View source for Conservation of energy

{{Short description|Law of physics and chemistry}}
{{About|the law of conservation of energy in physics |sustainable energy resources|Energy conservation}}
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{{Use dmy dates|date=October 2020}}
{{Continuum mechanics |laws}}

The '''law of conservation of energy''' states that the total [[energy]] of an [[isolated system]] remains constant; it is said to be [[Conservation law|''conserved'']] over time.<ref name=Feynman2Ch1S2>{{cite book |author=Richard Feynman |title=The Feynman Lectures on Physics Vol I |publisher=Addison Wesley  |year=1970 |isbn=978-0-201-02115-8  |url=https://feynmanlectures.caltech.edu/I_04.html}}</ref> In the case of a [[Closed_system#In_thermodynamics|closed system]] the principle says that the total amount of energy within the system can only be changed through energy entering or leaving the system. Energy can neither be created nor destroyed; rather, it can only be transformed or transferred from one form to another. For instance, [[chemical energy]] is [[Energy conversion|converted]] to [[kinetic energy]] when a stick of [[dynamite]] explodes. If one adds up all forms of energy that were released in the explosion, such as the kinetic energy and [[potential energy]] of the pieces, as well as heat and sound, one will get the exact decrease of chemical energy in the combustion of the dynamite.

Classically, conservation of energy was distinct from [[conservation of mass]]. However, [[special relativity]] shows that mass is related to energy and vice versa by <math qid=Q35875>E = mc^2</math>, the equation representing [[mass–energy equivalence]], and science now takes the view that mass-energy as a whole is conserved. Theoretically, this implies that any object with mass can itself be converted to pure energy, and vice versa. However, this is believed to be possible only under the most extreme of physical conditions, such as likely existed in the universe [[Grand unification epoch|very shortly after the Big Bang]] or when [[black hole]]s emit [[Hawking radiation]].

Given the [[stationary-action principle]], conservation of energy can be rigorously proven by [[Noether's theorem]] as a consequence of [[Continuous function|continuous]] [[time translation symmetry]]; that is, from the fact that the laws of physics do not change over time.

A consequence of the law of conservation of energy is that a [[Perpetual motion#Classification|perpetual motion machine of the first kind]] cannot exist; that is to say, no system without an external energy supply can deliver an unlimited amount of energy to its surroundings.<ref>Planck, M. (1923/1927). ''Treatise on Thermodynamics'', third English edition translated by A. Ogg from the seventh German edition, Longmans, Green & Co., London, page 40.</ref> Depending on the definition of energy, conservation of energy can arguably be violated by [[general relativity]] on the cosmological scale.<ref name="Energy Is Not Conserved">{{cite news |author1-link=Sean Carroll |title=Energy Is Not Conserved |url=https://www.discovermagazine.com/the-sciences/energy-is-not-conserved |access-date=25 September 2022 |work=Discover Magazine |date=2010 |language=en}}</ref>

==History==
{{More citations needed section|date=November 2015}}
[[Ancient philosophy|Ancient philosophers]] as far back as [[Thales of Miletus]] {{circa}}&nbsp;550 BCE had inklings of the conservation of some underlying substance of which everything is made. However, there is no particular reason to identify their theories with what we know today as "mass-energy" (for example, Thales thought it was water). [[Empedocles]] (490–430 BCE) wrote that in his universal system, composed of [[Classical element|four roots]] (earth, air, water, fire), "nothing comes to be or perishes";<ref>{{cite journal|last=Janko|first=Richard|title=Empedocles, "On Nature"|journal=Zeitschrift für Papyrologie und Epigraphik|year=2004 |volume=150 |pages=1–26|url=http://ancphil.lsa.umich.edu/-/downloads/faculty/janko/empedocles-nature.pdf }}</ref> instead, these elements suffer continual rearrangement. [[Epicurus]] ({{circa}}&nbsp;350 BCE) on the other hand believed everything in the universe to be composed of indivisible units of matter—the ancient precursor to 'atoms'—and he too had some idea of the necessity of conservation, stating that "the sum total of things was always such as it is now, and such it will ever remain."<ref>{{cite book|last=Laertius|first=Diogenes|title=Lives of Eminent Philosophers: Epicurus|url=https://www-loebclassics.com/view/diogenes_laertius-lives_eminent_philosophers_book_x_epicurus/1925/pb_LCL185.569.xml?result=1&rskey=YoU4V6}}{{Dead link|date=March 2021 |bot=InternetArchiveBot |fix-attempted=yes }}. This passage comes from a letter quoted in full by Diogenes, and purportedly written by Epicurus himself in which he lays out the tenets of his philosophy.</ref>

In 1605, the Flemish scientist [[Simon Stevin]] was able to solve a number of problems in statics based on the principle that [[perpetual motion]] was impossible.

In 1639, [[Galileo Galilei|Galileo]] published his analysis of several situations—including the celebrated "interrupted pendulum"—which can be described (in modern language) as conservatively converting potential energy to kinetic energy and back again. Essentially, he pointed out that the height a moving body rises is equal to the height from which it falls, and used this observation to infer the idea of inertia. The remarkable aspect of this observation is that the height to which a moving body ascends on a frictionless surface does not depend on the shape of the surface.

In 1669, [[Christiaan Huygens]] published his laws of collision. Among the quantities he listed as being invariant before and after the collision of bodies were both the sum of their [[linear momentum|linear momenta]] as well as the sum of their kinetic energies. However, the difference between elastic and inelastic collision was not understood at the time. This led to the dispute among later researchers as to which of these conserved quantities was the more fundamental. In his ''[[Horologium Oscillatorium]]'', he gave a much clearer statement regarding the height of ascent of a moving body, and connected this idea with the impossibility of perpetual motion. Huygens's study of the dynamics of pendulum motion was based on a single principle: that the center of gravity of a heavy object cannot lift itself.

[[File:Gottfried Wilhelm Leibniz.jpg|thumb|150px|[[Gottfried Leibniz]]]]
Between 1676 and 1689, [[Gottfried Leibniz]] first attempted a mathematical formulation of the kind of energy that is associated with ''motion'' (kinetic energy). Using Huygens's work on collision, Leibniz noticed that in many mechanical systems (of several [[mass]]es ''m<sub>i</sub>'', each with [[velocity]] ''v<sub>i</sub>''),
:<math>\sum_{i} m_i v_i^2</math>

was conserved so long as the masses did not interact. He called this quantity the ''[[vis viva]]'' or ''living force'' of the system. The principle represents an accurate statement of the approximate conservation of [[kinetic energy]] in situations where there is no friction. Many [[physicist]]s at that time, including [[Isaac Newton]], held that the [[conservation of momentum]], which holds even in systems with friction, as defined by the [[momentum]]:

:<math>\sum_{i} m_i v_i</math>

was the conserved ''vis viva''. It was later shown that both quantities are conserved simultaneously given the proper conditions, such as in an [[elastic collision]].

In 1687, [[Isaac Newton]] published his ''[[Philosophiæ Naturalis Principia Mathematica|Principia]]'', which set out his [[Newton's laws of motion|laws of motion]]. It was organized around the concept of force and momentum. However, the researchers were quick to recognize that the principles set out in the book, while fine for point masses, were not sufficient to tackle the motions of rigid and fluid bodies. Some other principles were also required.

By the 1690s, Leibniz was arguing that conservation of ''vis viva'' and conservation of momentum undermined the then-popular philosophical doctrine of [[interactionist dualism]]. (During the 19th century, when conservation of energy was better understood, Leibniz's basic argument would gain widespread acceptance. Some modern scholars continue to champion specifically conservation-based attacks on dualism, while others subsume the argument into a more general argument about [[causal closure]].)<ref>{{cite journal |last1=Pitts |first1=J. Brian |title=Conservation of Energy: Missing Features in Its Nature and Justification and Why They Matter |journal=Foundations of Science |date=September 2021 |volume=26 |issue=3 |pages=559–584 |doi=10.1007/s10699-020-09657-1|pmid=34759713 |pmc=8570307 }}</ref>

[[File:Daniel Bernoulli 001.jpg|thumb|left|150px|[[Daniel Bernoulli]]]]

The law of conservation of vis viva was championed by the father and son duo, [[Johann Bernoulli|Johann]] and [[Daniel Bernoulli]]. The former enunciated the principle of [[virtual work]] as used in statics in its full generality in 1715, while the latter based his ''[[Hydrodynamica]]'', published in 1738, on this single vis viva conservation principle. Daniel's study of loss of vis viva of flowing water led him to formulate the [[Bernoulli's principle]], which asserts the loss to be proportional to the change in hydrodynamic pressure. Daniel also formulated the notion of [[Work (physics)|work]] and efficiency for [[hydraulic]] machines; and he gave a kinetic theory of gases, and linked the kinetic energy of gas molecules with the temperature of the gas.

This focus on the vis viva by the continental physicists eventually led to the discovery of stationarity principles governing mechanics, such as the [[D'Alembert's principle]], [[Lagrangian mechanics|Lagrangian]], and [[Hamiltonian mechanics|Hamiltonian]] formulations of mechanics.
 
[[File:Emilie Chatelet portrait by Latour.jpg|thumb|right|150px|[[Emilie du Chatelet]]]]

[[Émilie du Châtelet]] (1706–1749) proposed and tested the hypothesis of the conservation of total energy, as distinct from momentum. Inspired by the theories of Gottfried Leibniz, she repeated and publicized an experiment originally devised by [[Willem 's Gravesande]] in 1722 in which balls were dropped from different heights into a sheet of soft clay. Each ball's kinetic energy—as indicated by the quantity of material displaced—was shown to be proportional to the square of the velocity. The deformation of the clay was found to be directly proportional to the height from which the balls were dropped, equal to the initial potential energy. Some earlier workers, including Newton and Voltaire, had believed that "energy" was not distinct from momentum and therefore proportional to velocity. According to this understanding, the deformation of the clay should have been proportional to the square root of the height from which the balls were dropped. In classical physics, the correct formula is <math>E_k = \frac12 mv^2</math>, where <math>E_k</math> is the kinetic energy of an object, <math>m</math> its mass and <math>v</math> its [[speed]]. On this basis, du Châtelet proposed that energy must always have the same dimensions in any form, which is necessary to be able to consider it in different forms (kinetic, potential, heat, ...).<ref name=Hagengruber>Hagengruber, Ruth, editor (2011) ''Émilie du Chatelet between Leibniz and Newton''. Springer. {{ISBN|978-94-007-2074-9}}.</ref><ref name=Arianrhod>{{cite book|last1=Arianrhod|first1=Robyn|title=Seduced by logic : Émilie du Châtelet, Mary Somerville, and the Newtonian revolution|date=2012|publisher=Oxford University Press|location=New York|isbn=978-0-19-993161-3|edition=US|url=http://journals.cambridge.org/action/displayAbstract?fromPage=online&aid=9444991}}</ref>

[[Engineer]]s such as [[John Smeaton]], [[Peter Ewart]], {{interlanguage link|Carl Holtzmann|de||ar|كارل هولتزمان}}, [[Gustave-Adolphe Hirn]], and [[Marc Seguin]] recognized that conservation of momentum alone was not adequate for practical calculation and made use of Leibniz's principle. The principle was also championed by some [[chemist]]s such as [[William Hyde Wollaston]]. Academics such as [[John Playfair]] were quick to point out that kinetic energy is clearly not conserved. This is obvious to a modern analysis based on the [[second law of thermodynamics]], but in the 18th and 19th centuries, the fate of the lost energy was still unknown.

Gradually it came to be suspected that the heat inevitably generated by motion under friction was another form of ''vis viva''. In 1783, [[Antoine Lavoisier]] and [[Pierre-Simon Laplace]] reviewed the two competing theories of ''vis viva'' and [[caloric theory]].<ref>Lavoisier, A.L. & Laplace, P.S. (1780) "Memoir on Heat", ''Académie Royale des Sciences'' pp.&nbsp;4–355</ref><ref>{{cite journal |last1=Guerlac |first1=Henry |title=Chemistry as a Branch of Physics: Laplace's Collaboration with Lavoisier |journal=Historical Studies in the Physical Sciences |date=1976 |volume=7 |pages=193–276 |doi=10.2307/27757357 |url=https://online.ucpress.edu/hsns/article-abstract/doi/10.2307/27757357/47949/Chemistry-as-a-Branch-of-Physics-Laplace-s?redirectedFrom=fulltext |access-date=24 March 2022 |publisher=University of California Press|jstor=27757357 }}</ref> [[Benjamin Thompson|Count Rumford]]'s 1798 observations of heat generation during the [[Boring (manufacturing)|boring]] of [[cannon]]s added more weight to the view that mechanical motion could be converted into heat and (that it was important) that the conversion was quantitative and could be predicted (allowing for a universal conversion constant between kinetic energy and heat). ''Vis viva'' then started to be known as ''energy'', after the term was first used in that sense by [[Thomas Young (scientist)|Thomas Young]] in 1807.

[[File:Gaspard-Gustave de Coriolis.jpg|thumb|150px|[[Gaspard-Gustave Coriolis]]]]

The recalibration of ''vis viva'' to

:<math>\frac {1} {2}\sum_{i} m_i v_i^2</math>

which can be understood as converting kinetic energy to [[Work (thermodynamics)|work]], was largely the result of [[Gaspard-Gustave Coriolis]] and [[Jean-Victor Poncelet]] over the period 1819–1839. The former called the quantity ''quantité de travail'' (quantity of work) and the latter, ''travail mécanique'' (mechanical work), and both championed its use in engineering calculations.

In the paper ''Über die Natur der Wärme'' (German "On the Nature of Heat/Warmth"), published in the {{Lang|de|[[Zeitschrift für Physik]]}} in 1837, [[Karl Friedrich Mohr]] gave one of the earliest general statements of the doctrine of the conservation of energy: "besides the 54 known chemical elements there is in the physical world one agent only, and this is called ''Kraft'' [energy or work]. It may appear, according to circumstances, as motion, chemical affinity, cohesion, electricity, light and magnetism; and from any one of these forms it can be transformed into any of the others."

===Mechanical equivalent of heat===
A key stage in the development of the modern conservation principle was the demonstration of the ''[[mechanical equivalent of heat]]''. The [[caloric theory]] maintained that heat could neither be created nor destroyed, whereas conservation of energy entails the contrary principle that heat and mechanical work are interchangeable.

In the middle of the eighteenth century, [[Mikhail Lomonosov]], a Russian scientist, postulated his corpusculo-kinetic theory of heat, which rejected the idea of a caloric. Through the results of empirical studies, Lomonosov came to the conclusion that heat was not transferred through the particles of the caloric fluid.

In 1798, Count Rumford ([[Benjamin Thompson]]) performed measurements of the frictional heat generated in boring cannons and developed the idea that heat is a form of kinetic energy; his measurements refuted caloric theory, but were imprecise enough to leave room for doubt.

[[File:SS-joule.jpg|thumb|left|130px|[[James Prescott Joule]]]]

The mechanical [[equivalence principle]] was first stated in its modern form by the German surgeon [[Julius Robert von Mayer]] in 1842.<ref>von Mayer, J.R. (1842) "Remarks on the forces of inorganic nature" in ''Annalen der Chemie und Pharmacie'', '''43''', 233</ref> Mayer reached his conclusion on a voyage to the [[Dutch East Indies]], where he found that his patients' blood was a deeper red because they were consuming less [[oxygen]], and therefore less energy, to maintain their body temperature in the hotter climate. He discovered that [[heat]] and [[mechanical work]] were both forms of energy, and in 1845, after improving his knowledge of physics, he published a monograph that stated a quantitative relationship between them.<ref>Mayer, J.R. (1845). ''Die organische Bewegung in ihrem Zusammenhange mit dem Stoffwechsel. Ein Beitrag zur Naturkunde'', Dechsler, Heilbronn.</ref>

[[File:Joule's Apparatus (Harper's Scan).png|thumb|right|[[James Prescott Joule|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.]]

Meanwhile, in 1843, [[James Prescott Joule]] independently discovered the mechanical equivalent in a series of experiments. In the most famous, now called the "Joule apparatus", a descending weight attached to a string caused a paddle immersed in water to rotate.  He showed that the [[gravitational energy|gravitational potential energy]] lost by the weight in descending was equal to the [[internal energy]] gained by the water through [[friction]] with the paddle.

Over the period 1840–1843, similar work was carried out by engineer [[Ludwig A. Colding]], although it was little known outside his native Denmark.

Both Joule's and Mayer's work suffered from resistance and neglect but it was Joule's that eventually drew the wider recognition.

{{For|the dispute between Joule and Mayer over priority|Mechanical equivalent of heat: Priority}}

In 1844, the Welsh scientist [[William Robert Grove]] postulated a relationship between mechanics, heat, [[light]], [[electricity]], and [[magnetism]] by treating them all as manifestations of a single "force" (''energy'' in modern terms). In 1846, Grove published his theories in his book ''The Correlation of Physical Forces''.<ref>{{cite book | author=Grove, W. R. | title=The Correlation of Physical Forces | url=https://archive.org/details/correlationphys06grovgoog | location=London | publisher=Longmans, Green | year=1874 | edition=6th }}</ref> In 1847, drawing on the earlier work of Joule, [[Nicolas Léonard Sadi Carnot|Sadi Carnot]], and [[Émile Clapeyron]], [[Hermann von Helmholtz]] arrived at conclusions similar to Grove's and published his theories in his book ''Über die Erhaltung der Kraft'' (''On the Conservation of Force'', 1847).<ref>{{cite web|title= On the Conservation of Force|url=http://www.bartleby.com/30/125.html|publisher=Bartleby|access-date= 6 April 2014}}</ref> The general modern acceptance of the principle stems from this publication.

In 1850, the Scottish mathematician [[William Rankine]] first used the phrase ''the law of the conservation of energy'' for the principle.<ref>William John Macquorn Rankine (1853) "On the General Law of the Transformation of Energy," ''Proceedings of the Philosophical Society of Glasgow'', vol. 3, no. 5, pages 276-280; reprinted in: (1) ''Philosophical Magazine'', series 4, vol. 5, no. 30, [https://books.google.com/books?id=3Ov22-gFMnEC&pg=PA106 pages 106-117] (February 1853); and (2) W. J. Millar, ed., ''Miscellaneous Scientific Papers: by W. J. Macquorn Rankine'', ... (London, England: Charles Griffin and Co., 1881), part II, [https://archive.org/stream/miscellaneoussci00rank#page/203/mode/1up pages 203-208]: "The law of the ''Conservation of Energy'' is already known—viz. that the sum of all the energies of the universe, actual and potential, is unchangeable."</ref>

In 1877, [[Peter Guthrie Tait]] claimed that the principle originated with Sir Isaac Newton, based on a creative reading of propositions 40 and 41 of the ''[[Philosophiae Naturalis Principia Mathematica]]''. This is now regarded as an example of [[Whig history]].<ref>{{cite book
|title=On the shoulders of merchants: exchange and the mathematical conception of nature in early modern Europe
|first1=Richard W. |last1=Hadden |publisher=SUNY Press
|year=1994 |isbn=978-0-7914-2011-9 |page=13
|url=https://books.google.com/books?id=7IxtC4Jw1YoC}}, [https://books.google.com/books?id=7IxtC4Jw1YoC&pg=PA13 Chapter&nbsp;1, p.&nbsp;13]
</ref>

===Mass–energy equivalence===
{{More citations needed section|date=November 2015}}
{{main|Mass–energy equivalence}}

Matter is composed of atoms and what makes up atoms. Matter has [[rest mass|''intrinsic'' or ''rest'' mass]]. In the limited range of recognized experience of the nineteenth century, it was found that such rest mass is conserved. Einstein's 1905 theory of [[special relativity]] showed that rest mass corresponds to an 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, as recognized in twentieth-century experience, 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.

For example, an [[electron]] and a [[positron]] each have rest mass.  They can perish together, converting their combined rest energy into [[photon]]s which have electromagnetic radiant energy but no rest mass.  If this occurs within an isolated system that does not release the photons or their energy into the external surroundings, then neither the total ''mass'' nor the total ''energy'' of the system will change.  The produced electromagnetic radiant energy contributes just as much to the inertia (and to any weight) of the system as did the rest mass of the electron and positron before their demise. Likewise, non-material forms of energy can perish into matter, which has rest mass.

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.

===Conservation of energy in beta decay===
{{Main|Beta decay#Neutrinos}}
The discovery in 1911 that electrons emitted in [[beta decay]] have a continuous rather than a discrete spectrum appeared to contradict conservation of energy, under the then-current assumption that beta decay is the simple emission of an electron from a nucleus.<ref>{{cite book |last1=Jensen |first1=Carsten |year=2000 |title=Controversy and Consensus: Nuclear Beta Decay 1911-1934 |url=https://www.springer.com/birkhauser/physics/book/978-3-7643-5313-1 |publisher=Birkhäuser Verlag |isbn=978-3-7643-5313-1 }}</ref><ref>{{cite journal |bibcode= 1978PhT....31i..23B |doi=10.1063/1.2995181 |title=The idea of the neutrino |journal=Physics Today |volume=31 |issue=9 |pages=23–8 |year=1978 |last1=Brown |first1=Laurie M. }}</ref> This problem was eventually resolved in 1933 by [[Enrico Fermi]] who proposed the correct [[Fermi's interaction|description of beta-decay]] as the emission of both an electron and an [[antineutrino]], which carries away the apparently missing energy.<ref>
{{cite journal
 |last=Wilson |first=F. L.
 |year=1968
 |title=Fermi's Theory of Beta Decay
 |url=http://microboone-docdb.fnal.gov/cgi-bin/RetrieveFile?docid=953;filename=FermiBetaDecay1934.pdf;version=1
 |journal=[[American Journal of Physics]]
 |volume=36 |issue=12 |pages=1150–1160
 |bibcode= 1968AmJPh..36.1150W
 |doi= 10.1119/1.1974382
}}</ref><ref>
{{cite book
 |last=Griffiths |first=D.
 |year=2009
 |title=Introduction to Elementary Particles
 |edition=2nd |pages=314–315
 |isbn=978-3-527-40601-2
}}</ref>

==First law of thermodynamics==
{{main|First law of thermodynamics}}
{{see also|First law of thermodynamics (fluid mechanics)}}

For a [[Closed system#In thermodynamics|closed thermodynamic system]], the first law of thermodynamics may be stated as:

:<math>\delta Q = \mathrm{d}U + \delta W</math>, or equivalently, <math>\mathrm{d}U = \delta Q - \delta W,</math>

where <math>\delta Q</math> is the quantity of [[energy]] added to the system by a [[heat]]ing process, <math>\delta W</math> is the quantity of energy lost by the system due to [[Work (thermodynamics)|work]] done by the system on its surroundings, and <math>\mathrm{d}U</math> is the change in the [[internal energy]] of the system.

The δ's before the heat and work terms are used to indicate that they describe an increment of energy which is to be interpreted somewhat differently than the <math>\mathrm{d}U</math> increment of internal energy (see [[Inexact differential]]). Work and heat refer to kinds of process which add or subtract energy to or from a system, while the internal energy <math>U</math> is a property of a particular state of the system when it is in unchanging thermodynamic equilibrium. Thus the term "heat energy" for <math>\delta Q</math> means "that amount of energy added as a result of heating" rather than referring to a particular form of energy. Likewise, the term "work energy" for <math>\delta W</math> means "that amount of energy lost as a result of work". Thus one can state the amount of internal energy possessed by a thermodynamic system that one knows is presently in a given state, but one cannot tell, just from knowledge of the given present state, how much energy has in the past flowed into or out of the system as a result of its being heated or cooled, nor as a result of work being performed on or by the system.

[[Entropy (classical thermodynamics)|Entropy]] is a function of the state of a system which tells of limitations of the possibility of conversion of heat into work.

For a simple compressible system, the work performed by the system may be written:

:<math>\delta W = P\,\mathrm{d}V,</math>

where <math>P</math> is the [[pressure]] and <math>dV</math> is a small change in the [[volume]] of the system, each of which are system variables. In the fictive case in which the process is idealized and infinitely slow, so as to be called ''quasi-static'', and regarded as reversible, the heat being transferred from a source with temperature infinitesimally above the system temperature, the heat energy may be written

:<math>\delta Q = T\,\mathrm{d}S,</math>

where <math>T</math> is the [[temperature]] and <math>\mathrm{d}S</math> is a small change in the entropy of the system. Temperature and entropy are variables of the state of a system.

If an open system (in which mass may be exchanged with the environment) has several walls such that the mass transfer is through rigid walls separate from the heat and work transfers, then the first law may be written as<ref>{{cite journal | url=https://pubs.acs.org/doi/full/10.1021/ed200405k | doi=10.1021/ed200405k | title=On the Relation between the Fundamental Equation of Thermodynamics and the Energy Balance Equation in the Context of Closed and Open Systems | year=2012 | last1=Knuiman | first1=Jan T. | last2=Barneveld | first2=Peter A. | last3=Besseling | first3=Nicolaas A. M. | journal=Journal of Chemical Education | volume=89 | issue=8 | pages=968–972 | bibcode=2012JChEd..89..968K }}</ref>

:<math>\mathrm{d}U = \delta Q - \delta W + \sum_i h_i\,dM_i,</math>

where <math>dM_i</math> is the added mass of species <math>i</math> and <math>h_i</math> is the corresponding enthalpy per unit mass. Note that generally <math>dS\neq\delta Q/T</math> in this case, as matter carries its own entropy. Instead, <math>dS=\delta Q/T+\textstyle{\sum_{i}}s_i\,dM_i</math>, where <math>s_i</math> is the entropy per unit mass of type <math>i</math>, from which we recover the [[fundamental thermodynamic relation]]

:<math>\mathrm{d}U = T\,dS - P\,dV + \sum_i\mu_i\,dN_i</math>

because the chemical potential <math>\mu_i</math> is the partial molar Gibbs free energy of species <math>i</math> and the Gibbs free energy <math>G\equiv H-TS</math>.

==Noether's theorem==
{{main|Noether's theorem}}
[[File:Noether.jpg|thumb|200px|[[Emmy Noether]] (1882-1935) was an influential [[mathematician]] known for her groundbreaking contributions to [[abstract algebra]] and [[theoretical physics]].]]
The conservation of energy is a common feature in many physical theories. From a mathematical point of view it is understood as a consequence of [[Noether's theorem]], developed by [[Emmy Noether]] in 1915 and first published in 1918. In any physical theory that obeys the stationary-action principle, the theorem states that every continuous symmetry has an associated conserved quantity; if the theory's symmetry is time invariance, then the conserved quantity is called "energy".<ref>Hanc, J., Tuleja, S., & Hancova, M. (2004). Symmetries and conservation laws: Consequences of Noether’s theorem. American Journal of Physics, 72(4), 428-435.</ref> The energy conservation law is a consequence of the shift [[Symmetry (physics)|symmetry]] of time; energy conservation is implied by the empirical fact that the [[laws of physics]] do not change with time itself. Philosophically this can be stated as "nothing depends on time per se". In other words, if the physical system is invariant under the [[continuous symmetry]] of [[time translation]], then its energy (which is the [[canonical conjugate]] quantity to time) is conserved.  Conversely, systems that are not invariant under shifts in time (e.g. systems with time-dependent potential energy) do not exhibit conservation of energy&nbsp;– unless we consider them to exchange energy with another, an external system so that the theory of the enlarged system becomes time-invariant again. Conservation of energy for finite systems is valid in physical theories such as special relativity and quantum theory (including [[Quantum electrodynamics|QED]]) in the flat [[space-time]].

==Special relativity==
With the discovery of special relativity by [[Henri Poincaré]] and [[Albert Einstein]], the energy was proposed to be a component of an [[four-momentum|energy-momentum 4-vector]]. Each of the four components (one of energy and three of momentum) of this vector is separately conserved across time, in any closed system, as seen from any given [[inertial reference frame]]. Also conserved is the vector length ([[Minkowski space|Minkowski norm]]), which is the [[rest mass]] for single particles, and the [[invariant mass]] for systems of particles (where momenta and energy are separately summed before the length is calculated).

The relativistic energy of a single [[mass]]ive particle contains a term related to its rest mass in addition to its kinetic energy of motion. In the limit of zero kinetic energy (or equivalently in the [[rest frame]]) of a massive particle, or else in the [[center of momentum frame]] for objects or systems which retain kinetic energy, the [[total energy]] of a particle or object (including internal kinetic energy in systems) is proportional to the rest mass or invariant mass, as described by the famous equation <math>E=mc^2</math>.

Thus, the rule of [[Mass in special relativity|''conservation of energy'' over time in special relativity]] continues to hold, so long as the [[frame of reference|reference frame]] of the observer is unchanged. This applies to the total energy of systems, although different observers disagree as to the energy value. Also conserved, and invariant to all observers, is the invariant mass, which is the minimal system mass and energy that can be seen by any observer, and which is defined by the [[energy–momentum relation]].

==General relativity==
General relativity introduces new phenomena. In an expanding universe, photons spontaneously redshift and tethers spontaneously gain tension; if vacuum energy is positive, the total vacuum energy of the universe appears to spontaneously increase as the volume of space increases. Some scholars claim that energy is no longer meaningfully conserved in any identifiable form.<ref>Harrison, E. R. (1995). Mining energy in an expanding universe. The Astrophysical Journal, 446, 63.</ref><ref name=leaking>{{cite news |author1=Tamara M. Davis |title=Is the Universe Leaking Energy? |url=https://www.scientificamerican.com/article/is-the-universe-leaking-energy/ |access-date=25 September 2022 |work=Scientific American |date=2010}}</ref>

[[John Baez]]'s view is that energy–momentum conservation is not well-defined except in certain special cases. Energy-momentum is typically expressed with the aid of a [[stress–energy–momentum pseudotensor]]. However, since pseudotensors are not tensors, they do not transform cleanly between reference frames. If the metric under consideration is static (that is, does not change with time) or asymptotically flat (that is, at an infinite distance away spacetime looks empty), then energy conservation holds without major pitfalls. In practice, some metrics, notably the [[Friedmann–Lemaître–Robertson–Walker metric]] that appears to govern the universe, do not satisfy these constraints and energy conservation is not well defined.<ref>{{cite web|url=http://math.ucr.edu/home/baez/physics/Relativity/GR/energy_gr.html|title=Is Energy Conserved in General Relativity?|author1=Michael Weiss |author2=John Baez |access-date=5 January 2017|archive-url=https://web.archive.org/web/20070605041426/http://math.ucr.edu/home/baez/physics/Relativity/GR/energy_gr.html|archive-date=5 June 2007|url-status=dead}}</ref> Besides being dependent on the coordinate system, pseudotensor energy is dependent on the type of pseudotensor in use; for example, the energy exterior to a [[Kerr–Newman black hole]] is twice as large when calculated from Møller's pseudotensor as it is when calculated using the [[Einstein pseudotensor]].<ref>{{cite journal |last1=Duerr |first1=Patrick M. |title=Fantastic Beasts and where (not) to find them: Local gravitational energy and energy conservation in general relativity |journal=Studies in History and Philosophy of Science Part B: Studies in History and Philosophy of Modern Physics |date=February 2019 |volume=65 |pages=1–14 |doi=10.1016/j.shpsb.2018.07.002|bibcode=2019SHPMP..65....1D |s2cid=126366668 }}</ref>

For asymptotically flat universes, Einstein and others salvage conservation of energy by introducing a specific global gravitational potential energy that cancels out mass-energy changes triggered by spacetime expansion or contraction. This global energy has no well-defined density and cannot technically be applied to a non-asymptotically flat universe; however, for practical purposes this can be finessed, and so by this view, energy is conserved in our universe.<ref>{{cite news |last1=Moskowitz |first1=Clara |title=Fact or Fiction?: Energy Can Neither Be Created Nor Destroyed |url=https://www.scientificamerican.com/article/energy-can-neither-be-created-nor-destroyed/ |access-date=25 September 2022 |work=Scientific American |date=2014 |language=en}}</ref><ref name="Energy Is Not Conserved"/> [[Alan Guth]] even famously stated that the universe might be "the ultimate free lunch", and theorized that, when accounting for gravitational potential energy, the net energy of the Universe is [[zero-energy universe|zero]].<ref>{{cite news |title=Confronting the Multiverse: What 'Infinite Universes' Would Mean |url=https://www.space.com/31465-is-our-universe-just-one-of-many-in-a-multiverse.html |access-date=25 September 2022 |work=Space.com |date=23 December 2015 |language=en}}</ref>

==Quantum theory==
In [[quantum mechanics]], the energy of a quantum system is described by a [[self-adjoint operator|self-adjoint]] (or Hermitian) operator called the [[Hamiltonian (quantum mechanics)|Hamiltonian]], which acts on the [[Hilbert space]] (or a space of [[wave function]]s) of the system. If the Hamiltonian is a time-independent operator, emergence probability of the measurement result does not change in time over the evolution of the system. Thus the expectation value of energy is also time independent. The local energy conservation in quantum field theory is ensured by the quantum [[Noether's theorem]] for the energy-momentum tensor operator. Thus energy is conserved by the normal unitary evolution of a quantum system.

However, when the non-unitary [[Born rule]] is applied, the system's energy is [[Measurement in quantum mechanics|measured]] with an energy that can be below or above the expectation value, if the system was not in an energy eigenstate. (For macroscopic systems, this effect is usually too small to measure.) The disposition of this energy gap is not well-understood; some physicists believe that the energy is transferred to or from the macroscopic environment in the course of the measurement process, while others believe that the observable energy is only conserved "on average".<ref>{{cite news |title=Puzzling Quantum Scenario Appears Not to Conserve Energy |url=https://www.quantamagazine.org/puzzling-quantum-scenario-appears-not-to-conserve-energy-20220516/ |access-date=25 September 2022 |work=[[Quanta Magazine]] |date=2022}}</ref><ref name="Energy Non-conservation in Quantum">{{cite journal |last1=Carroll |first1=Sean M. |last2=Lodman |first2=Jackie |title=Energy Non-conservation in Quantum Mechanics |journal=Foundations of Physics |date=August 2021 |volume=51 |issue=4 |pages=83 |doi=10.1007/s10701-021-00490-5|arxiv=2101.11052 |bibcode=2021FoPh...51...83C |s2cid=226664820 }}</ref> No experiment has been confirmed as definitive evidence of violations of the conservation of energy principle in quantum mechanics, but that doesn't rule out that some newer experiments, as proposed, may find evidence of violations of the conservation of energy principle in quantum mechanics.<ref name="Energy Non-conservation in Quantum"/>

==Status==
In the context of perpetual motion machines such as the [[Orbo]], Professor [[Eric Ash]] has argued at the [[BBC]]: "Denying [conservation of energy] would undermine not just little bits of science - the whole edifice would be no more. All of the technology on which we built the modern world would lie in ruins." It is because of conservation of energy that "we know - without having to examine details of a particular device - that Orbo cannot work."<ref>{{cite news |title=The perpetual myth of free energy |url=http://news.bbc.co.uk/2/hi/technology/6283374.stm |access-date=10 October 2022 | work=BBC |date=9 July 2007}}</ref>

Energy conservation has been a foundational physical principle for about two hundred years. From the point of view of modern general relativity, the lab environment can be well approximated by [[Minkowski space|Minkowski spacetime]], where energy is exactly conserved. The entire Earth can be well approximated by the [[Schwarzschild metric]], where again energy is exactly conserved. Given all the experimental evidence, any new theory (such as [[quantum gravity]]), in order to be successful, will have to explain why energy has appeared to always be exactly conserved in terrestrial experiments.<ref>{{cite journal |last1=Maudlin |first1=Tim |last2=Okon |first2=Elias |last3=Sudarsky |first3=Daniel |title=On the status of conservation laws in physics: Implications for semiclassical gravity |journal=Studies in History and Philosophy of Science Part B: Studies in History and Philosophy of Modern Physics |date=February 2020 |volume=69 |pages=67–81 |doi=10.1016/j.shpsb.2019.10.004|arxiv=1910.06473 |bibcode=2020SHPMP..69...67M |s2cid=204575731 }}</ref> In some speculative theories, corrections to quantum mechanics are too small to be detected at anywhere near the current [[TeV]] level accessible through particle accelerators. [[Doubly special relativity]] models may argue for a breakdown in energy-momentum conservation for sufficiently energetic particles; such models are constrained by observations that cosmic rays appear to travel for billions of years without displaying anomalous non-conservation behavior.<ref>{{cite journal |last1=Amelino-Camelia |first1=Giovanni |title=Quantum-Spacetime Phenomenology |journal=Living Reviews in Relativity |date=12 June 2013 |volume=16 |issue=1 |pages=5 |doi=10.12942/lrr-2013-5|pmid=28179844 |pmc=5255913 |arxiv=0806.0339 |bibcode=2013LRR....16....5A }}</ref> Some interpretations of quantum mechanics claim that observed energy tends to increase when the Born rule is applied due to localization of the wave function. If true, objects could be expected to spontaneously heat up; thus, such models are constrained by observations of large, cool astronomical objects as well as the observation of (often supercooled) laboratory experiments.<ref>{{cite journal |last1=Carlesso |first1=Matteo |last2=Donadi |first2=Sandro |last3=Ferialdi |first3=Luca |last4=Paternostro |first4=Mauro |last5=Ulbricht |first5=Hendrik |last6=Bassi |first6=Angelo |title=Present status and future challenges of non-interferometric tests of collapse models |journal=Nature Physics |date=March 2022 |volume=18 |issue=3 |pages=243–250 |doi=10.1038/s41567-021-01489-5|arxiv=2203.04231 |bibcode=2022NatPh..18..243C |s2cid=246949254 }}</ref>

[[Milton A. Rothman]] wrote that the law of conservation of energy has been verified by nuclear physics experiments to an accuracy of one part in a thousand million million (10<sup>15</sup>). He then defines its precision as "perfect for all practical purposes".<ref>{{Cite journal |last=Rothman |first=Milton |date=1989 |title=Myths About Science... And Belief in the Paranormal |journal=The Skeptical Inquirer |volume=14 |issue=1 |page=28}}</ref>

==See also==
{{portal|Energy}}
{{cols}}
* [[Energy quality]]
* [[Energy transformation]]
* [[Lagrangian mechanics]]
* [[Laws of thermodynamics]]
* [[Zero-energy universe]]
{{colend}}

==References==
{{reflist|30em}}

==Bibliography==

===Modern accounts===
* Goldstein, Martin, and Inge F., (1993). ''The Refrigerator and the Universe''. Harvard Univ. Press. A gentle introduction.
* {{cite book |author1=Kroemer, Herbert |author2=Kittel, Charles | title=Thermal Physics (2nd ed.) | publisher=W. H. Freeman Company | year=1980 | isbn=978-0-7167-1088-2 }}
* {{cite book | author=Nolan, Peter J. | title=Fundamentals of College Physics, 2nd ed. | publisher=William C. Brown Publishers | year=1996 }}
* {{cite book | author=Oxtoby & Nachtrieb | title=Principles of Modern Chemistry, 3rd ed. | publisher=Saunders College Publishing | year=1996 }}
* {{cite book | author=Papineau, D. | title=Thinking about Consciousness | location=Oxford | publisher=Oxford University Press | year=2002 }}
* {{cite book | author1=Serway, Raymond A. | author2=Jewett, John W. | title=Physics for Scientists and Engineers (6th ed.) | publisher=Brooks/Cole | year=2004 | isbn=978-0-534-40842-8 | url-access=registration | url=https://archive.org/details/physicssciengv2p00serw }}
* Stenger, Victor J. (2000). ''Timeless Reality''. Prometheus Books. Especially chpt. 12. Nontechnical.
* {{cite book | author=Tipler, Paul | title=Physics for Scientists and Engineers: Mechanics, Oscillations and Waves, Thermodynamics (5th ed.) | publisher=W. H. Freeman | year=2004 | isbn=978-0-7167-0809-4}}
* {{cite book | author=[[Lanczos]], Cornelius | title=The Variational Principles of Mechanics | location= Toronto | publisher=University of Toronto Press | year=1970 | isbn=978-0-8020-1743-7}}

===History of ideas===
* {{cite journal | author=Brown, T.M. | title=Resource letter EEC-1 on the evolution of energy concepts from Galileo to Helmholtz | journal=American Journal of Physics | year=1965 | volume=33 | pages=759–765  | doi = 10.1119/1.1970980|bibcode = 1965AmJPh..33..759B | issue=10 }}
* {{cite book | author=Cardwell, D.S.L. | title=From Watt to Clausius: The Rise of Thermodynamics in the Early Industrial Age | location=London | publisher=Heinemann | year=1971 | isbn=978-0-435-54150-7 }}
* {{cite book | author=Guillen, M. | title=[[Five Equations That Changed the World]] | publisher= Abacus| year=1999 | isbn=978-0-349-11064-6 | location=New York }}
* {{cite book | author=Hiebert, E.N. | title=Historical Roots of the Principle of Conservation of Energy | publisher=Ayer Co Pub | location=Madison, Wis. | year=1981 | isbn=978-0-405-13880-5 }}
* [[Thomas Kuhn|Kuhn, T.S.]] (1957) "Energy conservation as an example of simultaneous discovery", in M. Clagett (ed.) ''Critical Problems in the History of Science'' ''pp.''321–56
* {{cite journal | author=Sarton, G. | title=The discovery of the law of conservation of energy | journal=Isis | year=1929 | volume=13 | pages=18–49 | doi=10.1086/346430 | last2=Joule | first2=J. P. | last3=Carnot | first3=Sadi | s2cid=145585492 }}
* {{cite book | author=Smith, C. | title=The Science of Energy: Cultural History of Energy Physics in Victorian Britain | location=London | publisher=Heinemann | year=1998 | isbn=978-0-485-11431-7 }}
* {{cite book | author=[[Ernst Mach|Mach, E.]] | title=History and Root of the Principles of the Conservation of Energy | publisher=Open Court Pub. Co., Illinois | year=1872 | url=https://archive.org/details/historyandrootp00machgoog}}
* {{cite book | author=[[Henri Poincaré|Poincaré, H.]] | title=Science and Hypothesis | url=https://archive.org/details/scienceandhypoth00poinuoft| publisher=Walter Scott Publishing Co. Ltd; Dover reprint, 1952 | year=1905 | isbn=978-0-486-60221-9 }}, Chapter 8, "Energy and Thermo-dynamics"

==External links==
* [http://www.physnet.org/modules/pdf_modules/m158.pdf <small>MISN-0-158</small> ''The First Law of Thermodynamics''] ([[Portable Document Format|PDF file]]) by Jerzy Borysowicz for [http://www.physnet.org Project PHYSNET].

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{{DEFAULTSORT:Conservation Of Energy}}
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