TheInfoList

In
physics Physics is the that studies , its , its and behavior through , and the related entities of and . "Physical science is that department of knowledge which relates to the order of nature, or, in other words, to the regular succession of eve ...

and
chemistry Chemistry is the study of the properties and behavior of . It is a that covers the that make up matter to the composed of s, s and s: their composition, structure, properties, behavior and the changes they undergo during a with other . ...

, the law of conservation of energy states that the total
energy In , energy is the that must be to a or to perform on the body, or to it. Energy is a ; the law of states that energy can be in form, but not created or destroyed. The unit of measurement in the (SI) of energy is the , which is the ...

of an
isolated system In physical science, an isolated system is either of the following: # a physical system In physics Physics (from grc, φυσική (ἐπιστήμη), physikḗ (epistḗmē), knowledge of nature, from ''phýsis'' 'nature'), , i ...
remains constant; it is said to be ''conserved'' over time. This law, first proposed and tested by
Émilie du Châtelet Gabrielle Émilie Le Tonnelier de Breteuil, Marquise du Châtelet (; 17 December 1706 – 10 September 1749) was a French French (french: français(e), link=no) may refer to: * Something of, from, or related to France France (), officiall ...
, 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 substance A chemical substance is a form of matter In classical physics and general chemistry, matter is any substance that has mass and takes up space by having volume. All everyday objects th ...
is
converted Conversion or convert may refer to: Arts, entertainment, and media * Conversion (Doctor Who audio), "Conversion" (''Doctor Who'' audio), an episode of the audio drama ''Cyberman'' * Conversion (Stargate Atlantis), "Conversion" (''Stargate Atlantis ...
to
kinetic energy In physics, the kinetic energy of an object is the energy that it possesses due to its motion (physics), motion. It is defined as the work (physics), work needed to accelerate a body of a given mass from rest to its stated velocity. Having gaine ...
when a stick of
dynamite Dynamite is an explosive made of nitroglycerin, sorbents (such as powdered shells or clay) and Stabilizer (chemistry), stabilizers. It was invented by the Swedish people, Swedish chemist and engineer Alfred Nobel in Geesthacht, Northern Germany ...

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 (physics), motion. It is defined as the work (physics), work needed to accelerate a body of a given mass from rest to its stated velocity. Having gaine ...
and
potential energy In physics, potential energy is the energy In physics Physics (from grc, φυσική (ἐπιστήμη), physikḗ (epistḗmē), knowledge of nature, from ''phýsis'' 'nature'), , is the natural science that studies matter ...

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 In physics and chemistry, the law of conservation of mass or principle of mass conservation states that for any closed system, system closed to all transfers of matter and energy, the mass of the system must remain constant over time, as the syst ...
; however,
special relativity In physics Physics is the that studies , its , its and behavior through , and the related entities of and . "Physical science is that department of knowledge which relates to the order of nature, or, in other words, to the regular s ...
showed 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 emit
Hawking radiation Hawking radiation is black-body radiation that is predicted to be released by black holes, due to quantum effects near the black hole event horizon. It is named after the physicist Stephen Hawking, who provided a theoretical argument for its exis ...
. Conservation of energy can be rigorously proven by
Noether's theorem Noether's theorem or Noether's first theorem states that every differentiable In calculus (a branch of mathematics), a differentiable function of one Real number, real variable is a function whose derivative exists at each point in its Domai ...
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 ga ...
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 hypothesis that the laws of physics are unchang ...
; 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 cannot exist, that is to say, no system without an external energy supply can deliver an unlimited amount of energy to its surroundings. For systems which do not have
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 hypothesis that the laws of physics are unchang ...
, it may not be possible to define ''conservation of energy''. Examples include
curved space Curved space often refers to a spatial geometry which is not "flat", where a flat space is described by Euclidean geometry. Curved spaces can generally be described by Riemannian geometry though some simple cases can be described in other ways. Cur ...
times in
general relativity General relativity, also known as the general theory of relativity, is the of published by in 1915 and is the current description of gravitation in . General generalizes and refines , providing a unified description of gravity as a geome ...
or
time crystals In condensed matter physics, a time crystal refers to a system or subsystem whose lowest-energy states evolve periodically. This name was proposed theoretically by Frank Wilczek in 2012 as a temporal analog to common crystals, which are periodi ...
in
condensed matter physics Condensed matter physics is the field of that deals with the macroscopic and microscopic physical properties of , especially the and which arise from forces between s. More generally, the subject deals with "condensed" phases of matter: syst ...
.

# History

Ancient Ancient history is the aggregate of past eventsWordNet Search – 3.0
"History"
from ...

philosopher A philosopher is someone who practices philosophy Philosophy (from , ) is the study of general and fundamental questions, such as those about reason, Metaphysics, existence, Epistemology, knowledge, Ethics, values, Philosophy of mind, ...

s as far back as
Thales of Miletus Thales of Miletus ( ; el, Θαλῆς Thales of Miletus ( ; el, Θαλῆς (ὁ Μιλήσιος), ''Thalēs''; ) was a Greek mathematician A mathematician is someone who uses an extensive knowledge of mathematics Mathematics (fr ...

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 Greek may refer to: Greece Anything of, from, or related to Greece Greece ( el, Ελλάδα, , ), officially the Hellenic Republic, is a country located in So ...

(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 Epicurus, ''Epíkouros'', "ally, comrade" (341–270 BC) was an and who founded , a highly influential school of . He was born on the Greek island of to parents. Influenced by , , , and possibly the , he turned against the of his day and e ...

( 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 Stevinus Simon Stevin (; 1548–1620), sometimes called Stevinus, was a Flemish mathematician, physicist and military engineer. He made various contributions in many areas of science and engineering, both theoretical and practical. He also translated vario ...
was able to solve a number of problems in statics based on the principle that
perpetual motion 's 1618 "water screw" perpetual motion machine from a 1660 wood engraving. It is widely credited as the first attempt to describe such a device in order to produce useful work, that of driving millstones. Perpetual motion is the motion of bodies ...
was impossible. In 1639,
Galileo Galileo di Vincenzo Bonaiuti de' Galilei ( , ; 15 February 1564 – 8 January 1642), commonly referred to as Galileo, was an astronomer An astronomer is a scientist in the field of astronomy who focuses their studies on a specific qu ...

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 ( , also , ; la, Hugenius; 14 April 1629 – 8 July 1695), also spelled Huyghens, was a Dutch mathematician, physicist, astronomer and inventor, who is regarded as one of the greatest scientists of all time and a major fig ...

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 ''Horologium Oscillatorium: Sive de Motu Pendulorum ad Horologia Aptato Demonstrationes Geometricae'' (''The Pendulum Clock: or Geometrical Demonstrations Concerning the Motion of Pendula as Applied to Clocks'') is a book published by Christiaan H ...
'', 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' 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. It was Leibniz during 1676–1689 who first attempted a mathematical formulation of the kind of energy that is associated with ''motion'' (kinetic energy). Using Huygens' work on collision, Leibniz noticed that in many mechanical systems (of several
mass Mass is the quantity Quantity is a property that can exist as a multitude or magnitude, which illustrate discontinuity and continuity. Quantities can be compared in terms of "more", "less", or "equal", or by assigning a numerical value ...
es, ''mi'' each with
velocity The velocity of an object is the rate of change of its position with respect to a frame of reference In physics Physics is the that studies , its , its and behavior through , and the related entities of and . "Physical scie ...

''vi''), :$\sum_ m_i v_i^2$ was conserved so long as the masses did not interact. He called this quantity the ''
vis viva ''Vis viva'' (from the Latin Latin (, or , ) is a classical language belonging to the Italic languages, Italic branch of the Indo-European languages. Latin was originally spoken in the area around Rome, known as Latium. Through the power of the ...
'' 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 (physics), motion. It is defined as the work (physics), work needed to accelerate a body of a given mass from rest to its stated velocity. Having gaine ...
in situations where there is no friction. Many
physicist A physicist is a scientist A scientist is a person who conducts scientific research The scientific method is an Empirical evidence, empirical method of acquiring knowledge that has characterized the development of science since at leas ...

s at that time, such as Newton, held that the
conservation of momentum In Newtonian mechanics, linear momentum, translational momentum, or simply momentum is the product of the mass and velocity of an object. It is a Euclidean vector, vector quantity, possessing a magnitude and a direction. If is an object's ma ...
, which holds even in systems with friction, as defined by the
momentum In Newtonian mechanics, linear momentum, translational momentum, or simply momentum is the product of the mass Mass is the quantity Quantity is a property that can exist as a multitude or magnitude, which illustrate discontinui ...

: :$\sum_ m_i v_i$ 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 An elastic collision is an encounter between two bodies in which the total kinetic energy of the two bodies remains the same. In an ideal, perfectly elastic collision, there is no net conversion of kinetic energy into other forms such as heat, no ...

. In 1687,
Isaac Newton Sir Isaac Newton (25 December 1642 – 20 March 1726/27) was an English mathematician A mathematician is someone who uses an extensive knowledge of mathematics Mathematics (from Greek: ) includes the study of such topics a ...

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. The law of conservation of vis viva was championed by the father and son duo,
Johann Johann, typically a male given name A given name (also known as a first name or forename) is the part of a personal name A personal name, or full name, in onomastic Onomastics or onomatology is the study of the etymology, history, an ...

and
Daniel Bernoulli Daniel Bernoulli Fellows of the Royal Society, FRS (; – 27 March 1782) was a Swiss people, Swiss mathematician and physicist and was one of the many prominent mathematicians in the Bernoulli family from Basel. He is particularly remembered for ...
. The former enunciated the principle of
virtual work In mechanics Mechanics (Greek Greek may refer to: Greece Anything of, from, or related to Greece Greece ( el, Ελλάδα, , ), officially the Hellenic Republic, is a country located in Southeast Europe. Its population is approximate ...
as used in statics in its full generality in 1715, while the latter based his ''
Hydrodynamica ''Hydrodynamica'' (Latin Latin (, or , ) is a classical language belonging to the Italic languages, Italic branch of the Indo-European languages. Latin was originally spoken in the area around Rome, known as Latium. Through the power of the ...
'', published in 1738, on this single conservation principle. Daniel's study of loss of vis viva of flowing water led him to formulate the
Bernoulli's principle File:Venturi Tube en.webm, Video of a Venturi effect, venturi meter used in a lab experiment In fluid dynamics, Bernoulli's principle states that an increase in the speed of a fluid occurs simultaneously with a decrease in pressure, static press ...
, which asserts the loss to be proportional to the change in hydrodynamic pressure. Daniel also formulated the notion of
work Work may refer to: * Work (human activity) Work or labor is intentional activity people perform to support themselves, others, or the needs and wants of a wider community. Alternatively, work can be viewed as the human activity that cont ...

and efficiency for
hydraulic Hydraulics (from Greek#REDIRECT Greek Greek may refer to: Greece Anything of, from, or related to Greece Greece ( el, Ελλάδα, , ), officially the Hellenic Republic, is a country located in Southeast Europe. Its population is a ...

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 D'Alembert's principle, also known as the Lagrange–d'Alembert principle, is a statement of the fundamental classical Classical may refer to: European antiquity *Classical antiquity, a period of history from roughly the 7th or 8th century B. ...
, Lagrangian, and Hamiltonian formulations of mechanics.
Émilie du Châtelet Gabrielle Émilie Le Tonnelier de Breteuil, Marquise du Châtelet (; 17 December 1706 – 10 September 1749) was a French French (french: français(e), link=no) may refer to: * Something of, from, or related to France France (), officiall ...
(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 Willem Jacob 's Gravesande (26 September 1688 – 28 February 1742) was a Dutch mathematician and natural philosopher, chiefly remembered for developing experimental demonstrations of the laws of classical mechanics. As professor of mathematic ...
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 Kinematics is a subfield of physics, developed in classical mechanics, that describes the Motion (physics), motion of points, bodies (objects), and systems of bodies (groups of objects) without considerin ...

. 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 Engineering is the use of scientific principles to design and build machines, structures, and other items, including bridges, tunnels, roads, vehicles, and buildings. The discipline of enginee ...

s such as
John Smeaton John Smeaton (8 June 1724 – 28 October 1792) was a British civil engineer A civil engineer is a person who practices civil engineering Civil engineering is a Regulation and licensure in engineering, professional engineering discip ...

,
Peter Ewart Peter Ewart (14 May 1767 – 15 September 1842) was a British engineer Engineers, as practitioners of engineering, are Professional, professionals who Invention, invent, design, analyze, build and test Machine, machines, complex systems, ...
, Carl Holtzmann, and
Marc Seguin Marc Seguin (20 April 1786 – 24 February 1875) was a French engineer Engineers, as practitioners of engineering, are Professional, professionals who Invention, invent, design, analyze, build and test Machine, machines, complex systems, a ...
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 Medieval Latin was the form of Latin Latin (, or , ) is a classical language A classical language is a language A language is a structu ...

s such as
William Hyde Wollaston William Hyde Wollaston (; 6 August 1766 – 22 December 1828) was an English English usually refers to: * English language English is a West Germanic languages, West Germanic language first spoken in History of Anglo-Saxon England, ...

. Academics such as
John Playfair John Playfair FRSE, FRS (10 March 1748 – 20 July 1819) was a Church of Scotland minister, remembered as a scientist and mathematician, and a professor of natural philosophy at the University of Edinburgh. He is best known for his book ''Ill ...

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 establishes the concept of 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 an ...
, 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 Antoine-Laurent de Lavoisier ( , ,; 26 August 17438 May 1794), When reduced without charcoal, it gave off an air which supported respiration and combustion in an enhanced way. He concluded that this was just a pure form of common air and t ...

and
Pierre-Simon Laplace Pierre-Simon, marquis de Laplace (; ; 23 March 1749 – 5 March 1827) was a French scholar A scholar is a person who pursues academic and intellectual activities, particularly those that develop expertise in an area of Studying, study. A ...

reviewed the two competing theories of ''vis viva'' and
caloric theory The caloric theory is an obsolete scientific theory In Science#History, science, a theory is superseded when a scientific consensus once widely accepted it, but current science considers it inadequate, incomplete, or debunked (i.e., wrong). Such ...
.
Count Rumford Sir Benjamin Thompson, Count Rumford, (german: von Rumford; March 26, 1753August 21, 1814) was an American-born British and whose challenges to established were part of the 19th-century revolution in . He served as lieutenant-colonel of th ...

's 1798 observations of heat generation during the
boring Boring may refer to: *Something that causes boredom Engineering and science * Boring (earth), drilling a hole, tunnel, or well in the earth ** Tunnel boring machine, a machine used in boring tunnels * Boring (manufacturing), enlarging a hole tha ...
of
cannon A cannon is a large-caliber A 45 ACP hollowpoint (Federal Cartridge, Federal HST) with two .22 Long Rifle, 22 LR cartridges for comparison In gun A gun is a ranged weapon designed to use a shooting tube ( gun barrel) to launc ...

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 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 Work may refer to: * Work (human activity) Work or labor is intentional activity people perform to support themselves, others, or the needs and wants of a wider community. Alternatively, work can be viewed as the human activity that cont ...
, was largely the result of
Gaspard-Gustave Coriolis Gaspard-Gustave de Coriolis (; 21 May 1792 – 19 September 1843) was a French mathematician A mathematician is someone who uses an extensive knowledge of mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of s ...
and
Jean-Victor Poncelet Jean-Victor Poncelet (1 July 1788 – 22 December 1867) was a French engineer Engineers, as practitioners of engineering Engineering is the use of scientific principles to design and build machines, structures, and other items, inc ...

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 a 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 German scientific journal of physics established in 1920 by Springer Berlin Heidelberg. The series stopped publication in 1997, when they m ...
'' in 1837,
Karl Friedrich Mohr Karl Friedrich Mohr (November 4, 1806 – September 28, 1879) was a German chemist famous for his early statement of the principle of the conservation of energy. Ammonium iron(II) sulfate, (NH4)2Fe(SO4)2.6H2O, is named Mohr's salt after him. Life ...

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 history of science is the study of the development of science Science (from the Latin word ''scientia'', meaning "knowledge") is a systematic enterprise that Scientific method, builds and Taxonomy (general), ...
''. The
caloric theory The caloric theory is an obsolete scientific theory In Science#History, science, a theory is superseded when a scientific consensus once widely accepted it, but current science considers it inadequate, incomplete, or debunked (i.e., wrong). Such ...
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 Mikhail Vasilyevich Lomonosov (; russian: Михаил (Михайло) Васильевич Ломоносов, p=mʲɪxɐˈil vɐˈsʲilʲjɪvʲɪtɕ , a=Ru-Mikhail Vasilyevich Lomonosov.ogg; – ) was a Russian polymath A polymath ( e ...

, 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, Fellow of the Royal Society, FRS (german: Reichsgraf von Rumford; March 26, 1753August 21, 1814) was an American-born British physics, physicist and inventor whose challenges to established physical theory w ...

) 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 Julius Robert von Mayer (25 November 1814 – 20 March 1878) was a German physician A physician (American English), medical practitioner (English in the Commonwealth of Nations, Commonwealth English), medical doctor, or simply doctor, is a ...

in 1842. Mayer reached his conclusion on a voyage to the
Dutch East Indies The Dutch East Indies (or Netherlands East-Indies; nl, Nederlands(ch)-Indië; ) was a Dutch colony The Dutch colonial empire ( nl, Nederlandse koloniale rijk) comprised the overseas territories and trading posts controlled and administer ...
, where he found that his patients' blood was a deeper red because they were consuming less
oxygen Oxygen is the chemical element Image:Simple Periodic Table Chart-blocks.svg, 400px, Periodic table, The periodic table of the chemical elements In chemistry, an element is a pure substance consisting only of atoms that all have the same ...

, and therefore less energy, to maintain their body temperature in the hotter climate. He discovered that
heat In thermodynamics Thermodynamics is a branch of physics that deals with heat, Work (thermodynamics), work, and temperature, and their relation to energy, entropy, and the physical properties of matter and radiation. The behavior of these ...

and
mechanical work In physics Physics is the that studies , its , its and behavior through , and the related entities of and . "Physical science is that department of knowledge which relates to the order of nature, or, in other words, to the regular ...
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 James Prescott Joule (; 24 December 1818 11 October 1889) was an English physicist A physicist is a scientist A scientist is a person who conducts scientific research The scientific method is an Empirical evidence, empirical m ...

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 In physics, potential energy is the energy In physics Physics (from grc, φυσική (ἐπιστήμη), physikḗ (epistḗmē), knowledge of nature, from ''phýsis'' 'nature'), , is the natural science that studies matter ...

lost by the weight in descending was equal to the
internal energy The internal energy of a thermodynamic system A thermodynamic system is a body of matter In classical physics and general chemistry, matter is any substance that has mass and takes up space by having volume. All everyday objects that ca ...
gained by the water through
friction Friction is the force In physics, a force is an influence that can change the motion (physics), motion of an Physical object, object. A force can cause an object with mass to change its velocity (e.g. moving from a Newton's first law, st ...

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 In physics and chemistry Chemistry is the scientific discipl ...

postulated a relationship between mechanics, heat,
light Light or visible light is electromagnetic radiation within the portion of the electromagnetic spectrum that is visual perception, perceived by the human eye. Visible light is usually defined as having wavelengths in the range of 400–700 nan ...

,
electricity Electricity is the set of physical Physical may refer to: *Physical examination, a regular overall check-up with a doctor *Physical (album), ''Physical'' (album), a 1981 album by Olivia Newton-John **Physical (Olivia Newton-John song), "Physi ...

and
magnetism Magnetism is a class of physical attributes that are mediated by magnetic field A magnetic field is a vector field In vector calculus and physics, a vector field is an assignment of a vector to each point in a subset of space. For in ...

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 A physicist is a scientist A scientist is a person who conducts Scientific method, scientific research to advance knowledge in an Branc ...

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 William John Macquorn Rankine (; 5 July 1820 – 24 December 1872) was a Scottish mechanical engineer who also contributed to civil engineering, physics and mathematics. He was a founding contributor, with Rudolf Clausius and William Thomson, 1 ...
first used the phrase ''the law of the conservation of energy'' for the principle. In 1877,
Peter Guthrie Tait Peter Guthrie Tait FRSE Fellowship of the Royal Society of Edinburgh (FRSE) is an award granted to individuals that the Royal Society of Edinburgh, Scotland's national academy of science and Literature, letters, judged to be "eminently dist ...

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 Philosophy (from , ) is the study of general and fundamental questions, such as those about existence Existence is the ability of an entity to interact with physical reality Reality is the sum or aggregate of all that is real or existen ...
''. This is now regarded as an example of
Whig history Whig history (or Whig historiography) is an approach to historiography Historiography is the study of the methods of historians in developing history as an academic discipline, and by extension is any body of historical work on a particular su ...
.

## 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 Physics is the that studies , its , its and behavior through , and the related entities of and . "Physical science is that department of knowledge which relates to the order of nature, or, in other words, to the regular s ...
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 In physics Physics is the that studies , its , its and behavior through , and the related entities of and . "Physical science is that department of knowledge which relates to the order of nature, or, in other words, to the regular succ ...
. 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 is a subatomic particle (denoted by the symbol or ) whose electric charge is negative one elementary charge. Electrons belong to the first generation (particle physics), generation of the lepton particle family, and are general ...

and a
positron The positron or antielectron is the antiparticle s (left) and antiparticles (right). From top to bottom; electron The electron is a subatomic particle In physical sciences, subatomic particles are smaller than atom An atom is ...

each have rest mass. They can perish together, converting their combined rest energy into
photon The photon ( el, φῶς, phōs, light) is a type of elementary particle In , an elementary particle or fundamental particle is a that is not composed of other particles. Particles currently thought to be elementary include the fundamental s ...

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 In physics and chemistry, the law of conservation of mass or principle of mass conservation states that for any closed system, system closed to all transfers of matter and energy, the mass of the system must remain constant over time, as the syst ...
(''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 , beta decay (''β''-decay) is a type of in which a (fast energetic or ) is emitted from an , transforming the original to an of that nuclide. For example, beta decay of a transforms it into a by the emission of an electron accompanie ...

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 Enrico Fermi (; 29 September 1901 - 28 November 1954) was an Italian (later naturalized American) physicist and the creator of the world's first nuclear reactor, the Chicago Pile-1. He has been called the "architect of the nuclear age" and ...

who proposed the correct description of beta-decay as the emission of both an electron and an
antineutrino A neutrino ( or ) (denoted by the Greek letter Nu (letter), ) is a fermion (an elementary particle with spin-1/2, spin of ) that interacts only via the weak interaction and gravity. The neutrino is so named because it is electric charge, electri ...
, 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 , energy is the that must be to a or to perform on the body, or to it. Energy is a ; the law of states that energy can be in form, but not created or destroyed. The unit of measurement in the (SI) of energy is the , which is the ...

added to the system by a
heat In thermodynamics Thermodynamics is a branch of physics that deals with heat, Work (thermodynamics), work, and temperature, and their relation to energy, entropy, and the physical properties of matter and radiation. The behavior of these ...

ing process, $\delta W$ is the quantity of energy lost by the system due to
work Work may refer to: * Work (human activity) Work or labor is intentional activity people perform to support themselves, others, or the needs and wants of a wider community. Alternatively, work can be viewed as the human activity that cont ...
done by the system on its surroundings and $\mathrmU$ is the change in the
internal energy The internal energy of a thermodynamic system A thermodynamic system is a body of matter In classical physics and general chemistry, matter is any substance that has mass and takes up space by having volume. All everyday objects that ca ...
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 type of differential used in thermodynamics to express changes in path dependent quantities. In contrast, an integral of an exact differential (a differential of a function) is always path in ...
). 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 thermodynamic ...
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 In physics, a force is an influence that can change the motion (physics), motion of an Physical object, object. A force can cause an object with mass to change its velocity (e.g. moving fr ...

and $dV$ is a small change in the
volume Volume is a scalar quantity expressing the amount Quantity or amount is a property that can exist as a multitude Multitude is a term for a group of people who cannot be classed under any other distinct category, except for their shared fact ...

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 hot and cold. It is the manifestation of thermal energy Thermal radiation in visible light can be seen on this hot metalwork. Thermal energy refers to several distinct physical concept ...

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: :$\mathrmU = \delta Q - \delta W + u\text{'}\,dM,$ where $dM$ is the added mass and $u\text{'}$ is the internal energy per unit mass of the added mass, measured in the surroundings before the process.

# 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 Noether's theorem or Noether's first theorem states that every differentiable In calculus (a branch of mathematics), a differentiable function of one Real number, real variable is a function whose derivative exists at each point in its Domai ...
, developed by
Emmy Noether Amalie Emmy Noether Emmy is the '' Rufname'', the second of two official given names, intended for daily use. Cf. for example the résumé submitted by Noether to Erlangen University in 1907 (Erlangen University archive, ''Promotionsakt Emmy Noeth ...

in 1915 and first published in 1918. The theorem states that every continuous symmetry of a physical theory 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 Greek#REDIRECT Greek Greek may refer to: Greece Anything of, from, or related to Greece Greece ( el, Ελλάδα, , ), officially the Hellenic Republic, is a country located in Southeast Europe. Its population is appro ...
of time; energy conservation is implied by the empirical fact that the
laws of physics Scientific laws or laws of science are statements, based on repeated experiment An experiment is a procedure carried out to support, refute, or validate a hypothesis. Experiments provide insight into Causality, cause-and-effect by demonstrat ...
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 Symmetry in mathematics, symmetries as Motion (physics), motions, as opposed to discrete symmetry, e.g. reflection symmetry, which is invariant un ...
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 hypothesis that the laws of physics are uncha ...
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 – 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 In physics Physics is the that studies , its , its and behavior through , and the related entities of and . "Physical science is that department of knowledge which relates to the order of nature, or, in other words, to the regular suc ...
.

# Relativity

With the discovery of special relativity by
Henri Poincaré Jules Henri Poincaré ( S: stress final syllable ; 29 April 1854 – 17 July 1912) was a French mathematician A mathematician is someone who uses an extensive knowledge of mathematics Mathematics (from Greek: ) includes the s ...
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 physicists of all time. Einstein is known for developing the theory of relativity The theo ...

, the energy was proposed to be a component of an . 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 In classical physics Classical physics is a group of physics Physics (from grc, φυσική (ἐπιστήμη), physikḗ (epistḗmē), knowledge of nature, from ''phýsis'' 'nature'), , is the natural science that studies mat ...
. Also conserved is the vector length ( Minkowski norm), which is the
rest mass The invariant mass, rest mass, intrinsic mass, proper mass, or in the case of bound systems simply mass, is the portion of the total mass of an object Object may refer to: General meanings * Object (philosophy), a thing, being, or concept ** ...
for single particles, and the
invariant mass The invariant mass, rest mass, intrinsic mass, proper mass, or in the case of bound systems simply mass, is the portion of the total mass of an object Object may refer to: General meanings * Object (philosophy), a thing, being, or concept ** ...
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 the quantity Quantity is a property that can exist as a multitude or magnitude, which illustrate discontinuity and continuity. Quantities can be compared in terms of "more", "less", or "equal", or by assigning a numerical value ...
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 frameIn special relativity the rest frame of a particle is the coordinate system (frame of reference) in which the particle is at rest. The rest frame of compound objects (such as a fluid, or a solid made of many vibrating atoms) is taken to be the frame ...
) of a massive particle, or else in the
center of momentum frame In physics Physics (from grc, φυσική (ἐπιστήμη), physikḗ (epistḗmē), knowledge of nature, from ''phýsis'' 'nature'), , is the natural science that studies matter, its Motion (physics), motion and behavior through Sp ...
for objects or systems which retain kinetic energy, the
total energy In physics Physics is the that studies , its , its and behavior through , and the related entities of and . "Physical science is that department of knowledge which relates to the order of nature, or, in other words, to the regula ...

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, a frame of reference (or reference frame) consists of an abstract coordinate system In geometry Geometry (from the grc, γεωμετρία; ''wikt:γῆ, geo-'' "earth", ''wikt:μέτρον, -metron'' "measurement") is, wi ...

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 In physics Physics (from grc, φυσική (ἐπιστήμη), physikḗ (epistḗmē), knowledge of nature, from ''phýsis'' 'nature'), , is the natural science that studies matter, its Motion (physics), motion and behavior through Spa ...
. In general relativity, 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 pseudotensorIn 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 allows ...
. 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 such as the
Friedmann–Lemaître–Robertson–Walker metric The Friedmann–Lemaître–Robertson–Walker (FLRW; ) metric METRIC (Mapping EvapoTranspiration at high Resolution with Internalized Calibration) is a computer model Computer simulation is the process of mathematical modelling, performed on ...
do not satisfy these constraints and energy conservation is not well defined. The theory of general relativity leaves open the question of whether there is a conservation of energy for the entire universe.

# Quantum theory

In
quantum mechanics Quantum mechanics is a fundamental theory A theory is a reason, rational type of abstraction, abstract thinking about a phenomenon, or the results of such thinking. The process of contemplative and rational thinking is often associated with ...
, energy of a quantum system is described by a
self-adjointIn mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ha ...
(or Hermitian) operator called the Hamiltonian, which acts on the
Hilbert space In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...
(or a space of
wave functions A wave function in quantum physics Quantum mechanics is a fundamental theory A theory is a rational Rationality is the quality or state of being rational – that is, being based on or agreeable to reason Reason is the capacity ...

) 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 Noether's theorem or Noether's first theorem states that every differentiable In calculus (a branch of mathematics), a differentiable function of one Real number, real variable is a function whose derivative exists at each point in its Domai ...
for energy-momentum tensor operator. Due to the lack of the (universal) time operator in quantum theory, the uncertainty relations for time and energy are not fundamental in contrast to the position-momentum uncertainty principle, and merely holds in specific cases (see
Uncertainty principle In quantum mechanics Quantum mechanics is a fundamental theory A theory is a reason, rational type of abstraction, abstract thinking about a phenomenon, or the results of such thinking. The process of contemplative and rational thinking ...

). Energy at each fixed time can in principle be exactly measured without any trade-off in precision forced by the time-energy uncertainty relations. Thus the conservation of energy in time is a well defined concept even in quantum mechanics.

*
Energy quality , a form of energy that depends on an object's temperature, is partly 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, ...
*
Energy transformation Energy transformation, also known as energy conversion, is the process of changing energy from one form to another. In physics Physics is the natural science that studies matter, its Elementary particle, fundamental constituents, its Mo ...
* Eternity of the world *
Lagrangian mechanics Introduced by the Italian-French mathematician and astronomer Joseph-Louis Lagrange Joseph-Louis Lagrange (born Giuseppe Luigi Lagrangia
*
Laws of thermodynamics The laws of thermodynamics define a group of physical quantities, such as temperature Temperature ( ) is a physical quantity that expresses hot and cold. It is the manifestation of thermal energy Thermal radiation in visible light can be ...
* Zero-energy universe

# 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"