Conservation law

Inphysics,aconservation lawstates that a particular measurable property of an isolatedphysical systemdoes not change as the system evolves over time. Exact conservation laws includeconservation of mass-energy,conservation of linear momentum,conservation of angular momentum,andconservation of electric charge.There are also many approximate conservation laws, which apply to such quantities asmass,parity,lepton number,baryon number,strangeness,hypercharge,etc. These quantities are conserved in certain classes of physics processes, but not in all.

A local conservation law is usually expressed mathematically as acontinuity equation,apartial differential equationwhich gives a relation between the amount of the quantity and the "transport" of that quantity. It states that the amount of the conserved quantity at a point or within a volume can only change by the amount of the quantity which flows in or out of the volume.

FromNoether's theorem,every differentiablesymmetryleads to a conservation law.^[1]^[2]^[3]Other conserved quantities can exist as well.

Conservation laws as fundamental laws of nature

Conservation laws are fundamental to our understanding of the physical world, in that they describe which processes can or cannot occur in nature. For example, the conservation law of energy states that the total quantity of energy in an isolated system does not change, though it may change form. In general, the total quantity of the property governed by that law remains unchanged during physical processes. With respect to classical physics, conservation laws include conservation of energy, mass (or matter), linear momentum, angular momentum, and electric charge. With respect to particle physics, particles cannot be created or destroyed except in pairs, where one is ordinary and the other is an antiparticle. With respect to symmetries and invariance principles, three special conservation laws have been described, associated with inversion or reversal of space, time, and charge.

Conservation laws are considered to be fundamentallawsof nature, with broad application in physics, as well as in other fields such as chemistry, biology, geology, and engineering.

Most conservation laws are exact, or absolute, in the sense that they apply to all possible processes. Some conservation laws are partial, in that they hold for some processes but not for others.

One particularly important result concerning conservation laws isNoether's theorem,which states that there is a one-to-one correspondence between each one of them and a differentiablesymmetryof theUniverse.For example, the conservation of energy follows from theuniformity of timeand theconservation of angular momentumarises from theisotropyofspace,^[4]^[5]^[6]i.e. because there is no preferred direction of space in theUniverse.

Exact laws

A partial listing of physical conservation equationsdue to symmetrythat are said to beexact laws,or more preciselyhave never been proven to be violated:

Conservation law	Respective Noether symmetryinvariance		Number of independent parameters (i.e. dimension of phase space)
Conservation of mass-energyE	Time-translation invariance	Poincaré invariance	1	translation of time alongt-axis
Conservation of linear momentump	Space-translation invariance		3	translation of space alongx,y,zaxes
Conservation of angular momentumL=r×p	Rotation invariance		3	rotation of space aboutx,y,zaxes
Conservation of boost 3-vectorN=tp-Er	Lorentz-boost invariance		3	Lorentz-boost of space-time alongx,y,zaxes
Conservation of electric charge	U(1)_QGauge invariance		1	translation of electrodynamic scalar potential field alongV-axis (in phase space)
Conservation ofcolor charge	SU(3)_CGauge invariance		3	translation of chromodynamic potential field alongr,g,b-axes (in phase space)
Conservation ofweak isospin	SU(2)_LGauge invariance		1	translation of weak potential field along axis in phase space
Conservation of the difference between baryon and lepton numbersB - L	U(1)_B-LGauge invariance		1

Another exact symmetry isCPT symmetry,the simultaneous inversion of space and time coordinates, together with swapping all particles with their antiparticles; however being a discrete symmetry Noether's theorem does not apply to it. Accordingly, the conserved quantity, CPT parity, can usually not be meaningfully calculated or determined.

Approximate laws

There are alsoapproximateconservation laws. These are approximately true in particular situations, such as low speeds, short time scales, or certain interactions.

Conservation of mechanical energy
Conservation of mass(approximately true fornonrelativistic speeds)
Conservation ofbaryon number(Seechiral anomalyandsphaleron)
Conservation oflepton number(In theStandard Model)
Conservation offlavor(violated by theweak interaction)
Conservation ofstrangeness(violated by theweak interaction)
Conservation ofspace-parity(violated by theweak interaction)
Conservation ofcharge-parity(violated by theweak interaction)
Conservation oftime-parity(violated by theweak interaction)
Conservation ofCP parity(violated by theweak interaction); in the Standard Model, this is equivalent to conservation oftime-parity.

Global and local conservation laws

The total amount of some conserved quantity in the universe could remain unchanged if an equal amount were to appear at one pointAand simultaneously disappear from another separate pointB.For example, an amount of energy could appear on Earth without changing the total amount in the Universe if the same amount of energy were to disappear from some other region of the Universe. This weak form of "global" conservation is really not a conservation law because it is notLorentz invariant,so phenomena like the above do not occur in nature.^[7]^[8]Due tospecial relativity,if the appearance of the energy atAand disappearance of the energy atBare simultaneous in oneinertial reference frame,theywill not be simultaneousin other inertial reference frames moving with respect to the first. In a moving frame one will occur before the other; either the energy atAwill appearbeforeorafterthe energy atBdisappears. In both cases, during the interval energy will not be conserved.

A stronger form of conservation law requires that, for the amount of a conserved quantity at a point to change, there must be a flow, orfluxof the quantity into or out of the point. For example, the amount ofelectric chargeat a point is never found to change without anelectric currentinto or out of the point that carries the difference in charge. Since it only involves continuouslocalchanges, this stronger type of conservation law isLorentz invariant;a quantity conserved in one reference frame is conserved in all moving reference frames.^[7]^[8]This is called alocal conservationlaw.^[7]^[8]Local conservation also implies global conservation; that the total amount of the conserved quantity in the Universe remains constant. All of the conservation laws listed above are local conservation laws. A local conservation law is expressed mathematically by acontinuity equation,which states that the change in the quantity in a volume is equal to the total net "flux" of the quantity through the surface of the volume. The following sections discuss continuity equations in general.

Differential forms

Incontinuum mechanics,the most general form of an exact conservation law is given by acontinuity equation.For example, conservation of electric charge $q$ is ${\frac {\partial \rho }{\partial t}}=-\nabla \cdot \mathbf {j} \,$ where $\nabla\cdot$ is thedivergenceoperator, $ρ$ is the density of $q$ (amount per unit volume), $j$ is the flux of $q$ (amount crossing a unit area in unit time), and $t$ is time.

If we assume that the motionuof the charge is a continuous function of position and time, then ${\begin{aligned}\mathbf {j} &=\rho \mathbf {u} \\{\frac {\partial \rho }{\partial t}}&=-\nabla \cdot (\rho \mathbf {u} )\,.\end{aligned}}$

In one space dimension this can be put into the form of a homogeneous first-orderquasilinear hyperbolic equation:^[9]^: 43 $y_{t}+A(y)y_{x}=0$ where the dependent variable $y$ is called thedensityof aconserved quantity,and $A (y)$ is called thecurrent Jacobian,and thesubscript notation for partial derivativeshas been employed. The more general inhomogeneous case: $y_{t}+A(y)y_{x}=s$ is not a conservation equation but the general kind ofbalance equationdescribing adissipative system.The dependent variable $y$ is called anonconserved quantity,and the inhomogeneous term $s (y, x, t)$ is the-source,ordissipation.For example, balance equations of this kind are the momentum and energyNavier-Stokes equations,or theentropy balancefor a generalisolated system.

In theone-dimensional spacea conservation equation is a first-orderquasilinear hyperbolic equationthat can be put into theadvectionform: $y_{t}+a(y)y_{x}=0$ where the dependent variable $y (x, t)$ is called the density of theconserved(scalar) quantity, and $a (y)$ is called thecurrent coefficient,usually corresponding to thepartial derivativein the conserved quantity of acurrent densityof the conserved quantity $j (y)$ :^[9]^: 43 $a(y)=j_{y}(y)$

In this case since thechain ruleapplies: $j_{x}=j_{y}(y)y_{x}=a(y)y_{x}$ the conservation equation can be put into the current density form: $y_{t}+j_{x}(y)=0$

In aspace with more than one dimensionthe former definition can be extended to an equation that can be put into the form: $y_{t}+\mathbf {a} (y)\cdot \nabla y=0$

where theconserved quantityis $y (r, t)$ , $\cdot$ denotes thescalar product, $\nabla$ is thenablaoperator, here indicating agradient,and $a (y)$ is a vector of current coefficients, analogously corresponding to thedivergenceof a vector current density associated to the conserved quantity $j (y)$ : $y_{t}+\nabla \cdot \mathbf {j} (y)=0$

This is the case for thecontinuity equation: $\rho _{t}+\nabla \cdot (\rho \mathbf {u} )=0$

Here the conserved quantity is themass,withdensity $ρ (r, t)$ and current density $ρ u$ ,identical to themomentum density,while $u (r, t)$ is theflow velocity.

In thegeneral casea conservation equation can be also a system of this kind of equations (avector equation) in the form:^[9]^: 43 $\mathbf {y} _{t}+\mathbf {A} (\mathbf {y} )\cdot \nabla \mathbf {y} =\mathbf {0}$ where $y$ is called theconserved(vector) quantity, $\nabla y$ is itsgradient, $0$ is thezero vector,and $A (y)$ is called theJacobianof the current density. In fact as in the former scalar case, also in the vector caseA(y) usually corresponding to the Jacobian of acurrent density matrix $J (y)$ : $\mathbf {A} (\mathbf {y} )=\mathbf {J} _{\mathbf {y} }(\mathbf {y} )$ and the conservation equation can be put into the form: $\mathbf {y} _{t}+\nabla \cdot \mathbf {J} (\mathbf {y} )=\mathbf {0}$

For example, this the case for Euler equations (fluid dynamics). In the simple incompressible case they are: $\nabla \cdot \mathbf {u} =0\,,\qquad {\frac {\partial \mathbf {u} }{\partial t}}+\mathbf {u} \cdot \nabla \mathbf {u} +\nabla s=\mathbf {0},$

where:

$u$ is theflow velocity vector,with components in a N-dimensional space $u 1, u 2,..., u N$ ,
$s$ is the specificpressure(pressure per unitdensity) giving thesource term,

It can be shown that the conserved (vector) quantity and the current density matrix for these equations are respectively:

${\mathbf {y} }={\begin{pmatrix}1\\\mathbf {u} \end{pmatrix}};\qquad {\mathbf {J} }={\begin{pmatrix}\mathbf {u} \\\mathbf {u} \otimes \mathbf {u} +s\mathbf {I} \end{pmatrix}};\qquad$

where $\otimes$ denotes theouter product.

Integral and weak forms

Conservation equations can usually also be expressed in integral form: the advantage of the latter is substantially that it requires less smoothness of the solution, which paves the way toweak form,extending the class of admissible solutions to include discontinuous solutions.^[9]^: 62–63By integrating in any space-time domain the current density form in 1-D space: $y_{t}+j_{x}(y)=0$ and by usingGreen's theorem,the integral form is: $\int _{-\infty }^{\infty }y\,dx+\int _{0}^{\infty }j(y)\,dt=0$

In a similar fashion, for the scalar multidimensional space, the integral form is: $\oint \left[y\,d^{N}r+j(y)\,dt\right]=0$ where the line integration is performed along the boundary of the domain, in an anticlockwise manner.^[9]^: 62–63

Moreover, by defining atest functionφ(r,t) continuously differentiable both in time and space with compact support, theweak formcan be obtained pivoting on theinitial condition.In 1-D space it is: $\int _{0}^{\infty }\int _{-\infty }^{\infty }\phi _{t}y+\phi _{x}j(y)\,dx\,dt=-\int _{-\infty }^{\infty }\phi (x,0)y(x,0)\,dx$

In the weak form all the partial derivatives of the density and current density have been passed on to the test function, which with the former hypothesis is sufficiently smooth to admit these derivatives.^[9]^: 62–63

Notes

^Ibragimov, N. H. CRC HANDBOOK OF LIE GROUP ANALYSIS OF DIFFERENTIAL EQUATIONS VOLUME 1 -SYMMETRIES EXACT SOLUTIONS AND CONSERVATION LAWS. (CRC Press, 2023)
^Kosmann-Schwarzbach, Y. in The Philosophy and Physics of Noether’s Theorems: A Centenary Volume 4-24 (Cambridge University Press, 2022)
^Rao, A. K., Tripathi, A., Chauhan, B. & Malik, R. P. Noether Theorem and Nilpotency Property of the (Anti-)BRST Charges in the BRST Formalism: A Brief Review. Universe 8 (2022).https://doi.org/10.3390/universe8110566
^Ibragimov, N. H. CRC HANDBOOK OF LIE GROUP ANALYSIS OF DIFFERENTIAL EQUATIONS VOLUME 1 -SYMMETRIES EXACT SOLUTIONS AND CONSERVATION LAWS. (CRC Press, 2023).
^Kosmann-Schwarzbach, Y. in The Philosophy and Physics of Noether’s Theorems: A Centenary Volume 4-24 (Cambridge University Press, 2022).
^Rao, A. K., Tripathi, A., Chauhan, B. & Malik, R. P. Noether Theorem and Nilpotency Property of the (Anti-)BRST Charges in the BRST Formalism: A Brief Review. Universe 8 (2022).https://doi.org/10.3390/universe8110566
^^a ^b ^cAitchison, Ian J. R.; Hey, Anthony J.G. (2012).Gauge Theories in Particle Physics: A Practical Introduction: From Relativistic Quantum Mechanics to QED, Fourth Edition, Vol. 1.CRC Press. p. 43.ISBN 978-1466512993.Archivedfrom the original on 2018-05-04.
^^a ^b ^cWill, Clifford M. (1993).Theory and Experiment in Gravitational Physics.Cambridge Univ. Press. p. 105.ISBN 978-0521439732.Archivedfrom the original on 2017-02-20.
^^a ^b ^c ^d ^e ^fToro, E.F. (1999). "Chapter 2. Notions on Hyperbolic PDEs".Riemann Solvers and Numerical Methods for Fluid Dynamics.Springer-Verlag.ISBN 978-3-540-65966-2.

References

Philipson, Schuster,Modeling by Nonlinear Differential Equations: Dissipative and Conservative Processes,World Scientific Publishing Company 2009.
Victor J. Stenger,2000.Timeless Reality: Symmetry, Simplicity, and Multiple Universes.Buffalo NY: Prometheus Books. Chpt. 12 is a gentle introduction to symmetry, invariance, and conservation laws.
E. Godlewski and P.A. Raviart, Hyperbolic systems of conservation laws, Ellipses, 1991.

External links

Media related toConservation lawsat Wikimedia Commons
Conservation Laws– Ch. 11–15 in an online textbook

[1] Ibragimov, N. H. CRC HANDBOOK OF LIE GROUP ANALYSIS OF DIFFERENTIAL EQUATIONS VOLUME 1 -SYMMETRIES EXACT SOLUTIONS AND CONSERVATION LAWS. (CRC Press, 2023)

[2] Kosmann-Schwarzbach, Y. in The Philosophy and Physics of Noether’s Theorems: A Centenary Volume 4-24 (Cambridge University Press, 2022)

[3] Rao, A. K., Tripathi, A., Chauhan, B. & Malik, R. P. Noether Theorem and Nilpotency Property of the (Anti-)BRST Charges in the BRST Formalism: A Brief Review. Universe 8 (2022).https://doi.org/10.3390/universe8110566

[4] Ibragimov, N. H. CRC HANDBOOK OF LIE GROUP ANALYSIS OF DIFFERENTIAL EQUATIONS VOLUME 1 -SYMMETRIES EXACT SOLUTIONS AND CONSERVATION LAWS. (CRC Press, 2023).

[5] Kosmann-Schwarzbach, Y. in The Philosophy and Physics of Noether’s Theorems: A Centenary Volume 4-24 (Cambridge University Press, 2022).

[6] Rao, A. K., Tripathi, A., Chauhan, B. & Malik, R. P. Noether Theorem and Nilpotency Property of the (Anti-)BRST Charges in the BRST Formalism: A Brief Review. Universe 8 (2022).https://doi.org/10.3390/universe8110566

[Aitchison-7] Aitchison, Ian J. R.; Hey, Anthony J.G. (2012).Gauge Theories in Particle Physics: A Practical Introduction: From Relativistic Quantum Mechanics to QED, Fourth Edition, Vol. 1.CRC Press. p. 43.ISBN 978-1466512993.Archivedfrom the original on 2018-05-04.

[Will-8] Will, Clifford M. (1993).Theory and Experiment in Gravitational Physics.Cambridge Univ. Press. p. 105.ISBN 978-0521439732.Archivedfrom the original on 2017-02-20.

[Toro-9] Toro, E.F. (1999). "Chapter 2. Notions on Hyperbolic PDEs".Riemann Solvers and Numerical Methods for Fluid Dynamics.Springer-Verlag.ISBN 978-3-540-65966-2.

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Conservation law

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