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| title | chunk | source | category | tags | date_saved | instance |
|---|---|---|---|---|---|---|
| Conservation law | 2/3 | https://en.wikipedia.org/wiki/Conservation_law | reference | science, encyclopedia | 2026-05-05T13:31:25.992814+00:00 | kb-cron |
== 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. 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. This is called a local conservation law. 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
∂
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
∂
ρ
∂
t
=
−
∇
⋅
(
ρ
u
)
.
{\displaystyle {\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-order quasilinear hyperbolic equation:
y
t
+
A
(
y
)
y
x
=
0
{\displaystyle y_{t}+A(y)y_{x}=0}
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:
y
t
+
A
(
y
)
y
x
=
s
{\displaystyle y_{t}+A(y)y_{x}=s}
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 Navier-Stokes equations, or the entropy balance for a general isolated system. In the one-dimensional space a conservation equation is a first-order quasilinear hyperbolic equation that can be put into the advection form:
y
t
+
a
(
y
)
y
x
=
0
{\displaystyle y_{t}+a(y)y_{x}=0}
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):
a
(
y
)
=
j
y
(
y
)
{\displaystyle a(y)=j_{y}(y)}
In this case since the chain rule applies:
j
x
=
j
y
(
y
)
y
x
=
a
(
y
)
y
x
{\displaystyle 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
{\displaystyle y_{t}+j_{x}(y)=0}
In a space with more than one dimension the former definition can be extended to an equation that can be put into the form:
y
t
+
a
(
y
)
⋅
∇
y
=
0
{\displaystyle y_{t}+\mathbf {a} (y)\cdot \nabla y=0}
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):
y
t
+
∇
⋅
j
(
y
)
=
0
{\displaystyle y_{t}+\nabla \cdot \mathbf {j} (y)=0}
This is the case for the continuity equation:
ρ
t
+
∇
⋅
(
ρ
u
)
=
0
{\displaystyle \rho _{t}+\nabla \cdot (\rho \mathbf {u} )=0}