# Conservation of energy law

The law of conservation of energy expresses the fact that energy is a conservation quantity , i.e. that the total energy of a closed system does not change with time. Energy can be converted between different forms of energy , for example from kinetic energy to thermal energy. It can also be transported out of or into a system, but it is not possible to generate or destroy energy. Energy conservation is an important principle in all natural sciences .

The law of conservation of energy can theoretically be derived with the help of Noether's theorem from the assumption that the laws of physics that apply to the system do not depend on time.

## Colloquial language

In the physical sense of the law of conservation of energy, a “loss” of energy is not possible. Nevertheless, colloquial terms are used as “energy consumption”, “energy waste”, “energy saving” and “energy loss”. This is justifiable, because the earth is not a closed system and, moreover, humans and other living beings can only use energy in certain forms; The terms mentioned describe the transition of energy from technically easily usable or biologically usable forms of energy ( exergy ) to forms that are poorly or unusable ( anergy ). It is equally impossible to generate energy . The colloquial "energy generation" means the conversion of existing energy into a form that can be used by humans, usually electrical energy .

For most commonly used today types of energy conversion are energy sources with low or specific entropy converted into shapes with higher entropy. For example, a motor vehicle converts chemical energy that originally comes from petroleum or rapeseed oil into kinetic energy and thermal energy . Since petroleum cannot be regenerated, this can be seen as a loss of energy in the sense that this particular form of low-entropy chemical energy is lost for future generations or for other purposes.

With each of the types of conversion that are common today, only part of the energy present in the energy carrier is converted into usable energy. From energy conservation can therefore speaks when the efficiency of the energy conversion process or a device increases due to technological progress, so that less raw material provides more usable energy or the relevant purpose is achieved with less energy.

## history

As far as is known today, the law of conservation of energy was first formulated by the Heilbronn doctor Julius Robert von Mayer (1814–1878). In 1842 he proved by means of appropriate tests that a certain kinetic energy always produces the same amount of heat when it is completely converted into heat. He also determined the value of this "mechanical heat equivalent". Independently of Mayer, this was also done in 1843 by James Prescott Joule - whose work was far better known at the time - and other physicists and engineers such as Ludwig August Colding in Denmark (also in 1843). The law of conservation of energy was finally formulated in 1847 by Hermann von Helmholtz . He reported in Berlin on July 23, 1847 about the "constancy of power" and underpinned the law of conservation of energy.

Stephen Brush lists other scientists who more or less generally formulated a law of conservation of energy in the 19th century : Karl Friedrich Mohr , Sadi Carnot , Marc Seguin , Karl Holtzmann , Gustav Adolphe Hirn , William Robert Grove , Justus von Liebig , Michael Faraday .

The law of conservation of energy has not always been undisputed in the history of physics. The most famous example is Niels Bohr , who on several occasions only advocated statistical (averaged) conservation of energy in quantum processes, for example in the so-called BKS theory in 1924 with John C. Slater and Hendrik Anthony Kramers . This should bring the older quantum theory in line with the classical concept of electromagnetic fields. A little later this theory was refuted by experiments by Compton and also by Hans Geiger and Walther Bothe and the validity of the law of conservation of energy was also confirmed at the quantum level. Bohr later also tried to explain some of the initially puzzling quantum phenomena with only statistical validity of the law of conservation of energy, for example with beta decay ; the “missing” energy of the observed decay products there was explained by Wolfgang Pauli through the postulate of a new, only weakly interacting particle, the neutrino .

Today, the law of conservation of energy is considered established and is even often used to define energy.

## Conservation of energy law in Newtonian mechanics

Any two paths in a conservative force field

In the motion of a point mass in a conservative force field the sum of remains of kinetic energy and potential energy , the total energy received. The force is the negative gradient of the potential energy (often simply referred to as potential in jargon) ${\ displaystyle T}$ ${\ displaystyle V,}$${\ displaystyle E = T + V,}$

${\ displaystyle \ mathbf {F} = - \ nabla V}$.

If a point mass moves over time in such a force field on any path from a starting point to a destination , the path is irrelevant for the work that is done on the point mass. Regardless of the path, the work done is the difference between the potential energies at the start and finish. ${\ displaystyle t}$ ${\ displaystyle \ mathbf {x} (t)}$

For a point mass with constant mass in a potential , Newton's equations of motion apply in the following form: ${\ displaystyle m}$${\ displaystyle V}$

${\ displaystyle m {\ ddot {\ mathbf {x}}} = \ mathbf {F} = - \ nabla V}$

The scalar product with the velocity gives on the left side of the equation: ${\ displaystyle {\ dot {\ mathbf {x}}} (t)}$

${\ displaystyle m {\ ddot {\ mathbf {x}}} \ cdot {\ dot {\ mathbf {x}}} = {\ frac {\ mathrm {d}} {\ mathrm {d} t}} \ left ({\ frac {m} {2}} {\ dot {\ mathbf {x}}} \ cdot {\ dot {\ mathbf {x}}} \ right) = {\ frac {\ mathrm {d}} { \ mathrm {d} t}} \ left ({\ frac {m} {2}} | {\ dot {\ mathbf {x}}} | ^ {2} \ right) = {\ dot {T}}}$

Here is the time-derived kinetic energy that is changed by the work done by the force on the point mass. Using the chain rule results on the right side: ${\ displaystyle {\ dot {T}}}$

{\ displaystyle {\ begin {aligned} \ mathbf {F} \ cdot {\ dot {\ mathbf {x}}} (t) & = - \ nabla V (\ mathbf {x}) \ cdot {\ dot {\ mathbf {x}}} (t) \\ & = - \ sum _ {i = 1} ^ {3} {\ frac {\ partial V (\ mathbf {x})} {\ partial x_ {i}}} \ cdot {\ frac {\ mathrm {d} x_ {i} (t)} {\ mathrm {d} t}} \\ & = - {\ frac {\ mathrm {d}} {\ mathrm {d} t }} V (\ mathbf {x} (t)) = - {\ dot {V}} \ end {aligned}}}

An integration over time now provides the required work along any (piece-wise continuously differentiable) physical path with the respective potential energy at the start and at the destination: ${\ displaystyle V_ {1}}$${\ displaystyle V_ {2}}$

{\ displaystyle {\ begin {aligned} \ int _ {t_ {1}} ^ {t_ {2}} m {\ ddot {\ mathbf {x}}} (t) \ cdot {\ dot {\ mathbf {x }}} (t) \; \ mathrm {d} t & = \ int _ {t_ {1}} ^ {t_ {2}} {\ dot {T}} \, \ mathrm {d} t = T_ {2 } -T_ {1} \\ = \ int _ {t_ {1}} ^ {t_ {2}} \ mathbf {F} \ cdot {\ dot {\ mathbf {x}}} (t) \ \ mathrm { d} t & = - \ int _ {t_ {1}} ^ {t_ {2}} {\ dot {V}} \ \ mathrm {d} t = - \ int _ {V_ {1}} ^ {V_ { 2}} \ mathrm {d} V = -V_ {2} + V_ {1} \\\ rightarrow \ quad T_ {2} -T_ {1} & = - V_ {2} + V_ {1} \ end { aligned}}}

If you rearrange the terms, you get:

${\ displaystyle T_ {1} + V_ {1} = T_ {2} + V_ {2}}$

The sum of the kinetic and potential energy is still the same after a shift in the point mass. This is the law of conservation of energy for point masses.

If, for example, with a pendulum , the friction can be neglected, the sum of potential and kinetic energy does not change over time. If you deflect the pendulum, it swings between two reversal points and reaches its highest speed at the location of the potential minimum. At the reversal points, the kinetic energy is zero and the potential energy is maximum. Regardless of the position of the pendulum, the sum of the kinetic and potential energy has the value given by the initial deflection.

A force acting on a real body not only leads to an acceleration of its center of gravity, but also to a more or less pronounced deformation. In the hyper-elasticity there is a potential that the strain energy , the time derivative of the deformation performance is: ${\ displaystyle W}$${\ displaystyle L}$

${\ displaystyle L: = {\ dot {W}}}$

Deformation work is completely converted into deformation energy without dissipation and that is independent of the path. The deformation work performed is always the difference between the deformation energy at the start and finish. The power of the external forces acting on a hyperelastic body is divided into an acceleration (also an angular acceleration , which also contributes to the kinetic energy) and a (reversible) deformation:

{\ displaystyle {\ begin {aligned} - \ int _ {t_ {1}} ^ {t_ {2}} {\ dot {V}} \ \ mathrm {d} t = & \ int _ {t_ {1} } ^ {t_ {2}} {\ dot {T}} \ \ mathrm {d} t + \ int _ {t_ {1}} ^ {t_ {2}} {\ dot {W}} \ \ mathrm {d } t \\\ rightarrow \ quad -V_ {2} + V_ {1} = & \ T_ {2} -T_ {1} + W_ {2} -W_ {1} \ end {aligned}}}

In this system, the sum of kinetic, potential and deformation energy is constant over time:

${\ displaystyle T_ {1} + V_ {1} + W_ {1} = T_ {2} + V_ {2} + W_ {2}}$

This is the law of conservation of the mechanical energy of deformable, hyperelastic bodies in a conservative force field.

## Conservation of energy law in thermodynamics

Every thermodynamic system has a certain “store” of energy. This is made up of an outer part and an inner part ( inner energy ). The sum of the two components gives the total energy of a thermodynamic system, whereby in chemical thermodynamics the change in the external component is set to zero ( ). With this assumption one arrives at the first law of thermodynamics: ${\ displaystyle E _ {\ text {a}}}$${\ displaystyle E _ {\ text {i}}}$${\ displaystyle \ mathrm {d} E _ {\ mathrm {a}} = 0}$

“The internal energy is a property of the material components of a system and cannot be created or destroyed. The internal energy is a state variable . "

For closed systems it is therefore true that the internal energy is constant and consequently its change is zero. For closed systems, the first law of thermodynamics reads :

${\ displaystyle \ mathrm {d} U = \ delta Q + \ delta W}$

with internal energy , warmth and work . ${\ displaystyle U}$ ${\ displaystyle Q}$ ${\ displaystyle W}$

## Conservation of energy law in electrodynamics

Electromagnetic fields are often only a subsystem that is coupled to other systems, for example charged particles with a certain charge, mass and speed. The energy balance in electrodynamics , i.e. the energy flow in fields and the exchange with other subsystems, is described by Poynting's theorem .

## Conservation of energy law in relativity theory

A body of mass that moves with speed has energy in the special theory of relativity${\ displaystyle m}$${\ displaystyle v}$

${\ displaystyle E (v) = {\ frac {m \ c ^ {2}} {\ sqrt {1- \ left ({\ frac {v} {c}} \ right) ^ {2}}}}}$,

where is the speed of light. In rest he has the energy of rest${\ displaystyle c}$

${\ displaystyle E _ {\ text {calm}} = m \ c ^ {2}}$.

For small velocities ( , Taylor expansion in ) the energy is approximately equal to the sum of the rest energy and the kinetic energy according to Newtonian mechanics ${\ displaystyle v \ ll c}$${\ displaystyle \ left ({\ frac {v} {c}} \ right) ^ {2}}$

${\ displaystyle E \ approx m \ c ^ {2} + {\ frac {1} {2}} \ m \ v ^ {2}}$.

In the case of high-energy particles, this approximation is measurably incorrect. Only the sum of the relativistic energies remains in particle reactions.

The consideration of the universe by means of the general theory of relativity shows that the law of conservation of energy is not applicable to the universe as a whole. In particular, gravitational energy cannot always be clearly defined in a way that applies to the universe as a whole. The total energy of the universe is therefore neither preserved nor lost - it cannot be defined.

## Conservation of energy law in quantum mechanics

The energy of a quantum mechanical state is retained if the Hamilton operator does not depend on time. Quantum mechanical states that change measurably over time are not energy eigenstates ; in them, however, at least the expected value of the energy is retained.

## Energy balance

If a system can exchange energy with another system, for example through radiation or heat conduction , then one speaks of an energetically open system . Instead of energy conservation, the energy balance then applies: The energy that flows into a system minus the energy that leaves it is the change in the energy of the system and must be provided by the environment or absorbed by it. By looking at the energy flows in the system or between the system and its environment, one can draw conclusions about processes within the system, even if they cannot be observed themselves.

The energy of a system cannot be measured directly: If one disregards the equivalence of mass and energy , only differences in energy have a measurable effect .

The energy balance means more precisely, must of its environment in order to change the power of an open system work will be done to the system or transfer heat. In relation to a time interval, this means: The change over time in the total energy of an open system is equal to the power (including heat output) that is brought into or taken from the system by its surroundings. The law of conservation of energy is the special case of the energy balance in which this work or performance of the environment disappears and the energy content of the now closed system remains unchanged.

Possible interactions with the environment include:

## Noether theorem

In Lagrangian mechanics, energy conservation results from Noether's theorem if the effect is invariant under time shifts.

## literature

• Max Planck: The principle of the conservation of energy . BG Teubner, Leipzig 1887, p. 1–247 ( archive.org [PDF; 14.0 MB ]).

## Individual evidence

1. See e.g. B. Feynman lectures on physics. Volume 2: Electromagnetism and the Structure of Matter. 3rd edition, 2001, pp. 147, 162, 198.
2. Hermann von Helmholtz. ( Memento of the original from January 21, 2012 in the Internet Archive ) Info: The  archive link was automatically inserted and not yet checked. Please check the original and archive link according to the instructions and then remove this notice. In: Potsdam-Wiki.de. Retrieved July 23, 2011.
3. ^ Stephen Brush, Kinetic Theory, Pergamon Press, Volume 1, 1966, p. 20
4. Bohr, Kramers, Slater: The quantum theory of radiation. In: Philosophical Magazine. Vol. 47, 1924, pp. 785-802. German in: Zeitschr. for physics. Vol. 24, 1924, pp. 69-87.
5. TM Davis: Is the Universe Losing Energy? In: Spectrum of Science . November 2010, ISSN  0170-2971 , p. 23-29 .