Continuum mechanics  

Laws


In physics and chemistry, the law of conservation of energy states that the total energy of an isolated system remains constant; it is said to be conserved over time.^{[1]} This law 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 is converted to kinetic energy when a stick of dynamite explodes. If one adds up all forms of energy that were released in the explosion, such as the kinetic energy and potential energy of the pieces, as well as heat and sound, one will get the exact decrease of chemical energy in the combustion of the dynamite. Classically, conservation of energy was distinct from conservation of mass; however, special relativity showed that mass is related to energy and vice versa by E = mc^{2}, and science now takes the view that mass–energy as a whole is conserved. Theoretically, this implies that an object with mass can itself be converted to pure energy, and vice versa, though 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.
Conservation of energy can be rigorously proven by Noether's theorem 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 cannot exist, that is to say, no system without an external energy supply can deliver an unlimited amount of energy to its surroundings.^{[2]} For systems which do not have time translation symmetry, it may not be possible to define conservation of energy. Examples include curved spacetimes in general relativity^{[3]} or time crystals in condensed matter physics.^{[4]}^{[5]}^{[6]}^{[7]}
Ancient philosophers as far back as Thales of Miletus c. 550 BCE had inklings of the conservation of some underlying substance of which everything is made. However, there is no particular reason to identify this with what we know today as "massenergy" (for example, Thales thought it was water). Empedocles (490–430 BCE) wrote that in this universal system, composed of four roots (earth, air, water, fire), "nothing comes to be or perishes";^{[8]} instead, these elements suffer continual rearrangement.
In 1605, Simon Stevinus was able to solve a number of problems in statics based on the principle that perpetual motion was impossible.
In 1639, Galileo published his analysis of several situations—including the celebrated "interrupted pendulum"—which can be described (in modern language) as conservatively converting potential energy to kinetic energy and back again. Essentially, he pointed out that the height a moving body rises is equal to the height from which it falls, and used this observation to infer the idea of inertia. The remarkable aspect of this observation is that the height to which a moving body ascends on a frictionless surface does not depend on the shape of the surface.
In 1669, Christiaan Huygens published his laws of collision. Among the quantities he listed as being invariant before and after the collision of bodies were both the sum of their linear 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 a 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.
The fact that kinetic energy is scalar, unlike linear momentum which is a vector, and hence easier to work with did not escape the attention of Gottfried Wilhelm Leibniz. It was Leibniz during 1676–1689 who first attempted a mathematical formulation of the kind of energy which is connected with motion (kinetic energy). Using Huygens' work on collision, Leibniz noticed that in many mechanical systems (of several masses, m_{i} each with velocity v_{i}),
was conserved so long as the masses did not interact. He called this quantity the vis viva or living force of the system. The principle represents an accurate statement of the approximate conservation of kinetic energy in situations where there is no friction. Many physicists at that time, such as Newton, held that the conservation of momentum, which holds even in systems with friction, as defined by the momentum:
was the conserved vis viva. It was later shown that both quantities are conserved simultaneously, given the proper conditions such as an elastic collision.
In 1687, Isaac Newton 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 and Daniel Bernoulli. The former enunciated the principle of virtual work as used in statics in its full generality in 1715, while the latter based his Hydrodynamica, published in 1738, on this single conservation principle. Daniel's study of loss of vis viva of flowing water led him to formulate the Bernoulli's principle, which relates the loss to be proportional to the change in hydrodynamic pressure. Daniel also formulated the notion of work and efficiency for hydraulic machines; and he gave a kinetic theory of gases, and linked the kinetic energy of gas molecules with the temperature of the gas.
This focus on the vis viva by the continental physicists eventually led to the discovery of stationarity principles governing mechanics, such as the D'Alembert's principle, Lagrangian, and Hamiltonian formulations of mechanics.
Émilie du Châtelet (1706 – 1749) proposed and tested the hypothesis of the conservation of total energy, as distinct from momentum. Inspired by the theories of Gottfried Leibniz, she repeated and publicized an experiment originally devised by Willem 's Gravesande in 1722 in which balls were dropped from different heights into a sheet of soft clay. Each ball's kinetic energy  as indicated by the quantity of material displaced  was shown to be proportional to the square of the velocity. The deformation of the clay was found to be directly proportional to the height the balls were dropped from, 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 , where is the kinetic energy of an object, its mass and its speed. On this basis, Châtelet proposed that energy must always have the same dimensions in any form, which is necessary to be able to relate it in different forms (kinetic, potential, heat…).^{[9]}^{[10]}
Engineers such as John Smeaton, Peter Ewart, Carl Holtzmann, GustaveAdolphe Hirn and Marc Seguin recognized that conservation of momentum alone was not adequate for practical calculation and made use of Leibniz's principle. The principle was also championed by some chemists such as William Hyde Wollaston. Academics such as John Playfair were quick to point out that kinetic energy is clearly not conserved. This is obvious to a modern analysis based on the second law of thermodynamics, but in the 18th and 19th centuries the fate of the lost energy was still unknown.
Gradually it came to be suspected that the heat inevitably generated by motion under friction was another form of vis viva. In 1783, Antoine Lavoisier and PierreSimon Laplace reviewed the two competing theories of vis viva and caloric theory.^{[11]} Count Rumford's 1798 observations of heat generation during the boring of cannons added more weight to the view that mechanical motion could be converted into heat, and (as importantly) 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
which can be understood as converting kinetic energy to work, was largely the result of GaspardGustave Coriolis and JeanVictor Poncelet over the period 1819–1839. The former called the quantity quantité de travail (quantity of work) and the latter, travail mécanique (mechanical work), and both championed its use in engineering calculation.
In a paper Über die Natur der Wärme(German "On the Nature of Heat/Warmth"), published in the Zeitschrift für Physik in 1837, Karl Friedrich Mohr gave one of the earliest general statements of the doctrine of the conservation of energy in the words: "besides the 54 known chemical elements there is in the physical world one agent only, and this is called Kraft [energy or work]. It may appear, according to circumstances, as motion, chemical affinity, cohesion, electricity, light and magnetism; and from any one of these forms it can be transformed into any of the others."
A key stage in the development of the modern conservation principle was the demonstration of the mechanical equivalent of heat. The caloric theory maintained that heat could neither be created nor destroyed, whereas conservation of energy entails the contrary principle that heat and mechanical work are interchangeable.
In the middle of the eighteenth century, Mikhail Lomonosov, a Russian scientist, postulated his corpusculokinetic 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.^{[12]} Mayer reached his conclusion on a voyage to the Dutch East Indies, where he found that his patients' blood was a deeper red because they were consuming less oxygen, and therefore less energy, to maintain their body temperature in the hotter climate. He discovered that heat and mechanical work were both forms of energy and in 1845, after improving his knowledge of physics, he published a monograph that stated a quantitative relationship between them.^{[13]}
Meanwhile, in 1843, James Prescott Joule independently discovered the mechanical equivalent in a series of experiments. In the most famous, now called the "Joule apparatus", a descending weight attached to a string caused a paddle immersed in water to rotate. He showed that the gravitational potential energy lost by the weight in descending was equal to the internal energy gained by the water through friction 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 postulated a relationship between mechanics, heat, light, electricity and magnetism by treating them all as manifestations of a single "force" (energy in modern terms). In 1846, Grove published his theories in his book The Correlation of Physical Forces.^{[14]} In 1847, drawing on the earlier work of Joule, Sadi Carnot and Émile Clapeyron, Hermann von Helmholtz arrived at conclusions similar to Grove's and published his theories in his book Über die Erhaltung der Kraft (On the Conservation of Force, 1847).^{[15]} 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.^{[16]}
In 1877, Peter Guthrie Tait claimed that the principle originated with Sir Isaac Newton, based on a creative reading of propositions 40 and 41 of the Philosophiae Naturalis Principia Mathematica. This is now regarded as an example of Whig history.^{[17]}
Matter is composed of atoms and what makes up atoms. It 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 showed that it corresponds to an equivalent amount of rest energy. This means that it can be converted to or from equivalent amounts of other (nonmaterial) 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 and a positron each have rest mass. They can perish together, converting their combined rest energy into photons having 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, nonmaterial forms of energy can perish into matter, which has rest mass.
Thus, conservation of energy (total, including material or rest energy), and conservation of mass (total, not just rest), each still holds as an (equivalent) law. In the 18th century these had appeared as two seeminglydistinct laws.
The discovery in 1911 that electrons emitted in beta decay have a continuous rather than a discrete spectrum appeared to contradict conservation of energy, under the thencurrent assumption that beta decay is the simple emission of an electron from a nucleus.^{[18]}^{[19]} This problem was eventually resolved in 1933 by Enrico Fermi who proposed the correct description of betadecay as the emission of both an electron and an antineutrino, which carries away the apparently missing energy.^{[20]}^{[21]}
For a closed thermodynamic system, the first law of thermodynamics may be stated as:
where is the quantity of energy added to the system by a heating process, is the quantity of energy lost by the system due to work done by the system on its surroundings and is the change in the internal energy of the system.
The δ's before the heat and work terms are used to indicate that they describe an increment of energy which is to be interpreted somewhat differently than the increment of internal energy (see Inexact differential). Work and heat refer to kinds of process which add or subtract energy to or from a system, while the internal energy is a property of a particular state of the system when it is in unchanging thermodynamic equilibrium. Thus the term "heat energy" for means "that amount of energy added as the result of heating" rather than referring to a particular form of energy. Likewise, the term "work energy" for means "that amount of energy lost as the 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 the result of work being performed on or by the system.
Entropy is a function of the state of a system which tells of limitations of the possibility of conversion of heat into work.
For a simple compressible system, the work performed by the system may be written:
where is the pressure and is a small change in the volume of the system, each of which are system variables. In the fictive case in which the process is idealized and infinitely slow, so as to be called quasistatic, and regarded as reversible, the heat being transferred from a source with temperature infinitesimally above the system temperature, then the heat energy may be written
where is the temperature and is a small change in the entropy of the system. Temperature and entropy are variables of 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:^{[22]}
where is the added mass and is the internal energy per unit mass of the added mass, measured in the surroundings before the process.
The conservation of energy is a common feature in many physical theories. From a mathematical point of view it is understood as a consequence of Noether's theorem, developed by Emmy Noether in 1915 and first published in 1918. The theorem states 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 of time; energy conservation is implied by the empirical fact that the laws of physics do not change with time itself. Philosophically this can be stated as "nothing depends on time per se". In other words, if the physical system is invariant under the continuous symmetry of time translation then its energy (which is canonical conjugate quantity to time) is conserved. Conversely, systems which are not invariant under shifts in time (an example, systems with time dependent potential energy) do not exhibit conservation of energy – unless we consider them to exchange energy with another, external system so that the theory of the enlarged system becomes time invariant again. Conservation of energy for finite systems is valid in such physical theories as special relativity and quantum theory (including QED) in the flat spacetime.
With the discovery of special relativity by Henri Poincaré and Albert Einstein, energy was proposed to be one component of an energymomentum 4vector. Each of the four components (one of energy and three of momentum) of this vector is separately conserved across time, in any closed system, as seen from any given inertial reference frame. Also conserved is the vector length (Minkowski norm), which is the rest mass for single particles, and the invariant mass for systems of particles (where momenta and energy are separately summed before the length is calculated—see the article on invariant mass).
The relativistic energy of a single massive particle contains a term related to its rest mass in addition to its kinetic energy of motion. In the limit of zero kinetic energy (or equivalently in the rest frame) of a massive particle, or else in the center of momentum frame for objects or systems which retain kinetic energy, the total energy of particle or object (including internal kinetic energy in systems) is related to its rest mass or its invariant mass via the famous equation .
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.
In general relativity, energy–momentum conservation is not welldefined except in certain special cases. Energymomentum 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 such as the Friedmann–Lemaître–Robertson–Walker metric do not satisfy these constraints and energy conservation is not well defined.^{[23]} The theory of general relativity leaves open the question of whether there is a conservation of energy for the entire universe.
In quantum mechanics, energy of a quantum system is described by a selfadjoint (or Hermitian) operator called the Hamiltonian, which acts on the Hilbert space (or a space of wave functions) of the system. If the Hamiltonian is a timeindependent operator, emergence probability of the measurement result does not change in time over the evolution of the system. Thus the expectation value of energy is also time independent. The local energy conservation in quantum field theory is ensured by the quantum Noether's theorem for energymomentum 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 positionmomentum uncertainty principle, and merely holds in specific cases (see Uncertainty principle). Energy at each fixed time can in principle be exactly measured without any tradeoff in precision forced by the timeenergy uncertainty relations. Thus the conservation of energy in time is a well defined concept even in quantum mechanics.