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| [326 ]
§. 99.
Riemann’s Grundgesetz. Wir wollen auch mit Hülfe dieses zweiten Ausdruckes für
D
{\displaystyle D\,}
(1)
δ
∫
0
t
(
T
−
D
+
S
)
d
t
=
0
,
{\displaystyle \delta \int _{0}^{t}(T-D+S)dt=0,}
Lagrange sich ausspricht. Wir könnten nun wieder denselben Weg einschlagen wie in §. 97. Es ist aber auch erlaubt, sofort von der Formel (6) des §. 42 Gebrauch zu machen, welche hier lautet:
(2)
d
(
∂
(
T
−
D
)
)
∂
q
′
)
d
t
=
∂
(
T
−
D
+
S
)
∂
q
.
{\displaystyle {\frac {d\left({\frac {\partial (T-D))}{\partial q'}}\right)}{dt}}={\frac {\partial (T-D+S)}{\partial q}}.}
q
{\displaystyle q\,}
x
,
y
,
z
,
x
1
,
y
1
,
z
1
{\displaystyle x,\,y,\,z,\,x_{1},\,y_{1},\,z_{1}\ }
q
=
x
{\displaystyle q=x\,}
| [327 ]
T
=
1
2
m
(
x
′
2
+
y
′
2
+
z
′
2
)
+
1
2
m
1
(
x
1
′
2
+
y
1
′
2
+
z
2
′
2
)
,
{\displaystyle T={\frac {1}{2}}m(x'^{2}+y'^{2}+z'^{2})+{\frac {1}{2}}m_{1}(x_{1}'^{2}+y_{1}'^{2}+z_{2}'^{2}),\,}
∂
T
∂
x
′
=
m
x
′
,
∂
T
∂
x
=
0.
{\displaystyle {\frac {\partial T}{\partial x'}}=mx',\quad {\frac {\partial T}{\partial x}}=0.}
(3)
X
=
m
d
2
x
d
t
2
=
d
(
∂
D
∂
x
′
)
d
t
−
∂
D
∂
x
+
∂
S
∂
x
.
{\displaystyle X=m{\frac {d^{2}x}{dt^{2}}}={\frac {d\left({\frac {\partial D}{\partial x'}}\right)}{dt}}-{\frac {\partial D}{\partial x}}+{\frac {\partial S}{\partial x}}.}
∂
D
∂
x
′
=
2
ε
ε
′
c
2
1
r
(
d
x
d
t
−
x
1
d
t
)
,
{\displaystyle {\frac {\partial D}{\partial x'}}=2{\frac {\varepsilon \varepsilon '}{c^{2}}}{\frac {1}{r}}\left({\frac {dx}{dt}}-{\frac {x_{1}}{dt}}\right),}
∂
D
∂
x
=
−
ε
ε
′
c
2
1
r
2
∂
r
∂
x
{
(
d
x
d
t
−
d
x
1
d
t
)
2
+
(
d
y
d
t
−
d
y
1
d
t
)
2
+
(
d
z
d
t
−
d
z
1
d
t
)
2
}
.
{\displaystyle {\frac {\partial D}{\partial x}}=-{\frac {\varepsilon \varepsilon '}{c^{2}}}{\frac {1}{r^{2}}}{\frac {\partial r}{\partial x}}{\Bigg \lbrace }\left({\frac {dx}{dt}}-{\frac {dx_{1}}{dt}}\right)^{2}+\left({\frac {dy}{dt}}-{\frac {dy_{1}}{dt}}\right)^{2}+\left({\frac {dz}{dt}}-{\frac {dz_{1}}{dt}}\right)^{2}{\Bigg \rbrace }.\,}
∂
S
∂
x
=
ε
ε
′
r
2
∂
r
∂
x
.
{\displaystyle {\frac {\partial S}{\partial x}}={\frac {\varepsilon \varepsilon '}{r^{2}}}{\frac {\partial r}{\partial x}}.}
(4)
X
=
ε
ε
′
r
2
∂
r
∂
x
+
ε
ε
′
c
2
d
{
2
r
(
d
x
d
t
−
d
x
1
d
t
)
}
d
t
{\displaystyle X={\frac {\varepsilon \varepsilon '}{r^{2}}}{\frac {\partial r}{\partial x}}+{\frac {\varepsilon \varepsilon '}{c^{2}}}{\frac {d{\Bigg \lbrace }{\frac {2}{r}}\left({\frac {dx}{dt}}-{\frac {dx_{1}}{dt}}\right){\Bigg \rbrace }}{dt}}}
+
ε
ε
′
c
2
1
r
2
∂
r
∂
x
{
(
d
x
d
t
−
d
x
1
d
t
)
2
+
(
d
y
d
t
−
d
y
1
d
t
)
2
+
(
d
z
d
t
−
d
z
1
d
t
)
2
}
.
{\displaystyle +{\frac {\varepsilon \varepsilon '}{c^{2}}}{\frac {1}{r^{2}}}{\frac {\partial r}{\partial x}}{\Bigg \lbrace }\left({\frac {dx}{dt}}-{\frac {dx_{1}}{dt}}\right)^{2}+\left({\frac {dy}{dt}}-{\frac {dy_{1}}{dt}}\right)^{2}+\left({\frac {dz}{dt}}-{\frac {dz_{1}}{dt}}\right)^{2}{\Bigg \rbrace }.}
(5)
Y
=
ε
ε
′
r
2
∂
r
∂
y
+
ε
ε
′
c
2
d
{
2
r
(
d
y
d
t
−
d
y
1
d
t
)
}
d
t
{\displaystyle Y={\frac {\varepsilon \varepsilon '}{r^{2}}}{\frac {\partial r}{\partial y}}+{\frac {\varepsilon \varepsilon '}{c^{2}}}{\frac {d{\Bigg \lbrace }{\frac {2}{r}}\left({\frac {dy}{dt}}-{\frac {dy_{1}}{dt}}\right){\Bigg \rbrace }}{dt}}}
+
ε
ε
′
c
2
1
r
2
∂
r
∂
y
{
(
d
x
d
t
−
d
x
1
d
t
)
2
+
(
d
y
d
t
−
d
y
1
d
t
)
2
+
(
d
z
d
t
−
d
z
1
d
t
)
2
}
.
{\displaystyle +{\frac {\varepsilon \varepsilon '}{c^{2}}}{\frac {1}{r^{2}}}{\frac {\partial r}{\partial y}}{\Bigg \lbrace }\left({\frac {dx}{dt}}-{\frac {dx_{1}}{dt}}\right)^{2}+\left({\frac {dy}{dt}}-{\frac {dy_{1}}{dt}}\right)^{2}+\left({\frac {dz}{dt}}-{\frac {dz_{1}}{dt}}\right)^{2}{\Bigg \rbrace }.}
(6)
Z
=
ε
ε
′
r
2
∂
r
∂
z
+
ε
ε
′
c
2
d
{
2
r
(
d
z
d
t
−
d
z
1
d
t
)
}
d
t
{\displaystyle Z={\frac {\varepsilon \varepsilon '}{r^{2}}}{\frac {\partial r}{\partial z}}+{\frac {\varepsilon \varepsilon '}{c^{2}}}{\frac {d{\Bigg \lbrace }{\frac {2}{r}}\left({\frac {dz}{dt}}-{\frac {dz_{1}}{dt}}\right){\Bigg \rbrace }}{dt}}}
+
ε
ε
′
c
2
1
r
2
∂
r
∂
z
{
(
d
x
d
t
−
d
x
1
d
t
)
2
+
(
d
y
d
t
−
d
y
1
d
t
)
2
+
(
d
z
d
t
−
d
z
1
d
t
)
2
}
.
{\displaystyle +{\frac {\varepsilon \varepsilon '}{c^{2}}}{\frac {1}{r^{2}}}{\frac {\partial r}{\partial z}}{\Bigg \lbrace }\left({\frac {dx}{dt}}-{\frac {dx_{1}}{dt}}\right)^{2}+\left({\frac {dy}{dt}}-{\frac {dy_{1}}{dt}}\right)^{2}+\left({\frac {dz}{dt}}-{\frac {dz_{1}}{dt}}\right)^{2}{\Bigg \rbrace }.}
