Lead Article: Tables of Physics Formulae
This article is a summary of the laws, principles, defining quantities, and useful formulae in the analysis of Continuity and Conservation Equations.
To summarize essentials of physics, this section enumerates the classical conservation laws and continuity equations. All the following conservation laws carry through to modern physics, such as Quantum Mechanics, Relativity, Particle Physics and Quantum Relativity, though modifications to conserved quantities may be necessary. Particle physics introduces new conservation laws, many in a different way using quantum numbers.
For any isolated system (i.e. independent of external agents/influences) the following laws apply to the whole system. Constituents of the system possessing these quantities may experience changes, but the total amount of the quantity due to all constituents is constant.
Two equivalent ways of applying these in problems is by considering the quantities before and after an event, or considereing any two points in space and time, and equating the initial state of the system to the final, since the quantity is conserved.
Corresponding to conserved quantities are currents, current densities, or other time derivatives. These quantites must be conserved also since the amount of a conserved quantity associated with a system is invariant in space and time.
Conserved Quantity Constancy Equation System Equation/s Time Derivatives
Mass
Δ
m
=
0
{\displaystyle \Delta m=0\,\!}
M
s
y
s
t
e
m
=
∑
i
=
1
N
1
m
i
=
∑
j
=
1
N
2
m
j
{\displaystyle M_{\mathrm {system} }=\sum _{i=1}^{N_{1}}m_{i}=\sum _{j=1}^{N_{2}}m_{j}\,\!}
Mass current conservation
∑
i
=
1
N
1
(
I
m
)
i
=
∑
j
=
1
N
2
(
I
m
)
j
=
0
{\displaystyle \sum _{i=1}^{N_{1}}\left(I_{\mathrm {m} }\right)_{i}=\sum _{j=1}^{N_{2}}\left(I_{\mathrm {m} }\right)_{j}=0\,\!}
∑
i
=
1
N
1
(
j
m
)
i
=
∑
j
=
1
N
2
(
j
m
)
j
=
0
{\displaystyle \sum _{i=1}^{N_{1}}\left(\mathbf {j} _{\mathrm {m} }\right)_{i}=\sum _{j=1}^{N_{2}}\left(\mathbf {j} _{\mathrm {m} }\right)_{j}=\mathbf {0} \,\!}
Linear Momentum
Δ
p
=
0
{\displaystyle \Delta \mathbf {p} =\mathbf {0} \,\!}
∑
i
=
1
N
1
p
i
=
∑
i
=
1
N
2
p
j
{\displaystyle \sum _{i=1}^{N_{1}}\mathbf {p} _{i}=\sum _{i=1}^{N_{2}}\mathbf {p} _{j}\,\!}
which can be written in equivalent ways, most useful forms are:
∑
i
=
1
N
1
m
i
v
i
=
∑
j
=
1
N
2
m
j
v
j
{\displaystyle \sum _{i=1}^{N_{1}}m_{i}\mathbf {v} _{i}=\sum _{j=1}^{N_{2}}m_{j}\mathbf {v} _{j}}
Momentum current conservation
∑
i
=
1
N
1
(
I
p
)
i
=
∑
j
=
1
N
2
(
I
p
)
j
=
0
{\displaystyle \sum _{i=1}^{N_{1}}\left(I_{\mathrm {p} }\right)_{i}=\sum _{j=1}^{N_{2}}\left(I_{\mathrm {p} }\right)_{j}=0\,\!}
Momentum current density conservation
∑
i
=
1
N
1
(
j
p
)
i
=
∑
j
=
1
N
2
(
j
p
)
j
=
0
{\displaystyle \sum _{i=1}^{N_{1}}\left(\mathbf {j} _{\mathrm {p} }\right)_{i}=\sum _{j=1}^{N_{2}}\left(\mathbf {j} _{\mathrm {p} }\right)_{j}=\mathbf {0} \,\!}
Total Angular Momentum
Δ
L
t
o
t
a
l
=
0
{\displaystyle \Delta \mathbf {L} _{\mathrm {total} }=\mathbf {0} \,\!}
L
s
y
s
t
e
m
=
∑
i
=
1
N
1
L
i
=
∑
j
=
1
N
2
L
j
{\displaystyle \mathbf {L} _{\mathrm {system} }=\sum _{i=1}^{N_{1}}\mathbf {L} _{i}=\sum _{j=1}^{N_{2}}\mathbf {L} _{j}\,\!}
which can be written in equivalent ways, most useful forms are:
L
s
y
s
t
e
m
=
∑
i
=
1
N
1
(
I
a
b
ω
b
)
i
=
∑
j
=
1
N
2
(
I
a
b
ω
b
)
j
{\displaystyle \mathbf {L} _{\mathrm {system} }=\sum _{i=1}^{N_{1}}\left(\mathbf {I} _{\mathrm {ab} }{\boldsymbol {\omega }}_{\mathrm {b} }\right)_{i}=\sum _{j=1}^{N_{2}}\left(\mathbf {I} _{\mathrm {ab} }{\boldsymbol {\omega }}_{\mathrm {b} }\right)_{j}\,\!}
L
s
y
s
t
e
m
=
∑
i
=
1
N
1
r
i
×
p
i
=
∑
j
=
1
N
2
r
j
×
p
j
{\displaystyle \mathbf {L} _{\mathrm {system} }=\sum _{i=1}^{N_{1}}\mathbf {r} _{i}\times \mathbf {p} _{i}=\sum _{j=1}^{N_{2}}\mathbf {r} _{j}\times \mathbf {p} _{j}\,\!}
L
s
y
s
t
e
m
=
∑
i
=
1
N
1
m
i
(
r
i
×
v
i
)
=
∑
j
=
1
N
2
m
j
(
r
j
×
v
j
)
{\displaystyle \mathbf {L} _{\mathrm {system} }=\sum _{i=1}^{N_{1}}m_{i}\left(\mathbf {r} _{i}\times \mathbf {v} _{i}\right)=\sum _{j=1}^{N_{2}}m_{j}\left(\mathbf {r} _{j}\times \mathbf {v} _{j}\right)\,\!}
No analogue
Spin Angular Momentum
Δ
L
s
p
i
n
=
0
{\displaystyle \Delta \mathbf {L} _{\mathrm {spin} }=\mathbf {0} \,\!}
Same as above
Orbital Angular Momentum
Δ
L
o
r
b
i
t
a
l
=
0
{\displaystyle \Delta \mathbf {L} _{\mathrm {orbital} }=\mathbf {0} \,\!}
Same as above
Energy
Δ
E
=
0
{\displaystyle \Delta E=0\,\!}
E
s
y
s
t
e
m
=
∑
i
T
i
+
∑
j
V
j
{\displaystyle E_{\mathrm {system} }=\sum _{i}T_{i}+\sum _{j}V_{j}\,\!}
or simply
E
=
T
+
V
{\displaystyle E=T+V\,\!}
E
s
y
s
t
e
m
=
∑
i
=
1
N
1
(
T
i
+
V
i
)
=
∑
j
=
1
N
2
(
T
j
+
V
j
)
{\displaystyle E_{\mathrm {system} }=\sum _{i=1}^{N_{1}}\left(T_{i}+V_{i}\right)=\sum _{j=1}^{N_{2}}\left(T_{j}+V_{j}\right)\,\!}
Power conservation
∑
i
P
i
+
∑
j
P
j
=
0
{\displaystyle \sum _{i}P_{i}+\sum _{j}P_{j}=0\,\!}
Intensity conservation
∑
i
I
i
+
∑
j
I
j
=
0
{\displaystyle \sum _{i}I_{i}+\sum _{j}I_{j}=0\,\!}
Charge
Δ
q
=
0
{\displaystyle \Delta q=0\,\!}
Q
s
y
s
t
e
m
=
∑
i
=
1
N
1
q
i
=
∑
j
=
1
N
2
q
j
{\displaystyle Q_{\mathrm {system} }=\sum _{i=1}^{N_{1}}q_{i}=\sum _{j=1}^{N_{2}}q_{j}\,\!}
Electric current conservation
∑
i
=
1
N
1
I
i
=
∑
j
=
1
N
2
I
j
=
0
{\displaystyle \sum _{i=1}^{N_{1}}I_{i}=\sum _{j=1}^{N_{2}}I_{j}=0\,\!}
Electric current density conservation
∑
i
=
1
N
1
J
i
=
∑
j
=
1
N
2
J
j
=
0
{\displaystyle \sum _{i=1}^{N_{1}}\mathbf {J} _{i}=\sum _{j=1}^{N_{2}}\mathbf {J} _{j}=\mathbf {0} \,\!}
Continuity equations describe transport of conserved quantities though a local region of space. Note that these equations are not fundamental simply because of conservation; they can be derived.
Continuity Description Nomenclature General Equation Simple Case
Hydrodynamics, Fluid Flow
j
m
{\displaystyle j_{\mathrm {m} }\,\!}
ρ
{\displaystyle \rho \,\!}
u
{\displaystyle \mathbf {u} \,\!}
A
{\displaystyle \mathbf {A} \,\!}
∇
⋅
(
ρ
u
)
+
∂
ρ
∂
t
=
0
{\displaystyle \nabla \cdot (\rho \mathbf {u} )+{\partial \rho \over \partial t}=0\,\!}
j
m
=
ρ
1
A
1
⋅
u
1
=
ρ
2
A
2
⋅
u
2
{\displaystyle j_{\mathrm {m} }=\rho _{1}\mathbf {A} _{1}\cdot \mathbf {u} _{1}=\rho _{2}\mathbf {A} _{2}\cdot \mathbf {u} _{2}\,\!}
Electromagnetism, Charge
I
{\displaystyle I\,\!}
J
{\displaystyle \mathbf {J} \,\!}
ρ
{\displaystyle \rho \,\!}
u
{\displaystyle \mathbf {u} \,\!}
A
{\displaystyle \mathbf {A} \,\!}
∇
⋅
J
+
∂
ρ
∂
t
=
0
{\displaystyle \nabla \cdot \mathbf {J} +{\partial \rho \over \partial t}=0\,\!}
I
=
ρ
1
A
1
⋅
u
1
=
ρ
2
A
2
⋅
u
2
{\displaystyle I=\rho _{1}\mathbf {A} _{1}\cdot \mathbf {u} _{1}=\rho _{2}\mathbf {A} _{2}\cdot \mathbf {u} _{2}\,\!}
Quantum Mechnics, Probability
j
{\displaystyle \mathbf {j} \,\!}
P
=
P
(
x
,
t
)
{\displaystyle P=P(x,t)\,\!}
∇
⋅
j
+
∂
P
∂
t
=
0
{\displaystyle \nabla \cdot \mathbf {j} +{\frac {\partial P}{\partial t}}=0\,\!}
Conservation Laws
Continiuity Equations
