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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 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 conserva...

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...

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 mass-energy Time-translation invariance Lorentz invariance symmetry 1 translation along time axis Conservation of linear momentum Space-translation invariance 3 translation along x , y , z directions Conservation of angular momentum Rotation invariance 3 rotation about x , y , z axes Conservation of CM (center-of-momentum) velocity Lorentz-boost invariance 3 Lorentz-boost along x , y , z directions Conservation of electric charge 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 invaria...

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 Invariance under charge conjugation Invariance under time reversal CP symmetry, the combination of charge conjugation and parity (equivalent to time reversal if CPT holds)

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 a remote 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. 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...

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 ∂ ρ ∂ t = − ∇ ⋅ j {\displaystyle {\frac {\partial \rho }{\partial t}}=-\nabla \cdot \mathbf {j} \,} 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 j = ρ u {\displaystyle \mathbf {j} =\rho \mathbf {u} } ∂ ρ ∂ t = − ∇ ⋅ ( ρ u ) . {\displaystyle {\frac {\partial \rho }{\partial t}}=-\nabla \cdot (\rho \mathbf {u} )\,.} In one space dimension this can be put into the form of a homogeneous first-order quasilinear hyperbolic equation: y t + A ( y ) y x = 0 {\displaystyle y_{t}+A(y)y_{x}=0} where the dependent variable y ...

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. By integrating in any space-time domain the current density form in 1-D space: y t + j x ( y ) = 0 {\displaystyle y_{t}+j_{x}(y)=0} and by using Green's theorem, the integral form is: ∫ − ∞ ∞ y d x + ∫ 0 ∞ j ( y ) d t = 0 {\displaystyle \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: ∮ y d N r + j ( y ) d t = 0 {\displaystyle \oint y\,d^{N}r+j(y)\,dt=0} where the line integration is performed along the boundary of the domain, in an anticlock-wise manner. Moreover, by defining a test function φ ( r , t ) continuously differenti...