# Conservation Law

In physics, the law of conservation is the formulation of the observed fact that the value of a quantity , called the conservation quantity , does not change in certain physical processes. In a closed system , conserved quantities do not change.

The best known law of conservation is that of energy . Colloquially it reads: Whatever energy you put in at the front comes out again at the back; no energy is lost and none arises from nothing. The most general conservation laws apply to the quantities energy, momentum , angular momentum , electrical charge , baryon number and lepton number . For certain classes of physical processes (see basic forces of physics ) further conservation laws are added.

According to Noether's theorem , every continuous symmetry of action results in a conservation law, and conversely, every conservation law requires a continuous symmetry of action.

## States of a system

Conservation quantities can be calculated from the quantities that describe the state of a system, for example the locations and speeds of particles. While the state variables change over time with movement, the conserved variables calculated from them remain constant over time. So the energy of a particle of mass depends on the potential${\ displaystyle m}$${\ displaystyle V (x)}$

${\ displaystyle E (x, v) = {\ frac {1} {2}} \, m \, v ^ {2} + V (x)}$

on its speed and location . Even if both speed and location change over time , the energy remains ${\ displaystyle v (t)}$${\ displaystyle x (t)}$${\ displaystyle t}$

${\ displaystyle {\ frac {1} {2}} \, m \, v ^ {2} (t) + V (x (t)) = {\ frac {1} {2}} \, m \, v ^ {2} (0) + V (x (0))}$

unchanged over time.

Conservation quantities limit the conceivable movement of the physical system. For example, from the conservation of energy and momentum in Compton scattering , it follows how the energy of the scattered photon is related to its scattering angle and (depending on the scattering angle of the photon, which is not specified) with which energy and in which direction the originally resting electron follows the scatter moves.

Many conserved quantities are additive, that is, in two- and multi-particle systems the value of the additive conserved quantity is the sum of the individual values. The total pulse, for example, is the sum of the individual pulses. This apparent self-evident only applies to particles that do not or no longer interact with one another. During the interaction, fields can absorb energy and momentum and transfer it to other particles.

## Examples

• Conservation of energy : The total energy remains constant (associated symmetry: the physical processes do not depend on the choice of the time zero point, homogeneity of time).
• Conservation of momentum : The vector sum of all momentum remains constant (corresponding symmetry: the physical processes do not depend on the choice of the origin, homogeneity of the space).
• Conservation of angular momentum : The sum of all angular momentum remains constant (associated symmetry: the physical processes do not depend on the choice of reference directions, isotropy of space).
• Preservation of charge : The (electrical, color) charge remains constant (associated symmetry: the phase of the charged particle can be chosen at will). If a charge in a region is given as the integral of a charge density over this region, then it is a conservation quantity if it, together with a current density, gives the continuity equation${\ displaystyle \ rho (t, {\ vec {x}})}$ ${\ displaystyle {\ vec {j}} (t, {\ vec {x}})}$
${\ displaystyle {\ frac {\ partial} {\ partial t}} \ rho + \ nabla \ cdot {\ vec {j}} = 0}$
Fulfills. Then the charge in the area can change over time only because currents flow through the surface.
The conservation of the number of baryons and leptons cannot be traced back to a known symmetry requirement, but results from the interactions occurring in the standard model of elementary particle physics . In proposals for a Grand Unified Theory that go beyond the current Standard Model, the violation of both conservation laws is predicted, e.g. B. by the decay of the proton into leptons. Such a symmetry, which leads to the maintenance of the difference between the number of baryons and the number of leptons , is an additional one that is compatible with the standard model . Despite an intensive search, proton decay has not been observed to date .${\ displaystyle U (1)}$
• Conservation of mass : Conservation of mass is not a law of conservation in the strict sense of the word. It applies with high accuracy in classical physics (and in chemistry for all types of chemical reactions), but is only a limiting case of energy conservation, since mass is a form of energy. As soon as particles can transform into one another, the conservation of mass is measurably violated. For example, in the case of radioactive decay of atomic nuclei, the mass of the mother particle is greater than the sum of the masses of the daughter particles. There is no associated symmetry for the conservation of mass in fluid mechanics, since the equations of fluid mechanics do not originate from an operating principle.

## Conservation quantities and integrability

If the physical system under consideration has as many conserved quantities as there are degrees of freedom, the development over time can be specified by integrals . One speaks of an integrable system if they are in involution, i.e. the Poisson brackets${\ displaystyle E_ {i}}$${\ displaystyle E_ {i}}$

${\ displaystyle \ left \ {E_ {i}, E_ {j} \ right \} = 0}$

for everyone , becomes zero. ${\ displaystyle i}$${\ displaystyle j}$

This corresponds to the interchangeability of the symmetry transformations belonging to the conservation quantities when they are executed one after the other.

In the simplest case, energy-conserving movement of one degree of freedom , one solves the energy law ${\ displaystyle x}$

${\ displaystyle E = {\ frac {1} {2}} \, m \, v ^ {2} + V (x)}$

according to the speed

${\ displaystyle v = {\ frac {\ mathrm {d} x} {\ mathrm {d} t}} = \ pm {\ sqrt {{\ frac {2} {m}} \, (EV)}} \ ,.}$

The derivative of the inverse function , which indicates the time at which the particle passes through the location , is the reciprocal, ${\ displaystyle t (x)}$${\ displaystyle x}$

${\ displaystyle {\ frac {\ mathrm {d} t} {\ mathrm {d} x}} = \ pm \ left ({\ sqrt {{\ frac {2} {m}} (EV)}} \ right ) ^ {- 1} \ ,.}$

If this equation is integrated over from a lower limit to a freely selectable upper limit , the result is ${\ displaystyle x}$${\ displaystyle x_ {0}}$${\ displaystyle {\ bar {x}}}$

${\ displaystyle \ int _ {x_ {0}} ^ {\ bar {x}} \ mathrm {d} x {\ frac {\ mathrm {d} t} {\ mathrm {d} x}} = t ({ \ bar {x}}) - t (x_ {0}) = \ pm \ int _ {x_ {0}} ^ {\ bar {x}} \ mathrm {d} x \ left ({\ sqrt {{\ frac {2} {m}} (EV (x))}} \ right) ^ {- 1} \ ,.}$

So the inverse function is fixed as a function of the upper limit of an integral over the given function . The start time and the initial energy can be freely selected. ${\ displaystyle t ({\ bar {x}})}$${\ displaystyle \ textstyle \ left ({\ sqrt {{\ frac {2} {m}} (EV (x))}} \ right) ^ {- 1}}$${\ displaystyle t (x_ {0})}$${\ displaystyle E}$

## Conservation laws in the 19th century

Conservation laws are part of modern physics in the 20th century. At the end of the 19th century, the great German encyclopedias listed three subject areas under “Conservation”: With “Conservation of energy” they referred directly to the force , with “Conservation of surfaces” on the central movement , in which “the guide beam is the same at the same time Describes surface areas “(called today: conservation of angular momentum ). The only conservation law that took up more space as such was one that the authors declared as difficult, that of the "conservation of the world":

Preservation of the world , in church doctrine the act of divine will, through which the finished universe continues both according to its matter and its form. The prerequisite for the preservation of the world is creation, while the doctrine of the preservation of the world is initially followed by that of the world government directed towards humanity. The difficulty of the term lies in the ratio of those effects which are caused by the so-called. Second causes, the forces of nature and human beings, go out to the omnipotence of the first and last cause, God. "

- Meyers Konversationslexikon, Verlag des Bibliographisches Institut, Leipzig and Vienna, fourth edition, 1885–1892, pp. 779 f

## Individual evidence

1. Eugene J. Saletan and Alan H. Cromer: Theoretical Mechanics . John Wiley & Sons, 1971, ISBN 0-471-74986-9 , pp. 83-86 (English).
2. Mattew D. Schwartz: Quantum Field Theory and the standard model . Cambridge University Press, Cambridge 2014, ISBN 978-1-107-03473-0 , pp. 631-636 (English).