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physics Physics is the natural science that studies matter, its fundamental constituents, its motion and behavior through space and time, and the related entities of energy and force. "Physical science is that department of knowledge which ...
and
chemistry Chemistry is the scientific study of the properties and behavior of matter. It is a natural science that covers the elements that make up matter to the compounds made of atoms, molecules and ions: their composition, structure, proper ...
, the law of conservation of energy states that the total
energy In physics, energy (from Ancient Greek: ἐνέργεια, ''enérgeia'', “activity”) is the quantitative property that is transferred to a body or to a physical system, recognizable in the performance of work and in the form of ...
of an isolated system remains constant; it is said to be ''conserved'' over time. This law, first proposed and tested by Émilie du Châtelet, means that energy can neither be created nor destroyed; rather, it can only be transformed or transferred from one form to another. For instance,
chemical energy Chemical energy is the energy of chemical substances that is released when they undergo a chemical reaction and transform into other substances. Some examples of storage media of chemical energy include batteries, Schmidt-Rohr, K. (2018). "How ...
is
converted Conversion or convert may refer to: Arts, entertainment, and media * "Conversion" (''Doctor Who'' audio), an episode of the audio drama ''Cyberman'' * "Conversion" (''Stargate Atlantis''), an episode of the television series * "The Conversion" ...
to
kinetic energy In physics, the kinetic energy of an object is the energy that it possesses due to its motion. It is defined as the work needed to accelerate a body of a given mass from rest to its stated velocity. Having gained this energy during its acce ...
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 In physics, the kinetic energy of an object is the energy that it possesses due to its motion. It is defined as the work needed to accelerate a body of a given mass from rest to its stated velocity. Having gained this energy during its acce ...
and
potential energy In physics, potential energy is the energy held by an object because of its position relative to other objects, stresses within itself, its electric charge, or other factors. Common types of potential energy include the gravitational potenti ...
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 In physics, the special theory of relativity, or special relativity for short, is a scientific theory regarding the relationship between space and time. In Albert Einstein's original treatment, the theory is based on two postulates: # The law ...
shows that mass is related to energy and vice versa by ''E = mc2'', 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 very shortly after the Big Bang or when black holes emit Hawking radiation. Given the stationary-action principle, conservation of energy can be rigorously proven by Noether's theorem as a consequence of
continuous Continuity or continuous may refer to: Mathematics * Continuity (mathematics), the opposing concept to discreteness; common examples include ** Continuous probability distribution or random variable in probability and statistics ** Continuous g ...
time translation symmetry Time translation symmetry or temporal translation symmetry (TTS) is a mathematical transformation in physics that moves the times of events through a common interval. Time translation symmetry is the law that the laws of physics are unchanged ...
; 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 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. Depending on the definition of energy, conservation of energy can arguably be violated by
general relativity General relativity, also known as the general theory of relativity and Einstein's theory of gravity, is the geometric theory of gravitation published by Albert Einstein in 1915 and is the current description of gravitation in modern physics ...
on the cosmological scale.


History

Ancient philosophers as far back as
Thales of Miletus Thales of Miletus ( ; grc-gre, Θαλῆς; ) was a Greek mathematician, astronomer, statesman, and pre-Socratic philosopher from Miletus in Ionia, Asia Minor. He was one of the Seven Sages of Greece. Many, most notably Aristotle, regarded ...
 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 Empedocles (; grc-gre, Ἐμπεδοκλῆς; , 444–443 BC) was a Greek pre-Socratic philosopher and a native citizen of Akragas, a Greek city in Sicily. Empedocles' philosophy is best known for originating the cosmogonic theory of the ...
(490–430 BCE) wrote that in his universal system, composed of four roots (earth, air, water, fire), "nothing comes to be or perishes"; instead, these elements suffer continual rearrangement. Epicurus ( 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." In 1605, Simon Stevin was able to solve a number of problems in statics based on the principle that perpetual motion was impossible. In 1639,
Galileo Galileo di Vincenzo Bonaiuti de' Galilei (15 February 1564 – 8 January 1642) was an Italian astronomer, physicist and engineer, sometimes described as a polymath. Commonly referred to as Galileo, his name was pronounced (, ). He was ...
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 Christiaan Huygens, Lord of Zeelhem, ( , , ; also spelled Huyghens; la, Hugenius; 14 April 1629 – 8 July 1695) was a Dutch mathematician, physicist, engineer, astronomer, and inventor, who is regarded as one of the greatest scientists o ...
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 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. Between 1676–1689, 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 Mass is an intrinsic property of a body. It was traditionally believed to be related to the quantity of matter in a physical body, until the discovery of the atom and particle physics. It was found that different atoms and different ele ...
es ''mi'', each with
velocity Velocity is the directional speed of an object in motion as an indication of its rate of change in position as observed from a particular frame of reference and as measured by a particular standard of time (e.g. northbound). Velocity i ...
''vi''), :\sum_ m_i v_i^2 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 physics, the kinetic energy of an object is the energy that it possesses due to its motion. It is defined as the work needed to accelerate a body of a given mass from rest to its stated velocity. Having gained this energy during its acce ...
in situations where there is no friction. Many
physicist A physicist is a scientist who specializes in the field of physics, which encompasses the interactions of matter and energy at all length and time scales in the physical universe. Physicists generally are interested in the root or ultimate cau ...
s at that time, such as Newton, held that the conservation of momentum, which holds even in systems with friction, as defined by the
momentum In Newtonian mechanics, momentum (more specifically linear momentum or translational momentum) is the product of the mass and velocity of an object. It is a vector quantity, possessing a magnitude and a direction. If is an object's mass ...
: :\sum_ m_i v_i 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 Sir Isaac Newton (25 December 1642 – 20 March 1726/27) was an English mathematician, physicist, astronomer, alchemist, Theology, theologian, and author (described in his time as a "natural philosophy, natural philosopher"), widely ...
published his '' Principia'', which 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.) The law of conservation of vis viva was championed by the father and son duo,
Johann Johann, typically a male given name, is the German form of ''Iohannes'', which is the Latin form of the Greek name ''Iōánnēs'' (), itself derived from Hebrew name ''Yochanan'' () in turn from its extended form (), meaning " Yahweh is Gracio ...
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 ''Hydrodynamica'' (Latin for ''Hydrodynamics'') is a book published by Daniel Bernoulli in 1738.The book's full title is ''Hydrodynamica, sive de Viribus et Motibus Fluidorum Commentarii'' (Hydrodynamics, or commentaries on the forces and motio ...
'', 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 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, and Hamiltonian formulations of mechanics. 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. Earlier workers, including Newton and Voltaire, had all believed that "energy" (so far as they understood the concept at all) 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 E_k = \frac12 mv^2, where E_k is the kinetic energy of an object, m its mass and v its
speed In everyday use and in kinematics, the speed (commonly referred to as ''v'') of an object is the magnitude of the change of its position over time or the magnitude of the change of its position per unit of time; it is thus a scalar quant ...
. 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, ...).Hagengruber, Ruth, editor (2011) ''Émilie du Chatelet between Leibniz and Newton''. Springer. .
Engineer Engineers, as practitioners of engineering, are professionals who Invention, invent, design, analyze, build and test machines, complex systems, structures, gadgets and materials to fulfill functional objectives and requirements while considerin ...
s such as
John Smeaton John Smeaton (8 June 1724 – 28 October 1792) was a British civil engineer responsible for the design of bridges, canals, harbours and lighthouses. He was also a capable mechanical engineer and an eminent physicist. Smeaton was the fi ...
, Peter Ewart, , Gustave-Adolphe Hirn, and
Marc Seguin Marc Seguin (20 April 1786 – 24 February 1875) was a French engineer, inventor of the wire- cable suspension bridge and the multi-tubular steam-engine boiler. Early life Seguin was born in Annonay, Ardèche to Marc François Seguin, th ...
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 A chemist (from Greek ''chēm(ía)'' alchemy; replacing ''chymist'' from Medieval Latin ''alchemist'') is a scientist trained in the study of chemistry. Chemists study the composition of matter and its properties. Chemists carefully describe th ...
s such as
William Hyde Wollaston William Hyde Wollaston (; 6 August 1766 – 22 December 1828) was an English chemist and physicist who is famous for discovering the chemical elements palladium and rhodium. He also developed a way to process platinum ore into malleable ingot ...
. 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 The second law of thermodynamics is a physical law based on universal experience concerning heat and energy interconversions. One simple statement of the law is that heat always moves from hotter objects to colder objects (or "downhill"), unle ...
, 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 Pierre-Simon, marquis de Laplace (; ; 23 March 1749 – 5 March 1827) was a French scholar and polymath whose work was important to the development of engineering, mathematics, statistics, physics, astronomy, and philosophy. He summarize ...
reviewed the two competing theories of ''vis viva'' and
caloric theory The caloric theory is an obsolete scientific theory that heat consists of a self-repellent fluid called caloric that flows from hotter bodies to colder bodies. Caloric was also thought of as a weightless gas that could pass in and out of pores ...
. Count Rumford's 1798 observations of heat generation during the boring of
cannon A cannon is a large- caliber gun classified as a type of artillery, which usually launches a projectile using explosive chemical propellant. Gunpowder ("black powder") was the primary propellant before the invention of smokeless powder ...
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 in 1807. The recalibration of ''vis viva'' to :\frac \sum_ m_i v_i^2 which can be understood as converting kinetic energy to work, was largely the result of Gaspard-Gustave Coriolis and
Jean-Victor Poncelet Jean-Victor Poncelet (; 1 July 1788 – 22 December 1867) was a French engineer and mathematician who served most notably as the Commanding General of the École Polytechnique. He is considered a reviver of projective geometry, and his work '' ...
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 ''
Zeitschrift für Physik ''Zeitschrift für Physik'' (English: ''Journal for Physics'') is a defunct series of German peer-reviewed physics journals established in 1920 by Springer Berlin Heidelberg. The series stopped publication in 1997, when it merged with other jour ...
'' 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'' nergy 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 In the history of science, the mechanical equivalent of heat states that motion and heat are mutually interchangeable and that in every case, a given amount of work would generate the same amount of heat, provided the work done is totally conve ...
''. The
caloric theory The caloric theory is an obsolete scientific theory that heat consists of a self-repellent fluid called caloric that flows from hotter bodies to colder bodies. Caloric was also thought of as a weightless gas that could pass in and out of pores ...
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 Sir Benjamin Thompson, Count Rumford, FRS (german: Reichsgraf von Rumford; March 26, 1753August 21, 1814) was an American-born British physicist and inventor whose challenges to established physical theory were part of the 19th-century revolut ...
) 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. The mechanical equivalence principle was first stated in its modern form by the German surgeon Julius Robert von Mayer in 1842. Mayer reached his conclusion on a voyage to the
Dutch East Indies The Dutch East Indies, also known as the Netherlands East Indies ( nl, Nederlands(ch)-Indië; ), was a Dutch colony consisting of what is now Indonesia. It was formed from the nationalised trading posts of the Dutch East India Company, whic ...
, where he found that his patients' blood was a deeper red because they were consuming less
oxygen Oxygen is the chemical element with the symbol O and atomic number 8. It is a member of the chalcogen group in the periodic table, a highly reactive nonmetal, and an oxidizing agent that readily forms oxides with most elements ...
, and therefore less energy, to maintain their body temperature in the hotter climate. He discovered that
heat In thermodynamics, heat is defined as the form of energy crossing the boundary of a thermodynamic system by virtue of a temperature difference across the boundary. A thermodynamic system does not ''contain'' heat. Nevertheless, the term is ...
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. 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 potential energy lost by the weight in descending was equal to the internal energy gained by the water through
friction Friction is the force resisting the relative motion of solid surfaces, fluid layers, and material elements sliding against each other. There are several types of friction: *Dry friction is a force that opposes the relative lateral motion of ...
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. In 1844,
William Robert Grove Sir William Robert Grove, FRS FRSE (11 July 1811 – 1 August 1896) was a Welsh judge and physical scientist. He anticipated the general theory of the conservation of energy, and was a pioneer of fuel cell technology. He invented the Grove volt ...
postulated a relationship between mechanics, heat,
light Light or visible light is electromagnetic radiation that can be perceived by the human eye. Visible light is usually defined as having wavelengths in the range of 400–700 nanometres (nm), corresponding to frequencies of 750–420 t ...
,
electricity Electricity is the set of physical phenomena associated with the presence and motion of matter that has a property of electric charge. Electricity is related to magnetism, both being part of the phenomenon of electromagnetism, as describe ...
, and
magnetism Magnetism is the class of physical attributes that are mediated by a magnetic field, which refers to the capacity to induce attractive and repulsive phenomena in other entities. Electric currents and the magnetic moments of elementary particles ...
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''. In 1847, drawing on the earlier work of Joule, Sadi Carnot, and Émile Clapeyron,
Hermann von Helmholtz Hermann Ludwig Ferdinand von Helmholtz (31 August 1821 – 8 September 1894) was a German physicist and physician who made significant contributions in several scientific fields, particularly hydrodynamic stability. The Helmholtz Associat ...
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). The general modern acceptance of the principle stems from this publication. In 1850, William Rankine first used the phrase ''the law of the conservation of energy'' for the principle. 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.


Mass–energy equivalence

Matter is composed of atoms and what makes up atoms. Matter has ''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 In physics, the special theory of relativity, or special relativity for short, is a scientific theory regarding the relationship between space and time. In Albert Einstein's original treatment, the theory is based on two postulates: # The law ...
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 Radiant may refer to: Computers, software, and video games * Radiant (software), a content management system * GtkRadiant, a level editor created by id Software for their games * Radiant AI, a technology developed by Bethesda Softworks for ''Th ...
. When this happens, as recognized in twentieth-century experience, rest mass is not conserved, unlike the ''total'' mass or ''total'' energy. All forms of energy contribute to the total mass and total energy. For example, an
electron The electron ( or ) is a subatomic particle with a negative one elementary electric charge. Electrons belong to the first generation of the lepton particle family, and are generally thought to be elementary particles because they have n ...
and a positron each have rest mass. They can perish together, converting their combined rest energy into
photon A photon () is an elementary particle that is a quantum of the electromagnetic field, including electromagnetic radiation such as light and radio waves, and the force carrier for the electromagnetic force. Photons are massless, so they alwa ...
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

The discovery in 1911 that electrons emitted in
beta decay In nuclear physics, beta decay (β-decay) is a type of radioactive decay in which a beta particle (fast energetic electron or positron) is emitted from an atomic nucleus, transforming the original nuclide to an isobar of that nuclide. For ...
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. This problem was eventually resolved in 1933 by Enrico Fermi who proposed the correct description of beta-decay as the emission of both an electron and an antineutrino, which carries away the apparently missing energy.


First law of thermodynamics

For a closed thermodynamic system, the first law of thermodynamics may be stated as: :\delta Q = \mathrmU + \delta W, or equivalently, \mathrmU = \delta Q - \delta W, where \delta Q is the quantity of
energy In physics, energy (from Ancient Greek: ἐνέργεια, ''enérgeia'', “activity”) is the quantitative property that is transferred to a body or to a physical system, recognizable in the performance of work and in the form of ...
added to the system by a
heat In thermodynamics, heat is defined as the form of energy crossing the boundary of a thermodynamic system by virtue of a temperature difference across the boundary. A thermodynamic system does not ''contain'' heat. Nevertheless, the term is ...
ing process, \delta W is the quantity of energy lost by the system due to work done by the system on its surroundings, and \mathrmU 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 \mathrmU increment of internal energy (see
Inexact differential An inexact differential or imperfect differential is a differential whose integral is path dependent. It is most often used in thermodynamics to express changes in path dependent quantities such as heat and work, but is defined more generally wit ...
). Work and heat refer to kinds of process which add or subtract energy to or from a system, while the internal energy U is a property of a particular state of the system when it is in unchanging thermodynamic equilibrium. Thus the term "heat energy" for \delta Q 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 \delta W 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 Entropy is a scientific concept, as well as a measurable physical property, that is most commonly associated with a state of disorder, randomness, or uncertainty. The term and the concept are used in diverse fields, from classical thermodyna ...
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: :\delta W = P\,\mathrmV, where P is the
pressure Pressure (symbol: ''p'' or ''P'') is the force applied perpendicular to the surface of an object per unit area over which that force is distributed. Gauge pressure (also spelled ''gage'' pressure)The preferred spelling varies by country a ...
and dV is a small change in the
volume Volume is a measure of occupied three-dimensional space. It is often quantified numerically using SI derived units (such as the cubic metre and litre) or by various imperial or US customary units (such as the gallon, quart, cubic inch). ...
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 :\delta Q = T\,\mathrmS, where T is the
temperature Temperature is a physical quantity that expresses quantitatively the perceptions of hotness and coldness. Temperature is measured with a thermometer. Thermometers are calibrated in various temperature scales that historically have relied o ...
and \mathrmS 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 :\mathrmU = \delta Q - \delta W + \sum_i h_i\,dM_i, where dM_i is the added mass of species i and h_i is the corresponding enthalpy per unit mass. Note that generally dS\neq\delta Q/T in this case, as matter carries its own entropy. Instead, dS=\delta Q/T+\textstyles_i\,dM_i, where s_i is the entropy per unit mass of type i, from which we recover the fundamental thermodynamic relation :\mathrmU = T\,dS - P\,dV + \sum_i\mu_i\,dN_i because the chemical potential \mu_i is the partial molar Gibbs free energy of species i and the Gibbs free energy G\equiv H-TS.


Noether's theorem

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". The energy conservation law is a consequence of the shift
symmetry Symmetry (from grc, συμμετρία "agreement in dimensions, due proportion, arrangement") in everyday language refers to a sense of harmonious and beautiful proportion and balance. In mathematics, "symmetry" has a more precise definiti ...
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 In mathematics, continuous symmetry is an intuitive idea corresponding to the concept of viewing some symmetries as motions, as opposed to discrete symmetry, e.g. reflection symmetry, which is invariant under a kind of flip from one state to ano ...
of
time translation Time translation symmetry or temporal translation symmetry (TTS) is a mathematical transformation in physics that moves the times of events through a common interval. Time translation symmetry is the law that the laws of physics are unchanged ...
, then its energy (which is the
canonical conjugate Conjugate variables are pairs of variables mathematically defined in such a way that they become Fourier transform duals, or more generally are related through Pontryagin duality. The duality relations lead naturally to an uncertainty relation— ...
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 – 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 QED) in the flat space-time.


Special relativity

With the discovery of special relativity by Henri Poincaré and
Albert Einstein Albert Einstein ( ; ; 14 March 1879 – 18 April 1955) was a German-born theoretical physicist, widely acknowledged to be one of the greatest and most influential physicists of all time. Einstein is best known for developing the theor ...
, the energy was proposed to be a component of an 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 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 Mass is an intrinsic property of a body. It was traditionally believed to be related to the quantity of matter in a physical body, until the discovery of the atom and particle physics. It was found that different atoms and different ele ...
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 E=mc^2. Thus, the rule of ''conservation of energy'' over time in special relativity continues to hold, so long as the
reference frame In physics and astronomy, a frame of reference (or reference frame) is an abstract coordinate system whose origin, orientation, and scale are specified by a set of reference points― geometric points whose position is identified both math ...
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.
John Baez John Carlos Baez (; born June 12, 1961) is an American mathematical physicist and a professor of mathematics at the University of California, Riverside (UCR) in Riverside, California. He has worked on spin foams in loop quantum gravity, appli ...
'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 In the theory of general relativity, a stress–energy–momentum pseudotensor, such as the Landau–Lifshitz pseudotensor, is an extension of the non-gravitational stress–energy tensor that incorporates the energy–momentum of gravity. It allo ...
. 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. 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. 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. 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.


Quantum theory

In
quantum mechanics Quantum mechanics is a fundamental theory in physics that provides a description of the physical properties of nature at the scale of atoms and subatomic particles. It is the foundation of all quantum physics including quantum chemistry, ...
, the energy of a quantum system is described by a
self-adjoint In mathematics, and more specifically in abstract algebra, an element ''x'' of a *-algebra is self-adjoint if x^*=x. A self-adjoint element is also Hermitian, though the reverse doesn't necessarily hold. A collection ''C'' of elements of a st ...
(or Hermitian) operator called the Hamiltonian, which acts on the
Hilbert space In mathematics, Hilbert spaces (named after David Hilbert) allow generalizing the methods of linear algebra and calculus from (finite-dimensional) Euclidean vector spaces to spaces that may be infinite-dimensional. Hilbert spaces arise natural ...
(or a space of wave functions) 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 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". 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.


Status

In the context of perpetual motion machines such as the Orbo, Professor Eric Ash has argued at the BBC: "Denying onservation of energywould 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." 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 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. In some speculative theories, corrections to quantum mechanics are too small to be detected at anywhere near the current
TeV TEV may refer to: * Transient Earth Voltage: a term for voltages appearing on the metal work of switchgear due to internal partial discharges * TeV, or teraelectronvolt or trillion electron volt, a measure of energy * Total Enterprise Value, a ...
level accessible through particle accelerators.
Doubly special relativity Doubly special relativity (DSR) – also called deformed special relativity or, by some, extra-special relativity – is a modified theory of special relativity in which there is not only an observer-independent maximum velocity (the speed of ligh ...
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. 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.


See also

* Energy quality *
Energy transformation Energy transformation, also known as energy conversion, is the process of changing energy from one form to another. In physics, energy is a quantity that provides the capacity to perform work or moving, (e.g. Lifting an object) or provides he ...
* Lagrangian mechanics * Laws of thermodynamics * Zero-energy universe


References


Bibliography


Modern accounts

* Goldstein, Martin, and Inge F., (1993). ''The Refrigerator and the Universe''. Harvard Univ. Press. A gentle introduction. * * * * * * Stenger, Victor J. (2000). ''Timeless Reality''. Prometheus Books. Especially chpt. 12. Nontechnical. * *


History of ideas

* * * * * 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 * * * * , Chapter 8, "Energy and Thermo-dynamics"


External links


MISN-0-158 ''The First Law of Thermodynamics''
( PDF file) by Jerzy Borysowicz fo
Project PHYSNET
{{DEFAULTSORT:Conservation Of Energy Energy (physics) Laws of thermodynamics Conservation laws Articles containing video clips