aus Wikisource, der freien Quellensammlung
Andererseits haben wir aus (A ):
W
=
∫
∞
w
d
τ
{\displaystyle W=\int _{\infty }w\ d\tau }
wo
w
=
1
2
(
E
⋅
E
+
M
⋅
M
)
+
ε
0
μ
0
(
p
+
v
)
⋅
[
E
M
]
{\displaystyle w={\tfrac {1}{2}}({\mathsf {E}}\cdot {\mathfrak {E}}+{\mathsf {M}}\cdot {\mathfrak {M}})+\varepsilon _{0}\mu _{0}(p+v)\cdot {\mathsf {[EM]}}}
oder nach (C 1 ) auch
w
=
1
2
(
ε
E
2
+
μ
M
2
)
+
2
ε
0
μ
0
(
p
+
v
)
⋅
[
E
M
]
{\displaystyle w={\tfrac {1}{2}}(\varepsilon {\mathsf {E}}^{2}+\mu {\mathsf {M}}^{2})+2\varepsilon _{0}\mu _{0}(p+v)\cdot {\mathsf {[EM]}}}
(24)
Wir bilden
δ
w
δ
t
{\displaystyle {\frac {\delta w}{\delta t}}}
ε
{\displaystyle \varepsilon }
μ
{\displaystyle \mu }
δ
ε
δ
t
=
−
v
⋅
∇
ε
,
δ
μ
δ
t
=
−
v
⋅
∇
μ
{\displaystyle {\frac {\delta \varepsilon }{\delta t}}=-v\cdot \nabla \varepsilon ,\qquad {\frac {\delta \mu }{\delta t}}=-v\cdot \nabla \mu }
ist. So ergiebt sich aus (24):
δ
w
δ
t
=
E
⋅
δ
(
ε
E
)
δ
t
+
1
2
E
2
v
⋅
∇
ε
+
M
⋅
δ
(
μ
M
)
δ
t
+
1
2
M
2
v
⋅
∇
μ
+
2
ε
0
μ
0
(
p
+
v
)
⋅
δ
[
E
M
]
δ
t
+
2
ε
0
μ
0
δ
(
p
+
v
)
δ
t
⋅
[
E
M
]
}
{\displaystyle \left.{\begin{aligned}{\frac {\delta w}{\delta t}}={\mathsf {E}}\cdot {\frac {\delta (\varepsilon {\mathsf {E}})}{\delta t}}+{\tfrac {1}{2}}{\mathsf {E}}^{2}v\cdot \nabla \varepsilon +{\mathsf {M}}\cdot {\frac {\delta (\mu {\mathsf {M}})}{\delta t}}+{\tfrac {1}{2}}{\mathsf {M}}^{2}v\cdot \nabla \mu \\+2\varepsilon _{0}\mu _{0}(p+v)\cdot {\frac {\delta {\mathsf {[EM]}}}{\delta t}}+2\varepsilon _{0}\mu _{0}{\frac {\delta (p+v)}{\delta t}}\cdot {\mathsf {[EM]}}\end{aligned}}\right\}}
(25)
Aus (23) und (25) folgt:
−
Γ
[
(
E
−
[
v
M
]
)
(
M
+
[
v
E
]
)
]
=
δ
w
δ
t
+
E
⋅
Λ
−
p
⋅
ε
0
μ
0
δ
[
E
M
]
δ
t
+
v
⋅
f
{\displaystyle -{\mathsf {\Gamma }}{\bigl [}({\mathsf {E}}-[v{\mathfrak {M}}])({\mathsf {M}}+[v{\mathfrak {E}}]){\bigr ]}={\frac {\delta w}{\delta t}}+{\mathsf {E\cdot \Lambda }}-p\cdot \varepsilon _{0}\mu _{0}{\frac {\delta {\mathsf {[EM]}}}{\delta t}}+v\cdot f}
(26)
wo
f
=
Γ
(
E
)
E
−
1
2
E
2
∇
ε
+
Γ
(
M
)
M
−
1
2
M
2
∇
μ
+
[
Λ
M
]
+
δ
δ
t
{
[
E
M
]
−
ε
0
μ
0
[
E
M
]
}
+
[
Γ
(
E
)
v
⋅
M
]
−
[
Γ
(
M
)
v
⋅
E
]
.
}
{\displaystyle \left.{\begin{aligned}f={\mathsf {\Gamma }}({\mathfrak {E}}){\mathsf {E}}-{\tfrac {1}{2}}{\mathsf {E}}^{2}\nabla \varepsilon +{\mathsf {\Gamma }}({\mathfrak {M}}){\mathsf {M}}-{\tfrac {1}{2}}{\mathsf {M}}^{2}\nabla \mu +[{\mathsf {\Lambda }}{\mathfrak {M}}]\\+{\frac {\delta }{\delta t}}\{[{\mathfrak {EM}}]-\varepsilon _{0}\mu _{0}{\mathsf {[EM]}}\}+[{\mathsf {\Gamma }}({\mathfrak {E}})v\cdot {\mathfrak {M}}]-[{\mathsf {\Gamma }}({\mathfrak {M}})v\cdot {\mathfrak {E}}].\end{aligned}}\right\}}
(27)
Wir multipliciren die Gleichung (26) mit
d
τ
{\displaystyle d\tau }
δ
W
δ
t
=
∂
W
∂
t
{\displaystyle {\frac {\delta W}{\delta t}}={\frac {\partial W}{\partial t}}}
−
∂
W
∂
t
=
∫
∞
E
⋅
Λ
d
τ
−
p
⋅
∂
∂
t
∫
∞
ε
0
μ
0
[
E
M
]
d
τ
+
∫
∞
v
⋅
f
d
τ
{\displaystyle -{\frac {\partial W}{\partial t}}=\int _{\infty }{\mathsf {E\cdot \Lambda }}d\tau -p\cdot {\frac {\partial }{\partial t}}\int _{\infty }\varepsilon _{0}\mu _{0}{\mathsf {[EM]}}d\tau +\int _{\infty }v\cdot f\ d\tau }
(28)
Zunächst fassen wir die Partialgeschwindigkeiten gemäß (22) in eine zusammen, indem wir
p
=
0
{\displaystyle p=0}
v
=
u
{\displaystyle v=u}
−
∂
W
∂
t
=
∫
∞
E
⋅
Λ
d
τ
+
∫
∞
u
⋅
f
0
d
τ
{\displaystyle -{\frac {\partial W}{\partial t}}=\int _{\infty }{\mathsf {E\cdot \Lambda }}d\tau +\int _{\infty }u\cdot f_{0}\ d\tau }
(29)
![{\displaystyle w={\tfrac {1}{2}}({\mathsf {E}}\cdot {\mathfrak {E}}+{\mathsf {M}}\cdot {\mathfrak {M}})+\varepsilon _{0}\mu _{0}(p+v)\cdot {\mathsf {[EM]}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/57b4f73d948e1cd82db7a06fd4bc2624ac92927a)
![{\displaystyle w={\tfrac {1}{2}}(\varepsilon {\mathsf {E}}^{2}+\mu {\mathsf {M}}^{2})+2\varepsilon _{0}\mu _{0}(p+v)\cdot {\mathsf {[EM]}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/20f44cf52b75db9dd85d1c9ef3dbef6fc686cb88)
![{\displaystyle \left.{\begin{aligned}{\frac {\delta w}{\delta t}}={\mathsf {E}}\cdot {\frac {\delta (\varepsilon {\mathsf {E}})}{\delta t}}+{\tfrac {1}{2}}{\mathsf {E}}^{2}v\cdot \nabla \varepsilon +{\mathsf {M}}\cdot {\frac {\delta (\mu {\mathsf {M}})}{\delta t}}+{\tfrac {1}{2}}{\mathsf {M}}^{2}v\cdot \nabla \mu \\+2\varepsilon _{0}\mu _{0}(p+v)\cdot {\frac {\delta {\mathsf {[EM]}}}{\delta t}}+2\varepsilon _{0}\mu _{0}{\frac {\delta (p+v)}{\delta t}}\cdot {\mathsf {[EM]}}\end{aligned}}\right\}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/5dfe05173a9106342de44cd57d3b2d95b821a8aa)
![{\displaystyle -{\mathsf {\Gamma }}{\bigl [}({\mathsf {E}}-[v{\mathfrak {M}}])({\mathsf {M}}+[v{\mathfrak {E}}]){\bigr ]}={\frac {\delta w}{\delta t}}+{\mathsf {E\cdot \Lambda }}-p\cdot \varepsilon _{0}\mu _{0}{\frac {\delta {\mathsf {[EM]}}}{\delta t}}+v\cdot f}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d2ab096ee179a83523fec3575708643e47c1c135)
![{\displaystyle \left.{\begin{aligned}f={\mathsf {\Gamma }}({\mathfrak {E}}){\mathsf {E}}-{\tfrac {1}{2}}{\mathsf {E}}^{2}\nabla \varepsilon +{\mathsf {\Gamma }}({\mathfrak {M}}){\mathsf {M}}-{\tfrac {1}{2}}{\mathsf {M}}^{2}\nabla \mu +[{\mathsf {\Lambda }}{\mathfrak {M}}]\\+{\frac {\delta }{\delta t}}\{[{\mathfrak {EM}}]-\varepsilon _{0}\mu _{0}{\mathsf {[EM]}}\}+[{\mathsf {\Gamma }}({\mathfrak {E}})v\cdot {\mathfrak {M}}]-[{\mathsf {\Gamma }}({\mathfrak {M}})v\cdot {\mathfrak {E}}].\end{aligned}}\right\}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/0dd6514341e4384d04093a3c60a4e078b4400811)
![{\displaystyle -{\frac {\partial W}{\partial t}}=\int _{\infty }{\mathsf {E\cdot \Lambda }}d\tau -p\cdot {\frac {\partial }{\partial t}}\int _{\infty }\varepsilon _{0}\mu _{0}{\mathsf {[EM]}}d\tau +\int _{\infty }v\cdot f\ d\tau }](https://wikimedia.org/api/rest_v1/media/math/render/svg/8e3ec52cec20949666b340a7598c807b47417092)
