In ^{2}'', 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

_{i}'', each with _{i}''),
:$\backslash sum\_\; m\_i\; v\_i^2$
was conserved so long as the masses did not interact. He called this quantity the ''

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

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 rela ...

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, propertie ...

, 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 heat ...

of an isolated system
In physical science, an isolated system is either of the following:
# a physical system so far removed from other systems that it does not interact with them.
# a thermodynamic system enclosed by rigid immovable walls through which neither ...

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 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 acc ...

when a stick of dynamite
Dynamite is an explosive made of nitroglycerin, sorbents (such as powdered shells or clay), and stabilizers. It was invented by the Swedish chemist and engineer Alfred Nobel in Geesthacht, Northern Germany, and patented in 1867. It rapidly ...

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 acc ...

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 potentia ...

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 system closed to all transfers of matter and energy, the mass of the system must remain constant over time, as the system's 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 laws ...

shows that mass is related to energy and vice versa by ''E = mcblack hole
A black hole is a region of spacetime where gravity is so strong that nothing, including light or other electromagnetic waves, has enough energy to escape it. The theory of general relativity predicts that a sufficiently compact mass can def ...

s emit Hawking radiation
Hawking radiation is theoretical black body radiation that is theorized to be released outside a black hole's event horizon because of relativistic quantum effects. It is named after the physicist Stephen Hawking, who developed a theoretical arg ...

.
Given the stationary-action principle, conservation of energy can be rigorously proven by Noether's theorem
Noether's theorem or Noether's first theorem states that every differentiable symmetry of the action of a physical system with conservative forces has a corresponding conservation law. The theorem was proven by mathematician Emmy Noether ...

as a consequence of continuous time translation symmetry; that is, from the fact that the laws of physics do not change over time.
A consequence of the law of conservation of energy is that a perpetual motion machine of the first kind
Perpetual motion is the motion of bodies that continues forever in an unperturbed system. A perpetual motion machine is a hypothetical machine that can do work infinitely without an external energy source. This kind of machine is impossible, a ...

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 asThales 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 h ...

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 fo ...

(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 (; grc-gre, Ἐπίκουρος ; 341–270 BC) was an ancient Greek philosopher and sage who founded Epicureanism, a highly influential school of philosophy. He was born on the Greek island of Samos to Athenian parents. Influence ...

( 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
Simon Stevin (; 1548–1620), sometimes called Stevinus, was a Flemish mathematician, scientist and music theorist. He made various contributions in many areas of science and engineering, both theoretical and practical. He also translated var ...

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 ...

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
Gottfried Wilhelm (von) Leibniz . ( – 14 November 1716) was a German polymath active as a mathematician, philosopher, scientist and diplomat. He is one of the most prominent figures in both the history of philosophy and the history of mathem ...

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 eleme ...

es ''mvelocity
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 ...

''vvis viva
''Vis viva'' (from the Latin for "living force") is a historical term used for the first recorded description of what we now call kinetic energy in an early formulation of the principle of conservation of energy.
Overview
Proposed by Gottfried L ...

'' 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 acc ...

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 a ...

:
:$\backslash sum\_\; m\_i\; v\_i$
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 physics, an elastic collision is an encounter (collision) 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 int ...

.
In 1687, Isaac Newton
Sir Isaac Newton (25 December 1642 – 20 March 1726/27) was an English mathematician, physicist, astronomer, alchemist, theologian, and author (described in his time as a "natural philosopher"), widely recognised as one of the great ...

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
Physical causal closure is a metaphysical fallacy about the nature of causation in the physical realm with significant ramifications in the study of metaphysics and the mind. It's based on the misconception that the physical world is a combinato ...

.)
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 Gracious" ...

and Daniel Bernoulli
Daniel Bernoulli FRS (; – 27 March 1782) was a Swiss mathematician and physicist and was one of the many prominent mathematicians in the Bernoulli family from Basel. He is particularly remembered for his applications of mathematics to mech ...

. The former enunciated the principle of virtual work
In mechanics, virtual work arises in the application of the ''principle of least action'' to the study of forces and movement of a mechanical system. The work of a force acting on a particle as it moves along a displacement is different for ...

as used in statics in its full generality in 1715, while the latter based his '' Hydrodynamica'', published in 1738, on this single vis viva conservation principle. Daniel's study of loss of vis viva of flowing water led him to formulate the Bernoulli's principle
In fluid dynamics, Bernoulli's principle states that an increase in the speed of a fluid occurs simultaneously with a decrease in static pressure or a decrease in the fluid's potential energy. The principle is named after the Swiss mathematic ...

, 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), intentional activity people perform to support themselves, others, or the community
** Manual labour, physical work done by humans
** House work, housework, or homemaking
** Working animal, an animal ...

and efficiency for hydraulic
Hydraulics (from Greek: Υδραυλική) is a technology and applied science using engineering, chemistry, and other sciences involving the mechanical properties and use of liquids. At a very basic level, hydraulics is the liquid counte ...

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 laws of motion. It is named after its discoverer, the French physicist and mathematician Jean le Rond d'Alembert. D'Alember ...

, Lagrangian
Lagrangian may refer to:
Mathematics
* Lagrangian function, used to solve constrained minimization problems in optimization theory; see Lagrange multiplier
** Lagrangian relaxation, the method of approximating a difficult constrained problem wit ...

, 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 $E\_k\; =\; \backslash frac12\; mv^2$, 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 quanti ...

. 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 invent, design, analyze, build and test machines, complex systems, structures, gadgets and materials to fulfill functional objectives and requirements while considering the limi ...

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 firs ...

, Peter Ewart
Peter Ewart (14 May 1767 – 15 September 1842) was a British engineer who was influential in developing the technologies of turbines and theories of thermodynamics.
Biography
He was son of the Church of Scotland minister of Troqueer near D ...

, , 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, the fo ...

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 ingo ...

. Academics such as John Playfair were quick to point out that kinetic energy is clearly not conserved. This is obvious to a modern analysis based on the second law of thermodynamics, but in the 18th and 19th centuries, the fate of the lost energy was still unknown.
Gradually it came to be suspected that the heat inevitably generated by motion under friction was another form of ''vis viva''. In 1783, Antoine Lavoisier and Pierre-Simon Laplace
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 summarized ...

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 in ...

. 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 duri ...

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
:$\backslash frac\; \backslash sum\_\; m\_i\; v\_i^2$
which can be understood as converting kinetic energy to work
Work may refer to:
* Work (human activity), intentional activity people perform to support themselves, others, or the community
** Manual labour, physical work done by humans
** House work, housework, or homemaking
** Working animal, an animal ...

, 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 convert ...

''. 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 in ...

maintained that heat could neither be created nor destroyed, whereas conservation of energy entails the contrary principle that heat and mechanical work are interchangeable.
In the middle of the eighteenth century, Mikhail Lomonosov, a Russian scientist, postulated his corpusculo-kinetic theory of heat, which rejected the idea of a caloric. Through the results of empirical studies, Lomonosov came to the conclusion that heat was not transferred through the particles of the caloric fluid.
In 1798, Count Rumford ( Benjamin Thompson) performed measurements of the frictional heat generated in boring cannons and developed the idea that heat is a form of kinetic energy; his measurements refuted caloric theory, but were imprecise enough to leave room for doubt.
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, whi ...

, where he found that his patients' blood was a deeper red because they were consuming less oxygen
Oxygen is the chemical element with the chemical symbol, symbol O and atomic number 8. It is a member of the chalcogen Group (periodic table), group in the periodic table, a highly Chemical reaction, reactive nonmetal, and an oxidizing a ...

, and therefore less energy, to maintain their body temperature in the hotter climate. He discovered that heat and mechanical work were both forms of energy, and in 1845, after improving his knowledge of physics, he published a monograph that stated a quantitative relationship between them.
Meanwhile, in 1843, James Prescott Joule
James Prescott Joule (; 24 December 1818 11 October 1889) was an English physicist, mathematician and brewer, born in Salford, Lancashire. Joule studied the nature of heat, and discovered its relationship to mechanical work (see energy). Th ...

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
Gravitational energy or gravitational potential energy is the potential energy a massive object has in relation to another massive object due to gravity. It is the potential energy associated with the gravitational field, which is released (conve ...

lost by the weight in descending was equal to the internal energy
The internal energy of a thermodynamic system is the total energy contained within it. It is the energy necessary to create or prepare the system in its given internal state, and includes the contributions of potential energy and internal kinet ...

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 t ...

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 te ...

, 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 described ...

, 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 Associatio ...

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
Peter Guthrie Tait FRSE (28 April 1831 – 4 July 1901) was a Scottish mathematical physicist and early pioneer in thermodynamics. He is best known for the mathematical physics textbook '' Treatise on Natural Philosophy'', which he co-wrote ...

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 systematized study of general and fundamental questions, such as those about existence, reason, knowledge, values, mind, and language. Such questions are often posed as problems to be studied or resolved. Some ...

''. This is now regarded as an example of Whig history
Whig history (or Whig historiography) is an approach to historiography that presents history as a journey from an oppressive and benighted past to a "glorious present". The present described is generally one with modern forms of liberal democracy ...

.
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 ofspecial 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 laws ...

showed that rest mass corresponds to an equivalent amount of ''rest energy''. This means that ''rest mass'' can be converted to or from equivalent amounts of (non-material) forms of energy, for example, kinetic energy, potential energy, and electromagnetic radiant energy. When this happens, as recognized in twentieth-century experience, rest mass is not conserved, unlike the ''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 no kno ...

and a positron
The positron or antielectron is the antiparticle or the antimatter counterpart of the electron. It has an electric charge of +1 '' e'', a spin of 1/2 (the same as the electron), and the same mass as an electron. When a positron collides ...

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 always ...

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 system closed to all transfers of matter and energy, the mass of the system must remain constant over time, as the system's 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 inbeta 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
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 ( ; denoted by the Greek letter ) is a fermion (an elementary particle with spin of ) that interacts only via the weak interaction and gravity. The neutrino is so named because it is electrically neutral and because its rest mass i ...

, 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: :$\backslash delta\; Q\; =\; \backslash mathrmU\; +\; \backslash delta\; W$, or equivalently, $\backslash mathrmU\; =\; \backslash delta\; Q\; -\; \backslash delta\; W,$ where $\backslash delta\; Q$ is the quantity ofenergy
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 heat ...

added to the system by a heating process, $\backslash delta\; W$ is the quantity of energy lost by the system due to work
Work may refer to:
* Work (human activity), intentional activity people perform to support themselves, others, or the community
** Manual labour, physical work done by humans
** House work, housework, or homemaking
** Working animal, an animal ...

done by the system on its surroundings, and $\backslash mathrmU$ is the change in the internal energy
The internal energy of a thermodynamic system is the total energy contained within it. It is the energy necessary to create or prepare the system in its given internal state, and includes the contributions of potential energy and internal kinet ...

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 $\backslash mathrmU$ 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 with ...

). 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 $\backslash delta\; Q$ 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 $\backslash delta\; W$ 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:
:$\backslash delta\; W\; =\; P\backslash ,\backslash mathrmV,$
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 and ...

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). The de ...

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
:$\backslash delta\; Q\; =\; T\backslash ,\backslash mathrmS,$
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 $\backslash mathrmS$ 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
:$\backslash mathrmU\; =\; \backslash delta\; Q\; -\; \backslash delta\; W\; +\; \backslash sum\_i\; h\_i\backslash ,dM\_i,$
where $dM\_i$ is the added mass of species $i$ and $h\_i$ is the corresponding enthalpy per unit mass. Note that generally $dS\backslash neq\backslash delta\; Q/T$ in this case, as matter carries its own entropy. Instead, $dS=\backslash delta\; Q/T+\backslash textstyles\_i\backslash ,dM\_i$, where $s\_i$ is the entropy per unit mass of type $i$, from which we recover the fundamental thermodynamic relation
:$\backslash mathrmU\; =\; T\backslash ,dS\; -\; P\backslash ,dV\; +\; \backslash sum\_i\backslash mu\_i\backslash ,dN\_i$
because the chemical potential $\backslash mu\_i$ is the partial molar Gibbs free energy of species $i$ and the Gibbs free energy $G\backslash equiv\; H-TS$.
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 ofNoether's theorem
Noether's theorem or Noether's first theorem states that every differentiable symmetry of the action of a physical system with conservative forces has a corresponding conservation law. The theorem was proven by mathematician Emmy Noether ...

, developed by Emmy Noether
Amalie Emmy NoetherEmmy 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. 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 an ...

of time translation, then its energy (which is the canonical conjugate quantity to time) is conserved. Conversely, systems that are not invariant under shifts in time (e.g. systems with time-dependent potential energy) do not exhibit conservation of energy – 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 byHenri Poincaré
Jules Henri Poincaré ( S: stress final syllable ; 29 April 1854 – 17 July 1912) was a French mathematician, theoretical physicist, engineer, and philosopher of science. He is often described as a polymath, and in mathematics as "T ...

and Albert Einstein
Albert Einstein ( ; ; 14 March 1879 – 18 April 1955) was a German-born Theoretical physics, theoretical physicist, widely acknowledged to be one of the greatest and most influential physicists of all time. Einstein is best known for d ...

, 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
In classical physics and special relativity, an inertial frame of reference (also called inertial reference frame, inertial frame, inertial space, or Galilean reference frame) is a frame of reference that is not undergoing any acceleration. ...

. Also conserved is the vector length ( Minkowski norm), which is the rest mass 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 or system of objects that is independent of the overall motion of the system. More precisely, ...

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 eleme ...

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
In physics, the center-of-momentum frame (also zero-momentum frame or COM frame) of a system is the unique (up to velocity but not origin) inertial frame in which the total momentum of the system vanishes. The ''center of momentum'' of a system i ...

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 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's view is that energy–momentum conservation is not well-defined except in certain special cases. Energy-momentum is typically expressed with the aid of a stress–energy–momentum pseudotensor. However, since pseudotensors are not tensors, they do not transform cleanly between reference frames. If the metric under consideration is static (that is, does not change with time) or asymptotically flat (that is, at an infinite distance away spacetime looks empty), then energy conservation holds without major pitfalls. In practice, some metrics, notably theFriedmann–Lemaître–Robertson–Walker metric
The Friedmann–Lemaître–Robertson–Walker (FLRW; ) metric is a metric based on the exact solution of Einstein's field equations of general relativity; it describes a homogeneous, isotropic, expanding (or otherwise, contracting) universe t ...

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
0 (zero) is a number representing an empty quantity. In place-value notation such as the Hindu–Arabic numeral system, 0 also serves as a placeholder numerical digit, which works by multiplying digits to the left of 0 by the radix, usua ...

.
Quantum theory

Inquantum 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 sta ...

(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 naturally ...

(or a space of wave function
A wave function in quantum physics is a mathematical description of the quantum state of an isolated quantum system. The wave function is a complex-valued probability amplitude, and the probabilities for the possible results of measurements m ...

s) of the system. If the Hamiltonian is a time-independent operator, emergence probability of the measurement result does not change in time over the evolution of the system. Thus the expectation value of energy is also time independent. The local energy conservation in quantum field theory is ensured by the quantum Noether's theorem
Noether's theorem or Noether's first theorem states that every differentiable symmetry of the action of a physical system with conservative forces has a corresponding conservation law. The theorem was proven by mathematician Emmy Noether ...

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
The Born rule (also called Born's rule) is a key postulate of quantum mechanics which gives the probability that a measurement of a quantum system will yield a given result. In its simplest form, it states that the probability density of findi ...

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 asquantum gravity
Quantum gravity (QG) is a field of theoretical physics that seeks to describe gravity according to the principles of quantum mechanics; it deals with environments in which neither gravitational nor quantum effects can be ignored, such as in the v ...

), 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 level accessible through particle accelerators. Doubly special relativity models may argue for a breakdown in energy-momentum conservation for sufficiently energetic particles; such models are constrained by observations that cosmic rays appear to travel for billions of years without displaying anomalous non-conservation behavior. 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
In physics, Lagrangian mechanics is a formulation of classical mechanics founded on the stationary-action principle (also known as the principle of least action). It was introduced by the Italian-French mathematician and astronomer Joseph-Lo ...

* Laws of thermodynamics
The laws of thermodynamics are a set of scientific laws which define a group of physical quantities, such as temperature, energy, and entropy, that characterize thermodynamic systems in thermodynamic equilibrium. The laws also use various paramet ...

* 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''

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