Conservation law Information
In physics, a conservation law states that a particular measurable property of an isolated physical system does not change as the system evolves over time. Exact conservation laws include conservation of mass and energy, conservation of linear momentum, conservation of angular momentum, and conservation of electric charge. There are also many approximate conservation laws, which apply to such quantities as mass, 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 a continuity equation, a partial differential equation which 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.
From Noether's theorem, each conservation law is associated with a symmetry in the underlying physics.
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 fundamental laws of 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 is Noether theorem, which states that there is a onetoone correspondence between each one of them and a differentiable symmetry of nature. For example, the conservation of energy follows from the timeinvariance of physical systems, and the conservation of angular momentum arises from the fact that physical systems behave the same regardless of how they are oriented in space.
Exact laws
A partial listing of physical conservation equations due to symmetry that are said to be exact laws, or more precisely have never been proven to be violated:
Conservation law  Respective Noether symmetry invariance  Number of dimensions  

Conservation of massenergy  Timetranslation invariance  Poincaré invariance  1  translation along time axis 
Conservation of linear momentum  Spacetranslation invariance  3  translation along x,y,z directions  
Conservation of angular momentum  Rotation invariance  3  rotation about x,y,z axes  
Conservation of CM (centerofmomentum) velocity  Lorentzboost invariance  3  Lorentzboost along x,y,z directions  
Conservation of electric charge  U(1) Gauge invariance  1 ⊗ 4  scalar field (1D) in 4D spacetime (x,y,z + time evolution)  
Conservation of color charge  SU(3) Gauge invariance  3  r,g,b  
Conservation of weak isospin  SU(2)_{L} Gauge invariance  1  weak charge  
Conservation of probability  Probability invariance^{ [1]}  1 ⊗ 4  total probability always = 1 in whole x,y,z space, during time evolution 
Approximate laws
There are also approximate conservation laws. These are approximately true in particular situations, such as low speeds, short time scales, or certain interactions.
 Conservation of mechanical energy
 Conservation of rest mass
 Conservation of baryon number (See chiral anomaly and sphaleron)
 Conservation of lepton number (In the Standard Model)
 Conservation of flavor (violated by the weak interaction)
 Conservation of parity (violated by the weak interaction)
 Invariance under charge conjugation
 Invariance under time reversal
 CP symmetry, the combination of charge conjugation and parity (equivalent to time reversal if CPT holds)
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 point A and simultaneously disappear from another separate point B. 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 not Lorentz invariant, so phenomena like the above do not occur in nature.^{ [2]}^{ [3]} Due to special relativity, if the appearance of the energy at A and disappearance of the energy at B are simultaneous in one inertial reference frame, they will not be simultaneous in other inertial reference frames moving with respect to the first. In a moving frame one will occur before the other; either the energy at A will appear before or after the energy at B disappears. 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, or flux of the quantity into or out of the point. For example, the amount of electric charge at a point is never found to change without an electric current into or out of the point that carries the difference in charge. Since it only involves continuous local changes, this stronger type of conservation law is Lorentz invariant; a quantity conserved in one reference frame is conserved in all moving reference frames.^{ [2]}^{ [3]} This is called a local conservation law.^{ [2]}^{ [3]} 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 a continuity 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
In continuum mechanics, the most general form of an exact conservation law is given by a continuity equation. For example, conservation of electric charge q is
where ∇⋅ is the divergence operator, 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 motion u of the charge is a continuous function of position and time, then
In one space dimension this can be put into the form of a homogeneous firstorder quasilinear hyperbolic equation:^{ [4]}
where the dependent variable y is called the density of a conserved quantity, and A(y) is called the current Jacobian, and the subscript notation for partial derivatives has been employed. The more general inhomogeneous case:
is not a conservation equation but the general kind of balance equation describing a dissipative system. The dependent variable y is called a nonconserved quantity, and the inhomogeneous term s(y,x,t) is the source, or dissipation. For example, balance equations of this kind are the momentum and energy NavierStokes equations, or the entropy balance for a general isolated system.
In the onedimensional space a conservation equation is a firstorder quasilinear hyperbolic equation that can be put into the advection form:
where the dependent variable y(x,t) is called the density of the conserved (scalar) quantity, and a(y) is called the current coefficient, usually corresponding to the partial derivative in the conserved quantity of a current density of the conserved quantity j(y):^{ [4]}
In this case since the chain rule applies:
the conservation equation can be put into the current density form:
In a space with more than one dimension the former definition can be extended to an equation that can be put into the form:
where the conserved quantity is y(r,t), denotes the scalar product, ∇ is the nabla operator, here indicating a gradient, and a(y) is a vector of current coefficients, analogously corresponding to the divergence of a vector current density associated to the conserved quantity j(y):
This is the case for the continuity equation:
Here the conserved quantity is the mass, with density ρ(r,t) and current density ρu, identical to the momentum density, while u(r,t) is the flow velocity.
In the general case a conservation equation can be also a system of this kind of equations (a vector equation) in the form:^{ [4]}
where y is called the conserved (vector) quantity, ∇ y is its gradient, 0 is the zero vector, and A(y) is called the Jacobian of the current density. In fact as in the former scalar case, also in the vector case A(y) usually corresponding to the Jacobian of a current density matrix J(y):
and the conservation equation can be put into the form:
For example, this the case for Euler equations (fluid dynamics). In the simple incompressible case they are:
where:
 u is the flow velocity vector, with components in a Ndimensional space u_{1}, u_{2}, … u_{N},
 s is the specific pressure (pressure per unit density) giving the source term,
It can be shown that the conserved (vector) quantity and the current density matrix for these equations are respectively:
where denotes the outer product.
Integral and weak forms
Conservation equations can be also expressed in integral form: the advantage of the latter is substantially that it requires less smoothness of the solution, which paves the way to weak form, extending the class of admissible solutions to include discontinuous solutions.^{ [5]} By integrating in any spacetime domain the current density form in 1D space:
and by using Green's theorem, the integral form is:
In a similar fashion, for the scalar multidimensional space, the integral form is:
where the line integration is performed along the boundary of the domain, in an anticlockwise manner.^{ [5]}
Moreover, by defining a test function φ(r,t) continuously differentiable both in time and space with compact support, the weak form can be obtained pivoting on the initial condition. In 1D space it is:
Note that 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.^{ [5]}
See also
 Invariant (physics)
 Conservative system

Conserved quantity
 Some kinds of helicity are conserved in dissipationless limit: hydrodynamical helicity, magnetic helicity, crosshelicity.
 Principle of mutability
 Conservation law of the Stress–energy tensor
 Riemann invariant
 Philosophy of physics
 Totalitarian principle
 Convection–diffusion equation
 Uniformity of nature
Examples and applications
 Advection
 Mass conservation, or Continuity equation
 Charge conservation
 Euler equations (fluid dynamics)
 inviscid Burgers equation
 Kinematic wave
 Conservation of energy
 Traffic flow
Notes
 ^ "The gaugeinvariance of the probability current". Physics Stack Exchange. Archived from the original on 18 August 2017. Retrieved 4 May 2018.
 ^ ^{a} ^{b} ^{c} 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 9781466512993. Archived from the original on 20180504.
 ^ ^{a} ^{b} ^{c} Will, Clifford M. (1993). Theory and Experiment in Gravitational Physics. Cambridge Univ. Press. p. 105. ISBN 9780521439732. Archived from the original on 20170220.
 ^ ^{a} ^{b} ^{c} see Toro, p.43
 ^ ^{a} ^{b} ^{c} see Toro, p.6263
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.
 Toro, E.F. (1999). "Chapter 2. Notions on Hyperbolic PDEs". Riemann Solvers and Numerical Methods for Fluid Dynamics. SpringerVerlag. ISBN 9783540659662.
 E. Godlewski and P.A. Raviart, Hyperbolic systems of conservation laws, Ellipses, 1991.
External links
 Conservation Laws — Ch. 1115 in an online textbook