من ويكيبيديا، الموسوعة الحرة
في ديناميكا الموائع معادلات كيرشهوف تصف حركة جسم جاسئ في مائع مثالي .[1]
d
d
t
∂
T
∂
ω
→
=
∂
T
∂
ω
→
×
ω
→
+
∂
T
∂
v
→
×
v
→
+
Q
→
h
+
Q
→
,
d
d
t
∂
T
∂
v
→
=
∂
T
∂
v
→
×
ω
→
+
F
→
h
+
F
→
,
T
=
1
2
(
ω
→
T
I
~
ω
→
+
m
v
2
)
Q
→
h
=
−
∫
p
x
→
×
n
^
d
σ
,
F
→
h
=
−
∫
p
n
^
d
σ
{\displaystyle {\begin{aligned}{d \over {dt}}{{\partial T} \over {\partial {\vec {\omega }}}}&={{\partial T} \over {\partial {\vec {\omega }}}}\times {\vec {\omega }}+{{\partial T} \over {\partial {\vec {v}}}}\times {\vec {v}}+{\vec {Q}}_{h}+{\vec {Q}},\\[10pt]{d \over {dt}}{{\partial T} \over {\partial {\vec {v}}}}&={{\partial T} \over {\partial {\vec {v}}}}\times {\vec {\omega }}+{\vec {F}}_{h}+{\vec {F}},\\[10pt]T&={1 \over 2}\left({\vec {\omega }}^{T}{\tilde {I}}{\vec {\omega }}+mv^{2}\right)\\[10pt]{\vec {Q}}_{h}&=-\int p{\vec {x}}\times {\hat {n}}\,d\sigma ,\\[10pt]{\vec {F}}_{h}&=-\int p{\hat {n}}\,d\sigma \end{aligned}}}
ω
→
{\displaystyle {\vec {\omega }}}
v
→
{\displaystyle {\vec {v}}}
السرعة الزاوية والخطية على محور
x
→
{\displaystyle {\vec {x}}}
زخم موتّر العطالة
I
~
{\displaystyle {\tilde {I}}}
m
{\displaystyle m}
الكتلة ,
n
^
{\displaystyle {\hat {n}}}
x
→
{\displaystyle {\vec {x}}}
p
{\displaystyle p}
الضغط ,
Q
→
h
{\displaystyle {\vec {Q}}_{h}}
عزم الدوران
F
→
h
{\displaystyle {\vec {F}}_{h}}
القوة .
إذا كان الجسم مغمور كليا
d
d
t
∂
L
∂
ω
→
=
∂
L
∂
ω
→
×
ω
→
+
∂
L
∂
v
→
×
v
→
,
d
d
t
∂
L
∂
v
→
=
∂
L
∂
v
→
×
ω
→
,
{\displaystyle {d \over {dt}}{{\partial L} \over {\partial {\vec {\omega }}}}={{\partial L} \over {\partial {\vec {\omega }}}}\times {\vec {\omega }}+{{\partial L} \over {\partial {\vec {v}}}}\times {\vec {v}},\quad {d \over {dt}}{{\partial L} \over {\partial {\vec {v}}}}={{\partial L} \over {\partial {\vec {v}}}}\times {\vec {\omega }},}
L
(
ω
→
,
v
→
)
=
1
2
(
A
ω
→
,
ω
→
)
+
(
B
ω
→
,
v
→
)
+
1
2
(
C
v
→
,
v
→
)
+
(
k
→
,
ω
→
)
+
(
l
→
,
v
→
)
.
{\displaystyle L({\vec {\omega }},{\vec {v}})={1 \over 2}(A{\vec {\omega }},{\vec {\omega }})+(B{\vec {\omega }},{\vec {v}})+{1 \over 2}(C{\vec {v}},{\vec {v}})+({\vec {k}},{\vec {\omega }})+({\vec {l}},{\vec {v}}).}
تكون القراءة الأولى للتفاضل
J
0
=
(
∂
L
∂
ω
→
,
ω
→
)
+
(
∂
L
∂
v
→
,
v
→
)
−
L
,
J
1
=
(
∂
L
∂
ω
→
,
∂
L
∂
v
→
)
,
J
2
=
(
∂
L
∂
v
→
,
∂
L
∂
v
→
)
{\displaystyle J_{0}=\left({{\partial L} \over {\partial {\vec {\omega }}}},{\vec {\omega }}\right)+\left({{\partial L} \over {\partial {\vec {v}}}},{\vec {v}}\right)-L,\quad J_{1}=\left({{\partial L} \over {\partial {\vec {\omega }}}},{{\partial L} \over {\partial {\vec {v}}}}\right),\quad J_{2}=\left({{\partial L} \over {\partial {\vec {v}}}},{{\partial L} \over {\partial {\vec {v}}}}\right)}
المراجع Kirchhoff G. R. Vorlesungen ueber Mathematische Physik, Mechanik . Lecture 19. Leipzig: Teubner. 1877.
Lamb, H., Hydrodynamics . Sixth Edition Cambridge (UK): Cambridge University Press. 1932. 
![{\displaystyle {\begin{aligned}{d \over {dt}}{{\partial T} \over {\partial {\vec {\omega }}}}&={{\partial T} \over {\partial {\vec {\omega }}}}\times {\vec {\omega }}+{{\partial T} \over {\partial {\vec {v}}}}\times {\vec {v}}+{\vec {Q}}_{h}+{\vec {Q}},\\[10pt]{d \over {dt}}{{\partial T} \over {\partial {\vec {v}}}}&={{\partial T} \over {\partial {\vec {v}}}}\times {\vec {\omega }}+{\vec {F}}_{h}+{\vec {F}},\\[10pt]T&={1 \over 2}\left({\vec {\omega }}^{T}{\tilde {I}}{\vec {\omega }}+mv^{2}\right)\\[10pt]{\vec {Q}}_{h}&=-\int p{\vec {x}}\times {\hat {n}}\,d\sigma ,\\[10pt]{\vec {F}}_{h}&=-\int p{\hat {n}}\,d\sigma \end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a0cae909bc00f16818c74d53b772dbf33d8dc0cb)
