, kinetic energy, potential energy, and electromagnetic [[radiant energy]]. When this happens, rest mass is not conserved, unlike the [[mass in special relativity|''total'' mass]] or ''total'' energy. All forms of energy contribute to the total mass and total energy. Thus, conservation of energy (''total'', including material or ''rest'' energy) and [[conservation of mass]] (''total'', not just ''rest'') are one (equivalent) law. In the 18th century, these had appeared as two seemingly-distinct laws.<ref name=Egdall_2014>{{cite book | title=Einstein Relatively Simple: Our Universe Revealed In Everyday Language | first=Ira Mark | last=Egdall | publisher=World Scientific | year=2014 | isbn=978-981-4525-61-9 | page=99 | url=https://books.google.com/books?id=KUG7CgAAQBAJ&pg=PA99 }}</ref><ref name=Kolek_2024>{{cite book | title=On the general, the special and the general-special relativity theory | volume=1 | series=Chronicles of Business Informatics Physics (CBIP) | first=Erik | last=Kolek | edition=2nd | publisher=BoD – Books on Demand | year=2024 | isbn=978-3-7597-1218-9 | pages=126–128 | url=https://books.google.com/books?id=85wWEQAAQBAJ&pg=PA127 }}</ref>

The first evidence of quantization in atoms was the observation of [[spectral lines]] in light from the sun in the early 1800s by [[Joseph von Fraunhofer]] and [[William Hyde Wollaston]]. The notion of quantized energy levels was proposed in 1913 by Danish physicist [[Niels Bohr]] in the [[Bohr theory]] of the atom. The modern [[quantum mechanics|quantum mechanical theory]] giving an explanation of these energy levels in terms of the [[Schrödinger equation]] was advanced by [[Erwin Schrödinger]] and [[Werner Heisenberg]] in 1926.<ref name=Ruedenberg_Schwarz_2013>{{cite book | chapter=Three Millennia of Atoms and Molecules | first1=Klaus | last1=Ruedenberg | first2=W. H. Eugen | last2=Schwarz | title=Pioneers of Quantum Chemistry | pages=1–45 | series=ACS Symposium Series | volume=1122 | isbn=978-0-8412-2716-3 | date=February 13, 2013 | publisher=American Chemical Society | doi=10.1021/bk-2013-1122.ch001 }}</ref> Noether's theorem shows that the symmetry of this equation is equivalent to a ''[[conservation of probability]]''.<ref name=Phillips_2024>{{cite book | title=Einstein's Tutor: The Story of Emmy Noether and the Invention of Modern Physics | first=Lee | last=Phillips | publisher=PublicAffairs | year=2024 | isbn=978-1-5417-0297-4 | url=https://books.google.com/books?id=sbDrEAAAQBAJ&pg=PT280 }}</ref> At the quantum level, mass-energy interactions are all subject to this principle.<ref name=Lipkin_2014>{{cite book | title=Quantum Mechanics: New Approaches to Selected Topics | series=Dover books on physics | first=Harry J. | last=Lipkin | publisher=Courier Corporation | year=2014 | isbn=978-0-486-15185-4 | url=https://books.google.com/books?id=pUpMBAAAQBAJ&pg=PA205 }}</ref> During [[wave function collapse]], the conservation of energy does not hold at the local level, although statistically the principle holds on average for sufficiently large numbers of collapses.<ref name=Pearle_2000>{{cite journal | title=Wavefunction Collapse and Conservation Laws | last=Pearle | first=P. | journal=Foundations of Physics | date=August 2000 | volume=30 | issue=8 | pages=1145–1160 | doi=10.1023/A:1003677103804 | arxiv=quant-ph/0004067 | bibcode=2000FoPh...30.1145P }}</ref> Conservation of energy does apply during wave function collapse in [[Hugh Everett III|H. Everett]]'s [[many-worlds interpretation]] of quantum mechanics.<ref name=Carroll_Lodman_2021>{{cite journal | title=Energy Non-conservation in Quantum Mechanics | last1=Carroll | first1=S. M. | last2=Lodman | first2=J. | journal=Foundations of Physics | volume=51 | issue=83 | year=2021 | article-number=83 | doi=10.1007/s10701-021-00490-5 | arxiv=2101.11052 | bibcode=2021FoPh...51...83C }}</ref>

== Units of measure == [[File:Joule's Apparatus (Harper's Scan).png|thumb|right|Joule's apparatus for measuring the mechanical equivalent of heat. A descending weight attached to a string causes a paddle immersed in water to rotate.]] {{Main|Units of energy}}

In [[dimensional analysis]], the [[base units]] of energy are given by: [[Work (physics)|Work]] = Force × Distance = M L2 T−2, with the fundamental dimensions of Mass M, Length L, and time T.<ref name=Fuller_2014>{{cite book | title=Engineering Field Theory | series=The Commonwealth and international library. Applied electricity and electronics division | first=Ą. J. Baden | last=Fuller | editor-first=P. | editor-last=Hammon | publisher=Elsevier | year=2014 | isbn=978-1-4831-8700-6 | url=https://books.google.com/books?id=IJKjBQAAQBAJ&pg=PA7 }}</ref> In the [[International System of Units]] (SI), the unit of energy is the [[joule]]. It is a [[SI derived unit|derived unit]] that is equal to the energy expended, or work done, in applying a force of one [[Newton (unit)|newton]] through a distance of one meter.<ref name=Soman_2010>{{cite book | title=International System Of Units: A Handbook On S.I. Unit For Scientists And Engineers | first=K. | last=Soman | publisher=PHI Learning Pvt. Ltd. | year=2010 | isbn=978-81-203-3653-7 | page=15 | url=https://books.google.com/books?id=d-euEMU2cX8C&pg=PA15 }}</ref>

The SI unit of [[Power (physics)|power]], defined as energy per unit of time, is the [[watt]], which is one joule per second.<ref name=ISU_1964>{{cite book | title=The International System of Units: Physical Constants and Conversion Factors | volume=7012 | series=NASA SP | first=E. A. | last=Mechtly | publisher=Scientific and Technical Information Division, National Aeronautics and Space Administration | year=1964 | page=3 | url=https://books.google.com/books?id=xFUCAAAAIAAJ&pg=PA3 }}</ref> Thus, a [[kilowatt-hour]] (kWh), which can be realized as the energy delivered by one kilowatt of power for an hour, is equal to 3.6 million joules.<ref>{{cite book |last1=Thompson|first1=Ambler|last2=Taylor|first2=Barry N.|date=March 2008|title=Guide for the Use of the International System of Units (SI)|url=https://physics.nist.gov/cuu/pdf/sp811.pdf|publisher=[[National Institute of Standards and Technology]]|access-date=2025-09-16}}</ref> The [[centimetre gram second system of units|CGS]] energy unit is the [[erg]] and the [[imperial and US customary measurement systems|imperial and US customary]] unit is the [[foot-pound]].<ref name=Patience_2013>{{cite book | title=Experimental Methods and Instrumentation for Chemical Engineers | first=Gregory S. | last=Patience | publisher=Newnes | year=2013 | isbn=978-0-444-53805-5 | page=9 | url=https://books.google.com/books?id=nY-MQXBmZBAC&pg=PA9 }}</ref>

Other energy units such as the [[electronvolt]], [[food calorie]], thermodynamic [[kilocalorie]] and [[British thermal unit|BTU]] are used in specific areas of science and commerce.<ref name=Orecchini_Naso_2011>{{cite book | title=Energy Systems in the Era of Energy Vectors: A Key to Define, Analyze and Design Energy Systems Beyond Fossil Fuels | first1=Fabio | last1=Orecchini | first2=Vincenzo | last2=Naso | publisher=Springer Science & Business Media | year=2011 | isbn=978-0-85729-244-5 | pages=6–8 | url=https://books.google.com/books?id=vGiBKQ40c7QC&pg=PA7 }}</ref><ref name=Dougherty_Schissler_2020>{{cite book | title=SME Mining Reference Handbook | edition=2nd | editor1-first=Heather N. | editor1-last=Dougherty | editor2-first=Andrew P. | editor2-last=Schissler | publisher=Society for Mining, Metallurgy & Exploration | year=2020 | isbn=978-0-87335-435-6 | pages=2–3 | url=https://books.google.com/books?id=sVTHDwAAQBAJ&pg=PA2 }}</ref>

== 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.<ref name=Greiner_2006>{{cite book | title=Classical Mechanics: Point Particles and Relativity | series=Classical Theoretical Physics | first=Walter | last=Greiner | publisher=Springer Science & Business Media | year=2006 | isbn=978-0-387-21851-9 | page=109 | url=https://books.google.com/books?id=CynrBwAAQBAJ&pg=PA109 }}</ref>

: <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.<ref name=Obodovskiy_2019>{{cite book | title=Radiation: Fundamentals, Applications, Risks, and Safety | first=Ilya | last=Obodovskiy | publisher=Elsevier | year=2019 | isbn=978-0-444-63986-8 | url=https://books.google.com/books?id=xmOMDwAAQBAJ&pg=PA74 }}</ref>

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.<ref>{{cite web | title=Chapter 16.3 – The Hamiltonian | work=MIT OpenCourseWare website 18.013A | url=https://ocw.mit.edu/ans7870/18/18.013a/textbook/HTML/chapter16/section03.html | access-date=2025-07-28 }}</ref> These classical equations have direct analogs in nonrelativistic quantum mechanics.<ref name=Marchildon_2013>{{cite book | title=Quantum Mechanics: From Basic Principles to Numerical Methods and Applications | series=Advanced Texts in Physics | first=Louis | last=Marchildon | publisher=Springer Science & Business Media | year=2013 | isbn=978-3-662-04750-7 | page=38 | url=https://books.google.com/books?id=XibsCAAAQBAJ&pg=PA38 }}</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).<ref>{{cite web | title=Lecture L20 - Energy Methods: Lagrange's Equations | first=S. | last=Widnall | work=MIT OpenCourseWare website 16.07 Dynamics | year=2009 | version=3.0 | url=https://ocw.mit.edu/courses/16-07-dynamics-fall-2009/b39e882f1524a0f6a98553ee33ea6f35_MIT16_07F09_Lec20.pdf | access-date=2025-07-28 }}</ref>

[[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;<ref name=Berman_2025>{{cite book | title=Proofs and Logical Arguments Supporting the Foundational Laws of Physics: A Handy Guide for Students and Scientists | first=Jules J. | last=Berman | publisher=CRC Press | year=2025 | isbn=978-1-040-30073-2 | pages=144–146 | url=https://books.google.com/books?id=LbM1EQAAQBAJ&pg=PA144 }}</ref> for example, dissipative systems with continuous symmetries need not have a corresponding conservation law.

=== Chemistry ===  {{main|Chemical energy}} 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.<ref>See ''chemical change'' in: {{cite book | title=Hawley's Condensed Chemical Dictionary | first=Robert A. | last=Lewis | editor1-first=Michael D. | editor1-last=Larrañaga | editor2-first=Richard J. | editor2-last=Lewis, Sr. | edition=16 | publisher=John Wiley & Sons | year=2016 | isbn=978-1-119-26784-3 | url=https://books.google.com/books?id=KPrfCwAAQBAJ&pg=PA295 }}</ref>

[[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 ''T'') is related to the activation energy ''E'' by the Boltzmann population factor e−''E''/''kT''; that is, the probability of a molecule to have energy greater than or equal to ''E'' at a given temperature ''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.<ref name=Kondepudi_Prigogine_2014>{{cite book | title=Modern Thermodynamics: From Heat Engines to Dissipative Structures | first1=Dilip | last1=Kondepudi | first2=Ilya | last2=Prigogine | edition=2nd | publisher=John Wiley & Sons | year=2014 | isbn=978-1-118-69870-9 | pages=248–250 | url=https://books.google.com/books?id=SPU8BQAAQBAJ&pg=PA249 }}</ref>

=== Biology ===  {{Main|Bioenergetics|Food energy}} [[File:Energy and life.svg|thumb|Basic overview of [[Bioenergetics|energy and human life]]]] In [[biology#Energy|biology]], energy is an attribute of all biological systems, from the biosphere to the smallest living organism. It enables the growth, development, and functioning of a biological [[Cell (biology)|cell]] or [[organelle]] in an organism. 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. Energy provided through [[cellular respiration]] is stored in nutrients such as [[carbohydrate]]s (including sugars), [[lipid]]s, and [[protein]]s by [[Cell (biology)|cells]].<ref name=Hayamizu_2017>{{cite book | chapter=Amino Acids and Energy Metabolism: An Overview | first=Kohsuke | last=Hayamizu | title=Sustained Energy for Enhanced Human Functions and Activity | editor-first=Debasis | editor-last=Bagchi | publisher=Academic Press | year=2017 | isbn=978-0-12-809332-0 | page=339 | chapter-url=https://books.google.com/books?id=5epGDgAAQBAJ&pg=PA339 }}</ref>

Sunlight's radiant energy is 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.<ref name=Dimmitt_2016>{{cite book | chapter=Plant ecology of the Sonoran desert region | first=Mark A. | last=Dimmitt | title=Design with the Desert: Conservation and Sustainable Development | display-editors=1 | editor1-first=Richard | editor1-last=Malloy | editor2-first=John | editor2-last=Brock | editor3-first=Anthony | editor3-last=Floyd | editor4-first=Margaret | editor4-last=Livingston | editor5-first=Robert H. | editor5-last=Webb | publisher=CRC Press | year=2016 | isbn=978-1-000-21884-8 | pages=154–155 | chapter-url=https://books.google.com/books?id=Vef5DwAAQBAJ&pg=PA154 }}</ref> 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.<ref name=Sinn_2021>{{cite book | title=The Green Paradox: A Supply-Side Approach to Global Warming | first=Hans-Werner | last=Sinn | publisher=MIT Press | year=2012 | isbn=978-0-262-30058-2 | url=https://books.google.com/books?id=NnwCVOso9vgC&pg=PA85 }}</ref>

====Humans==== The [[basal metabolism rate]] measures the [[food energy]] expenditure per unit time by [[endotherm]]ic animals at rest.<ref name=McNab_1997>{{cite journal | first=Brian K. | last=McNab | title=On the Utility of Uniformity in the Definition of Basal Rate of Metabolism | journal=[[Physiological Zoology]] | year=1997 | volume=70 | issue=6 | pages=718–720 | doi=10.1086/515881 | pmid=9361146 | s2cid=34996894 }}</ref> In other words it is the energy required by body organs to perform normally. For humans, [[metabolic equivalent of task]] (MET) compares the energy expenditure per unit mass while performing a physical activity, relative to a baseline. By convention, this baseline is 3.5 mL of oxygen consumed per kg per minute, which is the energy consumed by a typical individual when sitting quietly.<ref name=Byrne_et_al_2005>{{cite journal | title=Metabolic equivalent: one size does not fit all | display-authors=1 | first1=Nuala M. | last1=Byrne | first2=Andrew P. | last2=Hills | first3=Gary R. | last3=Hunter | first4=Roland L. | last4=Weinsier | first5=Yves | last5=Schutz | journal=Applied Physiology | volume=99 | issue=3 | date=September 2005 | pages=1112–1119 | doi=10.1152/japplphysiol.00023.2004 | pmid=15831804 | bibcode=2005JAPh...99.1112B }}</ref>

{{citation needed span|date=July 2025|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 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 | title=Human Energy | url=http://www.uic.edu/aa/college/gallery400/notions/human%20energy.htm | publisher=Uic.edu | access-date=2010-12-12 | url-status=dead | 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>

The daily {{convert|1600|–|3000|Calorie|MJ|0}} recommended for a human adult are taken as food molecules,<ref name=MNT>{{cite web | title=How many calories should I eat in a day? | website=Medical News Today | date=12 February 2018 | url=https://www.medicalnewstoday.com/articles/245588 | access-date=2025-07-31 }}</ref> mostly carbohydrates and fats. 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 m race: 4 kJ : gain in gravitational potential energy of a 150 kg weight lifted through 2 metres: 3 kJ : daily food intake of a normal adult: 6–8 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.{{cn|date=July 2025}}

In growing organisms the energy that is converted to heat serves a vital purpose, as it allows the organism's 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 Pg/a of carbon that is [[carbon fixation|fixed]] by [[photosynthesis]], 64.3 Pg/a (52%) are used for the metabolism of green plants,<ref>{{cite book | chapter=Global Mapping of Terrestrial Primary Productivity and Light-Use Efficiency with a Process-Based Model | last1=Ito | first1=Akihito | last2=Oikawa | first2=Takehisa | year=2004 | display-editors=1 | editor1-first=Masae | editor1-last=Shiyomi | editor2-first=Hodaka | editor2-last=Kawahata | editor3-first=Hiroshi | editor3-last=Koizumi | editor4-first=Atsushi | editor4-last=Tsuda | editor5-first=Yoshio | editor5-last=Awaya | title=Global Environmental Change in the Ocean and on Land | pages=343–58 | chapter-url=http://www.terrapub.co.jp/e-library/kawahata/pdf/343.pdf | url-status=live | access-date=2006-10-02 | archive-url=https://web.archive.org/web/20061002083948/http://www.terrapub.co.jp/e-library/kawahata/pdf/343.pdf | archive-date=2006-10-02 }}</ref> i.e. reconverted into carbon dioxide and heat.

====Cell metabolism==== {{Main|Metabolism}}

Multicellular organisms such as humans have cell forms that are classified as [[Eukaryote]]. These cells include an [[organelle]] called the [[Mitochondrion|mitochondria]] that generates [[chemical energy]] for the rest of the hosting cell. Ninety percent of the oxygen intake by humans is utilized by the [[Mitochondrion|mitochondria]], especially for nutrient processing.<ref name=Liesa_et_al_2020>{{cite book | title=The Liver: Biology and Pathobiology | display-authors=1 | first1=Marc | last1=Liesa | first2=Ilan | last2=Benador | first3=Nathanael | last3=Miller | first4=Orian S. | last4=Shirihai | display-editors=1 | editor1-first=Irwin M. | editor1-last=Arias | editor2-first=Harvey J. | editor2-last=Alter | editor3-first=James L. | editor3-last=Boyer | editor4-first=David E. | editor4-last=Cohen | editor5-first=David A. | editor5-last=Shafritz | editor6-first=Snorri S. | editor6-last=Thorgeirsson | editor7-first=Allan W. | editor7-last=Wolkoff | edition=6th | publisher=John Wiley & Sons | year=2020 | isbn=978-1-119-43682-9 | pages=86–87 | url=https://books.google.com/books?id=80bLEAAAQBAJ&pg=PA87 }}</ref> The molecule [[adenosine triphosphate]] (ATP) is the primary energy transporter in living cells, providing an energy source for cellular processes. It is continually being broken down and synthesized as a component of cellular respiration.<ref name=Papachristodoulou_2014>{{cite book | title=Biochemistry and Molecular Biology | display-authors=1 | first1=Despo | last1=Papachristodoulou | first2=Alison | last2=Snape | first3=William H. | last3=Elliott | first4=Daphne C. | last4=Elliott | publisher=OUP Oxford | year=2014 | isbn=978-0-19-960949-9 | url=https://books.google.com/books?id=oPtzBAAAQBAJ&pg=PA5 }}</ref>

Two examples of nutrients consumed by animals are [[glucose]] (C6H12O6) and [[stearin]] (C57H110O6). These food molecules are oxidized to [[carbon dioxide]] and [[water (molecule)|water]] in the mitochondria:<ref name=Cohen_Hull_2020>{{cite book | title=Memmler's Structure & Function of the Human Body, Enhanced Edition | first1=Barbara Janson | last1=Cohen | first2=Kerry L. | last2=Hull | edition=12th | publisher=Jones & Bartlett Learning | year=2020 | isbn=978-1-284-59160-6 | page=375 }}</ref> <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]]:<ref name=Lehninger_1960>{{cite journal | title=The Enzymic and Morphological Organization of the Mitochondria | first=Albert L. | last=Lehninger | journal=Pediatrics | year=1960 | volume=26 | issue=3 | pages=466–475 | doi=10.1542/peds.26.3.466 }}</ref><ref name=Liesa_et_al_2020/> {{block indent|em=1.6|text=ADP + HPO42− → ATP + H2O}} The rest of the chemical energy of the nutrients 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).

=== 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, after accounting for interaction with the atmosphere.<ref name=Jackson_2019>{{cite book | title=Earth Science for Civil and Environmental Engineers | first=Richard E. | last=Jackson | publisher=Cambridge University Press | year=2019 | isbn=978-1-108-61581-5 | pages=123–125 | url=https://books.google.com/books?id=VJiHDwAAQBAJ&pg=PA123 }}</ref> 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).<ref name=Demirel_2012>{{cite book | title=Energy: Production, Conversion, Storage, Conservation, and Coupling | series=Green Energy and Technology | first=Yaşar | last=Demirel | publisher=Springer Science & Business Media | year=2012 | isbn=978-1-4471-2371-2 | page=305 | url=https://books.google.com/books?id=TsY8gJP7b58C&pg=PA305 }}</ref> 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.<ref name=Ahmadi_Dincer_2018>{{cite book | chapter=Energy Optimization | first1=Pouria | last1=Ahmadi | first2=Ibrahim | last2=Dincer | title=Comprehensive Energy Systems | editor-first=Ibrahim | editor-last=Dincer | publisher=Elsevier | year=2018 | isbn=978-0-12-814925-6 | volume=1 | pages=1139–1140 | chapter-url=https://books.google.com/books?id=foxODwAAQBAJ&pg=RA1-PA1139 }}</ref>

In a slower process, [[radioactive decay]] of atoms in the core of the Earth releases heat, which supplies more than half of the planet's [[Earth's internal heat budget|internal heat budget]].<ref name=Dye_2012>{{cite journal | title=Geoneutrinos and the radioactive power of the Earth | first=S. T. | last=Dye | journal=Reviews of Geophysics | volume=50 | issue=3 | date=September 2012 | article-number=2012RG000400 | doi=10.1029/2012RG000400 | arxiv=1111.6099 | bibcode=2012RvGeo..50.3007D }}</ref> In the present day, this [[radiogenic heat]] production was primarily driven by the decay of [[Uranium-235]], [[Potassium-40]], and [[Thorium-232]] some time in the past.<ref name=Nédélec_2025>{{cite book | title=Earth and Life: A History of Four Billion Years | first=Anne | last=Nédélec | publisher=Oxford University Press | year=2025 | pages=64–66 | isbn=978-0-19-894543-7 | url=https://books.google.com/books?id=ojZpEQAAQBAJ&pg=PA65 }}</ref> 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.<ref name=Kennett_2009>{{cite book | title=Seismic Wave Propagation in Stratified Media | series=DOAB Directory of Open Access Books | first=Brian | last=Kennett | publisher=ANU E Press | year=2009 | isbn=978-1-921536-73-1 | page=59 | url=https://books.google.com/books?id=Gn89socs0RcC&pg=PA59 }}</ref> 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).<ref name=Manuel_2007>{{cite book | title=Origin of Elements in the Solar System: Implications of Post-1957 Observations | first=Oliver K. | last=Manuel | publisher=Springer Science & Business Media | year=2007 | isbn=978-0-306-46927-5 | pages=589–634 | doi=10.1007/0-306-46927-8_44 | url=https://books.google.com/books?id=h-zxBwAAQBAJ&pg=PA626 }}</ref>

Early in a planet's history, the [[Accretion (astrophysics)|accretion]] process provides impact energy that can partially or completely melt the body. This allows a planet to become [[Planetary differentiation|differentiated]] by chemical element. Chemical phase changes of minerals during formation provide additional internal heating. Over time the internal heat is brought to the surface then radiated away into space, cooling the body. Accreted [[Radiogenic nuclide|radiogenic]] heat sources settle toward the core, providing thermal energy to the planet on a [[geologic time scale]].<ref name=Condie_2005>{{cite book | title=Earth as an Evolving Planetary System | first=Kent C. | last=Condie | publisher=Elsevier | year=2005 | isbn=978-0-08-049458-6 | pages=393–394 | url=https://books.google.com/books?id=I_t-hUWi5I8C&pg=PA394 }}</ref> Ongoing [[sedimentation]] provides a persistent internal energy source for [[gas giant]] planets like [[Jupiter]] and [[Saturn]].<ref name=Mankovich_Fortney_2019>{{cite journal | title=Evidence for a Dichotomy in the Interior Structures of Jupiter and Saturn from Helium Phase Separation | last1=Mankovich | first1=C. | last2=Fortney | first2=J. J. | journal=The Astrophysical Journal | volume=889 | issue=1 | date=December 2019 | page=51 | bibcode=2019AGUFM.P24B..02M | doi=10.3847/1538-4357/ab6210 | doi-access=free | arxiv=1912.01009 }}</ref>

=== Cosmology ===  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).<ref name=Hogan_2012>{{cite book | chapter=Energy flow in the universe | first=Craig J. | last=Hogan | title=Structure Formation in the Universe | editor1-first=Robert G. | editor1-last=Crittenden | editor2-first=Neil G. | editor2-last=Turok | pages=283–292 | isbn=978-94-010-0540-1 | series=NASA Science Series | publisher=Kluwer Academic Publishers | date=December 6, 2012 | chapter-url=https://books.google.com/books?id=HC7yCAAAQBAJ&pg=PA290 }}</ref>

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.<ref name=Rolfs_Rodney_1988>{{cite book | title=Cauldrons in the Cosmos: Nuclear Astrophysics | series=Theoretical Astrophysics | issn=1050-0510 | first1=Claus E. | last1=Rolfs | first2=William S. | last2=Rodney | publisher=University of Chicago Press | year=1988 | isbn=978-0-226-72457-7 | page=153 | url=https://books.google.com/books?id=BHKLFPUS1RcC&pg=PA153 }}</ref> {{anchor|Physics}}

The accretion of matter onto a [[compact object]] is a very efficient means of generating energy from [[gravitational potential]]. This behavior is responsible for some of the universe's brightest persistent energy sources.<ref name=Longair_1996>{{cite book | title=Our Evolving Universe | first=Malcolm S. | last=Longair | author-link=Malcolm Longair | publisher=CUP Archive | year=1996 | pages=86−88 | isbn=978-0-521-55091-8 | url=https://books.google.com/books?id=qyA4AAAAIAAJ&pg=PA86 }}</ref> The [[Penrose process]] is a theoretical method by which energy could be extracted from a rotating black hole.<ref>{{Cite journal | last1=Penrose | first1=R. | author-link=Roger Penrose | last2=Floyd | first2=R. M. | date=February 1971 | title=Extraction of Rotational Energy from a Black Hole | journal=Nature Physical Science | language=en | volume=229 | issue=6 | pages=177–179 | doi=10.1038/physci229177a0 | bibcode=1971NPhS..229..177P | issn=0300-8746 }}</ref> [[Hawking radiation]] is the emission of [[black-body radiation]] from a black hole, which results in a steady loss of mass and rotational energy. As the object evaporates, the temperature of this radiation is predicted to increase, speeding up the process.<ref name=Kreitler_2006>{{cite book | title=Trends in Black Hole Research | first=Paul V. | last=Kreitler | publisher=Nova Publishers | year=2006 | isbn=978-1-59454-475-0 | url=https://books.google.com/books?id=DGwYf8cOCq4C&pg=PA34 }}</ref>

=== 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]].<ref name=Parker_2018>{{cite book | title=Physics of Optoelectronics | series=Optical Science and Engineering | first=Michael A. | last=Parker | publisher=CRC Press | year=2018 | isbn=978-1-4200-2771-6 | page=299 | url=https://books.google.com/books?id=ReXLBQAAQBAJ&pg=PA299 }}</ref>

For electromagnetic waves in a vacuum, the energy states are related to the frequency by the [[Planck relation]]: <math>E = h\nu</math>, where <math>h</math> is the [[Planck constant]] and <math>\nu</math> the frequency. These energy states are called quanta of light or [[photon]]s.<ref name=Alenitsyn_et_al_2020>{{cite book | title=Concise Handbook of Mathematics and Physics | display-authors=1 | first1=Alexander G. | last1=Alenitsyn | first2=Eugene I. | last2=Butikov | first3=Alexander S. | last3=Kondratyev | publisher=CRC Press | year=2020 | isbn=978-1-000-16152-6 | page=462 | url=https://books.google.com/books?id=51gMEAAAQBAJ&pg=PA462 }}</ref>

=== 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:<ref name=Rahaman_2022>{{cite book | title=The Special Theory of Relativity: A Mathematical Approach | first=Farook | last=Rahaman | edition=2 | publisher=Springer Nature | year=2022 | isbn=978-981-19-0497-4 | page=143 | url=https://books.google.com/books?id=GL9pEAAAQBAJ&pg=PA143 }}</ref><math display="block"> E_0 = m_0 c^2 ,</math> where * ''m''0 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 the particles is created from a sufficiently energetic photon near a nucleus.<ref name=Norton_2021>{{cite book | title=Understanding the Universe: The Physics of the Cosmos from Quasars to Quarks | first=Andrew | last=Norton | publisher=CRC Press | year=2021 | isbn=978-1-000-38391-1 | url=https://books.google.com/books?id=N1kkEAAAQBAJ&pg=PA119 }}</ref>

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"/>{{Page needed|date=August 2025}}

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"),<ref name=Cao_1998>{{cite book | title=Conceptual Developments of 20th Century Field Theories | first=Tian Yu | last=Cao | publisher=Cambridge University Press | year=1998 | isbn=978-0-521-63420-5 | pages=60–63 | url=https://books.google.com/books?id=l4PtgYXpb_oC&pg=PA60 }}</ref> 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 | title=Gravitation | display-authors=1 | last1=Misner | first1=Charles W. | 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>{{Page needed|date=August 2025}} In other words, energy is invariant with respect to rotations of [[space]], but not invariant with respect to rotations of [[spacetime]] (= [[Lorentz boost|boosts]]).

== Transformation == {{Main|Energy transformation}}

{| class="wikitable" style="text-align:center;" |+Some forms of [[Energy transfer|transfer]] of energy ("energy in transit") from one object or system to another ! Type of transfer [[thermodynamic process|process]]!! Description |- |[[Heat]] |equal amount of [[Thermal energy#Differentiation from heat|thermal energy]] in transit spontaneously towards a lower-[[temperature]] object |- |[[Work (physics)|Work]] |equal amount of energy in transit due to a displacement in the direction of an applied [[force]] |- |Transfer of material |equal amount of energy carried by [[matter]] that is moving from one system to another |}

[[File:Turbogenerator01.jpg|thumb|A [[turbo generator]] transforms the energy of pressurized steam into electrical energy.]] Energy may be [[energy transformation|transformed]] between different forms at various [[energy conversion efficiency|efficiencies]]. Devices that usefully transform between these forms are called [[transducer]]s. Examples of transducers include a [[Battery (electric)|battery]] (from [[chemical energy]] to [[electric energy]]), a dam (from [[gravitational potential energy]] to the [[kinetic energy]] of water spinning the blades of a [[turbine]], and ultimately to [[electric energy]] through an [[electric generator]]), and a [[heat engine]] (from heat to work).<ref name=Blokhina_et_al_2016>{{cite book | chapter=Introduction to Vibration Energy Harvesting | title=Nonlinearity in Energy Harvesting Systems: Micro- and Nanoscale Applications | display-authors=1 | author1-first=Elena | author1-last=Blokhina | author2-first=Abdelali | author2-last=El Aroudi | author3-first=Eduard | author3-last=Alarcon | author4-first=Dimitri | author4-last=Galayko | publisher=Springer | year=2016 | isbn=978-3-319-20355-3 | pages=7–8 | chapter-url=https://books.google.com/books?id=Zc15DQAAQBAJ&pg=PA8 }}</ref><ref name=Stonier_2012>{{cite book | title=Information and the Internal Structure of the Universe: An Exploration into Information Physics | first=Tom | last=Stonier | publisher=Springer Science & Business Media | year=2012 | isbn=978-1-4471-3265-3 | pages=96–98 | url=https://books.google.com/books?id=5FPlBwAAQBAJ&pg=PA96 }}</ref>

Examples of energy transformation include generating [[electric energy]] from heat energy via a steam turbine,<ref name=Stonier_2012/> or lifting an object against gravity using electrical energy driving a crane motor. Lifting against gravity performs mechanical work on the object and stores gravitational potential energy in the object. If the object falls to the ground, gravity does mechanical work on the object which transforms the potential energy in the gravitational field to the kinetic energy released as heat on impact with the ground.<ref name=Smith_Holroyd_2013>{{cite book | title=Engineering Principles for Electrical Technicians | series=The Commonwealth and International Library: Electrical Engineering Division | first1=K. M. | last1=Smith | first2=P. | last2=Holroyd | editor-first=N. | editor-last=Hiller | publisher=Elsevier | year=2013 | isbn=978-1-4831-4030-8 | pages=63–64 | url=https://books.google.com/books?id=-D4fAwAAQBAJ&pg=PA63 }}</ref> The Sun transforms [[nuclear potential energy]] to other forms of energy; its total mass does not decrease due to that itself (since it still contains the same total energy even in different forms) but its mass does decrease when the energy escapes out to its surroundings, largely as [[radiant energy]].<ref name=Pinsonneault_Ryden_2023>{{cite book | title=Stellar Structure and Evolution | volume=2 | series=The Ohio State astrophysics series | first1=Marc | last1=Pinsonneault | first2=Barbara | last2=Ryden | publisher=Cambridge University Press | year=2023 | isbn=978-1-108-83581-7 | pages=30 | url=https://books.google.com/books?id=Fv-xEAAAQBAJ&pg=PA30 }}</ref>

There are strict limits to how efficiently heat can be converted into [[Work (physics)|work]] in a cyclic process, e.g. in a heat engine, as described by [[Carnot's theorem (thermodynamics)|Carnot's theorem]] and the [[second law of thermodynamics]].<ref name=Fleck_2023>{{cite book | title=Entropy and the Second Law of Thermodynamics: ... or Why Things Tend to Go Wrong and Seem to Get Worse | first=Robert | last=Fleck | publisher=Springer Nature | year=2023 | isbn=978-3-031-34950-8 | url=https://books.google.com/books?id=aDfZEAAAQBAJ&pg=PA64 }}</ref> However, some energy transformations can be quite efficient.<ref name=Wald_1992>{{cite book | title=Space, Time, and Gravity: The Theory of the Big Bang and Black Holes | first=Robert M. | last=Wald | publisher=University of Chicago Press | year=1992 | isbn=978-0-226-87029-8 | url=https://books.google.com/books?id=sk5a8ieI91kC&pg=PA113 }}</ref> The direction of transformations in energy (what kind of energy is transformed to what other kind) is often determined by [[entropy]] (equal energy spread among all available [[degrees of freedom (physics and chemistry)|degrees of freedom]]) considerations. In practice all energy transformations are permitted on a sufficiently small scale, but certain larger transformations are highly improbable because it is statistically unlikely that energy or matter will randomly move into more concentrated forms or smaller spaces.<ref name=Dincer_Rosen_2002>{{cite book | title=Thermal Energy Storage: Systems and Applications | first1=Ibrahim | last1=Dincer | first2=Marc | last2=Rosen | publisher=John Wiley & Sons | year=2002 | isbn=978-0-471-49573-4 | url=https://books.google.com/books?id=EsfcWE5lX40C&pg=PA20 }}</ref>

Energy transformations in the universe over time are characterized by various kinds of potential energy, that has been available since the [[Big Bang]], being "released" (transformed to more active types of energy such as kinetic or radiant energy) when a triggering mechanism is available.<ref name=Neubauer_2011>{{cite book | title=Evolution and the Emergent Self: The Rise of Complexity and Behavioral Versatility in Nature | first=Raymond L. | last=Neubauer | publisher=Columbia University Press | year=2011 | isbn=978-0-231-52168-0 | pages=263–266 | url=https://books.google.com/books?id=5VH_hCSHMCkC&pg=PA263 }}</ref> Familiar examples of such processes include [[nucleosynthesis]], a process ultimately using the gravitational potential energy released from the [[gravitational collapse]] of [[supernova]]e to "store" energy in the creation of heavy isotopes (such as [[uranium]] and [[thorium]]), and [[nuclear decay]], a process in which energy is released that was originally stored in these heavy elements, before they were incorporated into the Solar System and the Earth.<ref name=Shahvisi_2021>{{cite book | chapter=Entropy Asymmetry | first=Arianne | last=Shahvisi | title=The Routledge Companion to Philosophy of Physics | series=Routledge Philosophy Companions | editor1-first=Eleanor | editor1-last=Knox | editor2-first=Alastair | editor2-last=Wilson | publisher=Routledge | year=2021 | isbn=978-1-317-22713-7 | chapter-url=https://books.google.com/books?id=dvc8EAAAQBAJ&pg=PT626 }}</ref> This energy is triggered and released in nuclear [[fission bomb]]s or in civil nuclear power generation. Similarly, in the case of a [[Chemical explosive|chemical explosion]], [[chemical potential]] energy is transformed to [[kinetic energy|kinetic]] and [[thermal energy]] in a very short time.<ref name=Baum_et_al_1959>{{cite web | title=Physics of an explosion | date=December 1959 | display-authors=1 | last1=Baum | first1=F. A. | last2=Stanyukovich | first2=K. P. | last3=Shekhter | first3=B. I. | publisher=Defense Technical Information Center | location=Arlington, VA | url=https://apps.dtic.mil/sti/pdfs/AD0400151.pdf | access-date=2025-08-02 }}</ref>

Yet another example of energy transformation is that of a simple gravity [[pendulum]]. At its highest points the [[kinetic energy]] is zero and the [[gravitational potential energy]] is at its maximum. At its lowest point the [[kinetic energy]] is at its maximum and is equal to the decrease in [[potential energy]]. If one (unrealistically) assumes that there is no [[friction]] or other losses, the conversion of energy between these processes would be perfect, and the [[pendulum]] would continue swinging forever. Energy is transferred from potential energy (<math>E_p</math>) to kinetic energy (<math>E_k</math>) and then back to potential energy constantly. This is referred to as conservation of energy.

In this [[isolated system]], energy cannot be created or destroyed; therefore, the initial energy and the final energy will be equal to each other. This can be demonstrated by the following: {{NumBlk||<math display="block">E_{pi} + E_{ki} = E_{pF} + E_{kF}</math>|{{EquationRef|4}}}}

The equation can then be simplified further since <math>E_p = mgh</math> (mass times acceleration due to gravity times the height) and <math display="inline">E_k = \frac{1}{2} mv^2</math> (half mass times velocity squared). Then the total amount of energy can be found by adding <math>E_p + E_k = E_\text{total}</math>.<ref name=Vázquez_Corona-Corona_2018>{{cite journal | last1=Vázquez | first1=A. L. | last2=Corona-Corona | first2=G. | year=2018 | title=Period of the Simple Pendulum without Differential Equations | journal=American Scientific Research Journal for Engineering, Technology, and Sciences | volume=40 | pages=125–131 | url=https://core.ac.uk/download/pdf/235050526.pdf | access-date=2025-08-02 }}</ref>

=== Conservation of energy and mass in transformation === Within a gravitational field, both mass and energy give rise to a measureable weight when trapped in a system with zero momentum. The formula ''E'' = ''mc''2, derived by [[Albert Einstein]] (1905) quantifies this [[mass–energy equivalence]] between [[relativistic mass]] and energy within the concept of special relativity. In different theoretical frameworks, similar formulas were derived by [[J. J. Thomson]] (1881), [[Henri Poincaré]] (1900), [[Friedrich Hasenöhrl]] (1904), and others (see [[Mass–energy equivalence#History]] for further information).

Part of the rest energy (equivalent to rest mass) of [[matter]] may be converted to other forms of energy (still exhibiting mass), but neither energy nor mass can be destroyed; rather, both remain constant during any process. However, since <math>c^2</math> is extremely large relative to ordinary human scales, the conversion of an everyday amount of rest mass from rest energy to other forms of energy (such as kinetic energy, thermal energy, or the radiant energy carried by light and other radiation) can liberate tremendous amounts of energy, as can be seen in [[nuclear reactor]]s and nuclear weapons.<ref name=Ristinen_et_al_2022>{{cite book | title=Energy and the Environment | display-authors=1 | first1=Robert A. | last1=Ristinen | first2=Jack J. | last2=Kraushaar | first3=Jeffrey T. | last3=Brack | edition=4th | publisher=John Wiley & Sons | year=2022 | isbn=978-1-119-80025-5 | pages=8–9 | url=https://books.google.com/books?id=o8V6EAAAQBAJ&pg=PA8 }}</ref> For example, 1 kg of rest mass equals {{val|9|e=16|u=joules}}, equivalent to 21.5 megatonnes of TNT.<ref>The energy from the rest mass is given by the mass-energy equivalence: * E = mc2 = 1 kg × ({{Val|3|e=8|u=m/s}})2 = {{Val|9|e=16|u=J}} * TNT energy = {{Val|4.184|e=9|u=J/tonne}}† Hence, * E = ({{Val|9|e=16|u=J}})/({{Val|4.184|e=9|u=J/tonne}}) = 21.5 megatonnes †: {{cite web | title=NIST Guide to the SI, Appendix B.8: Factors for Units Listed Alphabetically | publisher=NIST | date=February 17, 2022 | url=https://www.nist.gov/pml/special-publication-811/nist-guide-si-appendix-b-conversion-factors/nist-guide-si-appendix-b8 | access-date=2025-08-04 }}</ref>

Conversely, the mass equivalent of an everyday amount energy is minuscule. Examples of large-scale transformations between the rest energy of matter and other forms of energy are found in [[nuclear physics]] and [[particle physics]]. The complete conversion of matter, such as atoms, to non-matter, such as photons, occurs during interaction with [[antimatter]].<ref name=Schmidt_et_al_2000>{{cite journal | title=Antimatter Requirements and Energy Costs for Near-Term Propulsion Applications | display-authors=1 | first1=G. R. | last1=Schmidt | first2=H. P. | last2=Gerrish | first3=J. J. | last3=Martin | first4=G. A. | last4=Smith | first5=K. J. | last5=Meyer | journal=Journal of Propulsion and Power | volume=16 | issue=5 | page=923 | date=September 2000 | doi=10.2514/2.5661 | hdl=2060/19990110316 | hdl-access=free }}</ref>

=== Reversible and non-reversible transformations === Thermodynamics divides energy transformation into two kinds: [[Reversible process (thermodynamics)|reversible processes]] and [[irreversible process]]es. An irreversible process is one in which energy is dissipated (spread) into empty energy states available in a volume, from which it cannot be recovered into more concentrated forms (fewer quantum states), without degradation of even more energy. A reversible process is one in which this sort of dissipation does not happen. For example, conversion of energy from one type of potential field to another is reversible, as in the pendulum system described above.<ref name=Eu_Al-ghoul_2018>{{cite book | title=Chemical Thermodynamics: Reversible And Irreversible Thermodynamics | edition=Second | first1=Byung Chan | last1=Eu | first2=Mazen | last2=Al-ghoul | publisher=World Scientific Publishing Company | year=2018 | isbn=978-981-322-607-4 | url=https://books.google.com/books?id=uWdhDwAAQBAJ&pg=PT45 }}</ref>

At the atomic scale, thermal energy is present in the form of motion and vibrations of individual atoms and molecules. When heat is generated, radiation excites lower energy states of these atoms and their surrounding fields. This heating process acts as a reservoir for part of the applied energy, from which it cannot be converted with 100% efficiency into other forms of energy.<ref name=Avison_2014>{{cite book | title=The World of Physics | edition=2nd | first=John | last=Avison | publisher=Nelson Thornes | year=2014 | isbn=978-0-17-438733-6 | page=414 | url=https://books.google.com/books?id=DojwZzKAvN8C&pg=PA414 }}</ref> According to the second law of thermodynamics, this heat can only be completely recovered as usable energy at the price of an increase in some other kind of heat-like disorder in quantum states.

As the universe evolves with time, more and more of its energy becomes trapped in irreversible states (i.e., as heat or as other kinds of increases in disorder). This has led to the hypothesis of the inevitable thermodynamic [[heat death of the universe]]. In this heat death the energy of the universe does not change, but the fraction of energy which is available to do work through a [[heat engine]], or be transformed to other usable forms of energy (through the use of generators attached to heat engines), continues to decrease.<ref name=Luscombe_2018>{{cite book | title=Thermodynamics | first=James | last=Luscombe | publisher=CRC Press | year=2018 | isbn=978-0-429-01788-9 | pages=60–62 | url=https://books.google.com/books?id=fFwPEAAAQBAJ&pg=PA60 }}</ref>

== Conservation of energy == {{Main|Conservation of energy}}

The fact that energy can be neither created nor destroyed is called the law of [[conservation of energy]]. In the form of the [[first law of thermodynamics]], this states that a [[closed system]]'s energy is constant unless energy is transferred in or out as [[Work (thermodynamics)|work]] or [[heat]], and that no energy is lost in transfer. The total inflow of energy into a system must equal the total outflow of energy from the system, plus the change in the energy contained within the system. 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.<ref>{{cite book | first1=Charles | last1=Kittel | first2=Walter D. | last2=Knight | first3=Malvin A. | last3=Ruderman | title=Berkeley Physics Course | volume=1 | year=1965 | publisher=McGraw-Hill }}</ref>

While heat can always be fully converted into work in a reversible isothermal expansion of an ideal gas, for cyclic processes of practical interest in [[heat engine]]s the [[second law of thermodynamics]] states that the system doing work always loses some energy as [[waste heat]]. This creates a limit to the amount of heat energy that can do work in a cyclic process, a limit called the [[available energy]]. Mechanical and other forms of energy can be transformed in the other direction into [[thermal energy]] without such limitations.<ref name="thermo-laws"/> The total energy of a system can be calculated by adding up all forms of energy in the system.

[[Richard Feynman]] said during a 1961 lecture:<ref name="RPF1"/> {{Blockquote|There is a fact, or if you wish, a ''law'', governing all natural phenomena that are known to date. There is no known exception to this law – it is exact so far as we know. The law is called the ''[[conservation of energy]]''. It states that there is a certain quantity, which we call energy, that does not change in manifold changes which nature undergoes. That is a most abstract idea, because it is a mathematical principle; it says that there is a numerical quantity which does not change when something happens. It is not a description of a mechanism, or anything concrete; it is just a strange fact that we can calculate some number and when we finish watching nature go through her tricks and calculate the number again, it is the same.|''[[The Feynman Lectures on Physics]]''}}

Most kinds of energy (with gravitational energy being a notable exception)<ref>{{cite web|url=http://www.physics.ucla.edu/~cwp/articles/noether.asg/noether.html |title=E. Noether's Discovery of the Deep Connection Between Symmetries and Conservation Laws |publisher=UCLA Physics & Astronomy |date=December 1996 |first1=Nina |last1=Byers |access-date=2010-12-12 |url-status=dead |archive-url=https://web.archive.org/web/20110514080739/http://www.physics.ucla.edu/~cwp/articles/noether.asg/noether.html |archive-date=2011-05-14 }}</ref> are subject to strict local conservation laws as well. In this case, energy can only be exchanged between adjacent regions of space, and all observers agree as to the volumetric density of energy in any given space. There is also a global law of conservation of energy, stating that the total energy of the universe cannot change; this is a corollary of the local law, but not vice versa.<ref name="thermo-laws">[http://www.av8n.com/physics/thermo-laws.htm ''The Laws of Thermodynamics'']. {{webarchive|url=https://web.archive.org/web/20061215201900/http://www.av8n.com/physics/thermo-laws.htm|date=2006-12-15}} including careful definitions of energy, free energy, et cetera.</ref><ref name="RPF1">{{Cite book | first=Richard | last=Feynman | title=The Feynman Lectures on Physics; Volume 1 | chapter=Ch. 4: Conservation of Energy | chapter-url=https://feynmanlectures.caltech.edu/I_04.html#Ch4-S1-p2 | year=1964 | publisher=Addison Wesley | location=US | isbn=((978-0-201-02115-8)) | access-date=2022-05-04 | archive-date=2022-07-30 | archive-url=https://web.archive.org/web/20220730093042/https://www.feynmanlectures.caltech.edu/I_04.html#Ch4-S1-p2 | url-status=live }}</ref>

This law is a fundamental principle of physics. As shown rigorously by [[Noether's theorem]], the conservation of energy is a mathematical consequence of [[translational symmetry]] of time,<ref>{{cite web |url=http://ptolemy.eecs.berkeley.edu/eecs20/week9/timeinvariance.html |title=Time Invariance |publisher=Ptolemy Project |work=EECS20N |access-date=2010-12-12 |url-status=live |archive-url=https://web.archive.org/web/20110717210455/http://ptolemy.eecs.berkeley.edu/eecs20/week9/timeinvariance.html |archive-date=2011-07-17 }}</ref> a property of most phenomena below the cosmic scale that makes them independent of their locations on the time coordinate. Put differently, yesterday, today, and tomorrow are physically indistinguishable. This is because energy is the quantity which is [[canonical conjugate]] to time. This mathematical entanglement of energy and time also results in the uncertainty principle – it is impossible to define the exact amount of energy during any definite time interval (though this is practically significant only for very short time intervals). The uncertainty principle should not be confused with [[energy conservation]] – rather it provides mathematical limits to which energy can in principle be defined and measured.

Each of the basic forces of nature is associated with a different type of potential energy, and all types of potential energy (like all other types of energy) appear as system [[mass]], whenever present. For example, a compressed spring will be slightly more massive than before it was compressed. Likewise, whenever energy is transferred between systems by any mechanism, an associated mass is transferred with it.<ref name=Schmitz_2019>{{cite book | title=Particles, Fields and Forces: A Conceptual Guide to Quantum Field Theory and the Standard Model | series=The Frontiers Collection | first=Wouter | last=Schmitz | publisher=Springer | year=2019 | isbn=978-3-030-12878-4 | page=245 | url=https://books.google.com/books?id=wXeUDwAAQBAJ&pg=PA245 }}</ref>

In [[quantum mechanics]] energy is expressed using the [[Hamiltonian operator]]. On any time scale, the uncertainty in the energy is given by: : <math>\Delta E \Delta t \ge \frac { \hbar } {2 } </math> which is similar in form to the [[Heisenberg Uncertainty Principle]],<ref name=Wigner_1997>{{Cite book | last=Wigner | first=E. P. |author-link=Eugene Wigner | chapter=On the Time–Energy Uncertainty Relation | year=1997 | editor-last=Wightman | editor-first=Arthur S. | title=Part I: Particles and Fields. Part II: Foundations of Quantum Mechanics | chapter-url=http://link.springer.com/10.1007/978-3-662-09203-3_58 | language=en | location=Berlin, Heidelberg | publisher=Springer Berlin Heidelberg | pages=538–548 | doi=10.1007/978-3-662-09203-3_58 | isbn=978-3-642-08179-8 }}</ref> but not really mathematically equivalent thereto, since ''E'' and ''t'' are not dynamically [[conjugate variables]], neither in classical nor in quantum mechanics.<ref name=Desai_2010>{{cite book | title=Quantum Mechanics with Basic Field Theory | first=Bipin R. | last=Desai | publisher=Cambridge University Press | year=2010 | isbn=978-0-521-87760-2 | pages=63–66 | url=https://books.google.com/books?id=cScBCJ6wLpYC&pg=PA65 }}</ref>

In [[particle physics]], this inequality permits a qualitative understanding of [[virtual particles]], which carry [[momentum]].<ref name=Desai_2010/> The exchange of virtual particles with real particles is responsible for the creation of all known [[fundamental forces]] (more accurately known as [[fundamental interactions]]).<ref name=Braibant_et_al_2011>{{cite book | title=Particles and Fundamental Interactions: An Introduction to Particle Physics | series=Undergraduate Lecture Notes in Physics | first1=Sylvie | last1=Braibant | first2=Giorgio | last2=Giacomelli | first3=Maurizio | last3=Spurio | publisher=Springer Science & Business Media | year=2011 | isbn=978-94-007-2463-1 | url=https://books.google.com/books?id=0Pp-f0G9_9sC&pg=PA101 }}</ref>{{rp|p=101}} [[Virtual photons]] are also responsible for the electrostatic interaction between [[electric charge]]s (which results in [[Coulomb's law]]),<ref name=Braibant_et_al_2011/>{{rp|p=336}} for [[Spontaneous fission|spontaneous]] radiative decay of excited atomic and nuclear states, for the [[Casimir force]],<ref name=Madou_2011>{{cite book | title=Solid-State Physics, Fluidics, and Analytical Techniques in Micro- and Nanotechnology | first=Marc J. | last=Madou | publisher=CRC Press | year=2011 | isbn=978-1-4398-9534-4 | page=542 | url=https://books.google.com/books?id=sRvSBQAAQBAJ&pg=PA542 }}</ref> for the [[Van der Waals force]],<ref name=Volokitin_Persson_2007>{{cite book | chapter=Theory of Noncontact Friction | first1=A. I. | last1=Volokitin | first2=B. N. J. | last2=Persson | title=Fundamentals of Friction and Wear on the Nanoscale | series=NanoScience and Technology | editor1-first=Enrico | editor1-last=Gnecco | editor2-first=Ernst | editor2-last=Meyer | publisher=Springer Science & Business Media | year=2007 | isbn=978-3-540-36807-6 | page=394 | chapter-url=https://books.google.com/books?id=v2Pe5thhNiwC&pg=PA394 }}</ref> and some other observable phenomena.<ref name=Scully_et_al_2018>{{cite journal | last1=Scully | first1=M. | last2=Sokolov | first2=A. | last3=Svidzinsky | first3=A. | year=2018 | title=Virtual photons: From the Lamb shift to black holes | journal=Optics and Photonics News | volume=29 | issue=2 | pages=34–40 | doi=10.1364/OPN.29.2.000034 | bibcode=2018OptPN..29...34S | url=https://www.fonlo.org/2018/additionalreading/Scully-Marlan/OPN-VirtualPhotonsBH.pdf | access-date=2025-08-04 }}</ref>

== Energy transfer == {{redirect|Energy transfer|the pipeline company|Energy Transfer Partners}}

=== Closed systems === Energy transfer can be considered for the special case of systems which are [[closed system|closed]] to transfers of matter. The portion of the energy which is transferred by [[conservative force]]s over a distance is measured as the [[Work (thermodynamics)|work]] the source system does on the receiving system. The portion of the energy which does not do work during the transfer is called [[heat]].<ref group=note>Although heat is "wasted" energy for a specific energy transfer (see: [[waste heat]]), it can often be harnessed to do useful work in subsequent interactions. However, the maximum energy that can be "recycled" from such recovery processes is limited by the [[second law of thermodynamics]].</ref> Energy can be transferred between systems in a variety of ways. Examples include the transmission of [[electromagnetic energy]] via photons, physical collisions which transfer [[kinetic energy]],<ref group=note>The mechanism for most macroscopic physical collisions is actually [[Electromagnetism|electromagnetic]], but it is very common to simplify the interaction by ignoring the mechanism of collision and just calculate the beginning and end result.</ref> [[tidal interactions]],<ref>{{cite book | title=The Physics of Energy | first1=Robert L. | last1=Jaffe | first2=Washington | last2=Taylor | date=2018 | isbn=978-1-107-01665-1 | page=611 | publisher=Cambridge University Press | url=https://books.google.com/books?id=drZDDwAAQBAJ&pg=PA611 }}</ref> and the conductive transfer of [[thermal energy]].<ref name=Lucien_Daniel_2010>{{cite book | title=Thermodynamics and Energy Systems Analysis: From Energy to Exergy | series=Engineering Sciences | first1=Lucien | last1=Borel | first2=Daniel | last2=Favrat | publisher=EPFL Press | year=2010 | isbn=978-1-4398-3516-6 | url=https://books.google.com/books?id=bnyCpHkqQ_0C&pg=PA16 }}</ref>

Energy is strictly conserved and is also locally conserved wherever it can be defined. In thermodynamics, for closed systems, the process of energy transfer is described by the [[first law of thermodynamics|first law]]:<ref group=note>There are several [[First law of thermodynamics#Description|sign conventions for this equation]]. Here, the signs in this equation follow the IUPAC convention.</ref><ref name=Lucien_Daniel_2010/>

{{NumBlk|:|<math>\Delta{}E = W + Q </math>|{{EquationRef|1}}}}

where <math>E</math> is the amount of energy transferred, <math>W</math>  represents the work done on or by the system, and <math>Q</math> represents the heat flow into or out of the system. As a simplification, the heat term, <math>Q</math>, can sometimes be ignored, especially for fast processes involving gases, which are poor conductors of heat, or when the [[thermal efficiency]] of the transfer is high. For such [[adiabatic process]]es,

{{NumBlk|:|<math>\Delta{}E = W</math>|{{EquationRef|2}}}}

This simplified equation is the one used to define the [[joule]], for example.

=== Open systems === Beyond the constraints of closed systems, [[Thermodynamic system#Open system|open systems]] can gain or lose energy in association with matter transfer (this process is illustrated by injection of an air-fuel mixture into a car engine, a system which gains in energy thereby, without addition of either work or heat). Denoting this energy by <math>E_\text{matter}</math>, one may write:<ref name=Rathakrishnan_2019>{{cite book | title=Applied Gas Dynamics | first=Ethirajan | last=Rathakrishnan | edition=2nd | publisher=John Wiley & Sons | year=2019 | isbn=978-1-119-50038-4 | pages=12–13 | url=https://books.google.com/books?id=esaKDwAAQBAJ&pg=PA12 }}</ref>

{{NumBlk|:|<math>\Delta E = W + Q + E_\text{matter} .</math>|{{EquationRef|3}}}}

== Thermodynamics == {{Thermodynamics}}

=== Internal energy === [[Internal energy]] is the sum of all microscopic forms of energy of a system. It is the energy needed to create the system. It is related to the potential energy, e.g., molecular structure, crystal structure, and other geometric aspects, as well as the motion of the particles, in form of kinetic energy. Thermodynamics is chiefly concerned with changes in internal energy and not its absolute value, which is impossible to determine with thermodynamics alone.<ref name=klotz>I. Klotz, R. Rosenberg, ''Chemical Thermodynamics – Basic Concepts and Methods'', 7th ed., Wiley (2008), p. 39</ref>

=== First law of thermodynamics === The [[first law of thermodynamics]] asserts that the total energy of a system and its surroundings (but not necessarily [[thermodynamic free energy]]) is always conserved<ref name="KK">{{Cite book|author=Kittel and Kroemer|title=Thermal Physics |year=1980|publisher=W. H. Freeman |location=New York| isbn=978-0-7167-1088-2}}</ref> and that heat flow is a form of energy transfer. For homogeneous systems, with a well-defined temperature and pressure, 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) without chemical changes, the differential change in the internal energy of the system (with a ''gain'' in energy signified by a positive quantity) is given as:<ref name=Chen_2022>{{cite book | title=Thermodynamic Equilibrium and Stability of Materials |author1-link=Long-Qing Chen | first1=Long-Qing | last1=Chen | publisher=Springer Nature | year=2022 | isbn=978-981-13-8691-6 | pages=30–31 | url=https://books.google.com/books?id=82RXEAAAQBAJ&pg=PA30 }}</ref>

: <math>\mathrm{d}E = T\mathrm{d}S - P\mathrm{d}V\,,</math> where the first term on the right is the heat transferred into the system, expressed in terms of [[temperature]] ''T'' and [[entropy]] ''S'' (in which entropy increases and its change d''S'' is positive when heat is added to the system), 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 requires work to be done on it and so the volume change, d''V'', is negative when work is done on the system).

This equation is highly specific, ignoring all chemical, electrical, nuclear, and gravitational forces, effects such as [[advection]] of any form of energy other than heat and ''PV''-work. The general formulation of the first law (i.e., conservation of energy) is valid even in situations in which the system is not homogeneous. For these cases the change in internal energy of a ''closed'' system is expressed in a general form by:<ref name=Lucien_Daniel_2010/> : <math>\mathrm{d}E=\delta Q+\delta W</math> where <math>\delta Q</math> is the heat supplied to the system and <math>\delta W</math> is the work applied to the system.

=== Equipartition of energy === The energy of a mechanical [[harmonic oscillator]] (a mass on a spring) is alternately [[kinetic energy|kinetic]] and [[potential energy]]. At two points in the oscillation [[Frequency|cycle]] it is entirely kinetic, and at two points it is entirely potential.<ref name=Vázquez_Corona-Corona_2018/> Over a whole cycle, or over many cycles, average energy is equally split between kinetic and potential. This is an example of the [[equipartition principle]]: the total energy of a system with many degrees of freedom is equally split among all available degrees of freedom, on average.<ref name=Dill_Bromberg_2010>{{cite book | title=Molecular Driving Forces: Statistical Thermodynamics in Biology, Chemistry, Physics, and Nanoscience | first1=Ken |last1=Dill | first2=Sarina | last2=Bromberg | edition=2nd | publisher=Garland Science | year=2010 | isbn=978-1-1366-7299-6 | pages=212–213 | url=https://books.google.com/books?id=1gYPBAAAQBAJ&pg=PA212 }}</ref>

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 (mathematics)|distribution]] of energy between parts of a system. When an isolated system is given more degrees of freedom (i.e., given new available [[energy state]]s that 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 part of the [[second law of thermodynamics]]. The second law of thermodynamics is simple only for systems which are near or in a physical [[equilibrium state]]. For non-equilibrium systems, the laws governing the systems' behavior are still debatable. One of the guiding principles for these systems is the principle of [[principle of maximum entropy|maximum entropy production]].<ref>{{cite journal | last=Onsager | first=L. | title=Reciprocal relations in irreversible processes | journal=Physical Review | volume=37 | issue=4 | date=1931 | pages=405–26 | bibcode=1931PhRv...37..405O | doi=10.1103/PhysRev.37.405 | doi-access=free }}</ref><ref>{{cite journal | last1=Martyushev | first1=L. M. | last2=Seleznev | first2=V. D. | date=2006 | title=Maximum entropy production principle in physics, chemistry and biology | journal=Physics Reports | volume=426 | issue=1 | pages=1–45 | bibcode=2006PhR...426....1M | doi=10.1016/j.physrep.2005.12.001 }}</ref> It states that nonequilibrium systems behave in such a way as to maximize their entropy production.<ref>{{cite journal | last1=Belkin | first1=A. | last2=Hubler | first2=A. | last3=Bezryadin | first3=A. | title=Self-Assembled Wiggling Nano-Structures and the Principle of Maximum Entropy Production | journal=Scientific Reports | volume=5 | pages=8323 | date=2015 | issue=1 | doi=10.1038/srep08323 | pmid=25662746 | pmc=4321171 | bibcode=2015NatSR...5.8323B }}</ref>

== See also == {{Portal|Energy|Physics|Renewable energy}} {{Div col|colwidth=15em}} * [[Combustion]] * [[Efficient energy use]] * [[Energy democracy]] * [[Energy crisis]] * [[Energy recovery]] * [[Energy recycling]] * [[Index of energy articles]] * [[Index of wave articles]] * [[List of low-energy building techniques]] * [[Orders of magnitude (energy)]] * [[Power station]] * [[Sustainable energy]] * [[Spaceflight#Transfer energy|Transfer energy]] * [[Waste-to-energy]] * [[Waste-to-energy plant]] * [[Zero-energy building]] {{Div col end}} {{clear|left}}

== Notes == {{reflist|group=note|30em}}

== References == {{reflist}}

== External links== {{Sister project links|d=Q11379|collapsible=collapsed}} {{Prone to spam|date=July 2013}}  * [http://www.biocab.org/Heat.html Differences between Heat and Thermal energy] ({{Webarchive|url=https://web.archive.org/web/20160827224418/http://www.biocab.org/Heat.html |date=2016-08-27 }}) – BioCab * [http://www.energyhistory.eu/en ''The Journal of Energy History / Revue d'histoire de l'énergie'' (JEHRHE), 2018–]{{Webarchive|url=https://web.archive.org/web/20211113191823/http://www.energyhistory.eu/en |date=2021-11-13 }}

[[Category:Energy (physics)| ]] [[Category:Main topic articles]] [[Category:Universe]] [[Category:Scalar physical quantities]]