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* Page found: Drehimpuls und Bewegungsgleichungen (eq math.2005.0)

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Drehimpuls und Bewegungsgleichungen Eq: math.2005.0

Hash: 75b38fb04946de0c2672290f32ecba7e

TeX (original user input):

\begin{align}
  & \bar{l}=\sum\limits_{i=1}^{n}{{}}{{m}_{i}}{{{\bar{r}}}_{i}}\times {{{\dot{\bar{r}}}}_{i}}=\sum\limits_{i=1}^{n}{{}}{{m}_{i}}\left( {{{\bar{r}}}_{S}}+{{{\bar{x}}}^{(i)}} \right)\times \left( \bar{V}+\bar{\omega }\times {{{\bar{x}}}^{(i)}} \right) \\
 & \bar{l}=\sum\limits_{i=1}^{n}{{}}{{m}_{i}}\left( {{{\bar{r}}}_{S}}\times \bar{V} \right)+\sum\limits_{i=1}^{n}{{}}{{m}_{i}}{{{\bar{x}}}^{(i)}}\times \bar{V}+{{{\bar{r}}}_{S}}\times \left( \bar{\omega }\times \sum\limits_{i}{{{m}_{i}}}{{{\bar{x}}}^{(i)}} \right)+\sum\limits_{i}{{{m}_{i}}}{{{\bar{x}}}^{(i)}}\times \left( \bar{\omega }\times {{{\bar{x}}}^{(i)}} \right) \\
 & \sum\limits_{i=1}^{n}{{}}{{m}_{i}}{{{\bar{x}}}^{(i)}}\times \bar{V}=\sum\limits_{i=1}^{n}{{}}{{m}_{i}}\left( {{{\bar{x}}}^{(i)}} \right)=0 \\
 & \bar{l}=\sum\limits_{i=1}^{n}{{}}{{m}_{i}}\left( {{{\bar{r}}}_{S}}\times \bar{V} \right)+\sum\limits_{i}{{{m}_{i}}}{{{\bar{x}}}^{(i)}}\times \left( \bar{\omega }\times {{{\bar{x}}}^{(i)}} \right)=M\left( {{{\bar{r}}}_{S}}\times \bar{V} \right)+\sum\limits_{i}{{{m}_{i}}}{{{\bar{x}}}^{(i)}}\times \left( \bar{\omega }\times {{{\bar{x}}}^{(i)}} \right) \\
\end{align}

TeX (checked):

{\begin{aligned}&{\bar {l}}=\sum \limits _{i=1}^{n}{}{{m}_{i}}{{\bar {r}}_{i}}\times {{\dot {\bar {r}}}_{i}}=\sum \limits _{i=1}^{n}{}{{m}_{i}}\left({{\bar {r}}_{S}}+{{\bar {x}}^{(i)}}\right)\times \left({\bar {V}}+{\bar {\omega }}\times {{\bar {x}}^{(i)}}\right)\\&{\bar {l}}=\sum \limits _{i=1}^{n}{}{{m}_{i}}\left({{\bar {r}}_{S}}\times {\bar {V}}\right)+\sum \limits _{i=1}^{n}{}{{m}_{i}}{{\bar {x}}^{(i)}}\times {\bar {V}}+{{\bar {r}}_{S}}\times \left({\bar {\omega }}\times \sum \limits _{i}{{m}_{i}}{{\bar {x}}^{(i)}}\right)+\sum \limits _{i}{{m}_{i}}{{\bar {x}}^{(i)}}\times \left({\bar {\omega }}\times {{\bar {x}}^{(i)}}\right)\\&\sum \limits _{i=1}^{n}{}{{m}_{i}}{{\bar {x}}^{(i)}}\times {\bar {V}}=\sum \limits _{i=1}^{n}{}{{m}_{i}}\left({{\bar {x}}^{(i)}}\right)=0\\&{\bar {l}}=\sum \limits _{i=1}^{n}{}{{m}_{i}}\left({{\bar {r}}_{S}}\times {\bar {V}}\right)+\sum \limits _{i}{{m}_{i}}{{\bar {x}}^{(i)}}\times \left({\bar {\omega }}\times {{\bar {x}}^{(i)}}\right)=M\left({{\bar {r}}_{S}}\times {\bar {V}}\right)+\sum \limits _{i}{{m}_{i}}{{\bar {x}}^{(i)}}\times \left({\bar {\omega }}\times {{\bar {x}}^{(i)}}\right)\\\end{aligned}}

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\begin{matrix} \bar{l} = \sum_{i = 1}^{n} m_{i} {\bar{r}}_{i} \times {\dot{\bar{r}}}_{i} = \sum_{i = 1}^{n} m_{i} ({\bar{r}}_{S} + {\bar{x}}^{(i)}) \times (\bar{V} + \bar{ω} \times {\bar{x}}^{(i)}) \\ \bar{l} = \sum_{i = 1}^{n} m_{i} ({\bar{r}}_{S} \times \bar{V}) + \sum_{i = 1}^{n} m_{i} {\bar{x}}^{(i)} \times \bar{V} + {\bar{r}}_{S} \times (\bar{ω} \times \sum_{i} m_{i} {\bar{x}}^{(i)}) + \sum_{i} m_{i} {\bar{x}}^{(i)} \times (\bar{ω} \times {\bar{x}}^{(i)}) \\ \sum_{i = 1}^{n} m_{i} {\bar{x}}^{(i)} \times \bar{V} = \sum_{i = 1}^{n} m_{i} ({\bar{x}}^{(i)}) = 0 \\ \bar{l} = \sum_{i = 1}^{n} m_{i} ({\bar{r}}_{S} \times \bar{V}) + \sum_{i} m_{i} {\bar{x}}^{(i)} \times (\bar{ω} \times {\bar{x}}^{(i)}) = M ({\bar{r}}_{S} \times \bar{V}) + \sum_{i} m_{i} {\bar{x}}^{(i)} \times (\bar{ω} \times {\bar{x}}^{(i)}) \end{matrix}

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Identifiers

$\bar{l}$

$i$

$n$

$m_{i}$

${\bar{r}}_{i}$

${\dot{\bar{r}}}_{i}$

$i$

$n$

$m_{i}$

${\bar{r}}_{S}$

$\bar{x}$

$i$

$\bar{V}$

$\bar{ω}$

$\bar{x}$

$i$

$\bar{l}$

$i$

$n$

$m_{i}$

${\bar{r}}_{S}$

$\bar{V}$

$i$

$n$

$m_{i}$

$\bar{x}$

$i$

$\bar{V}$

${\bar{r}}_{S}$

$\bar{ω}$

$i$

$m_{i}$

$\bar{x}$

$i$

$i$

$m_{i}$

$\bar{x}$

$i$

$\bar{ω}$

$\bar{x}$

$i$

$i$

$n$

$m_{i}$

$\bar{x}$

$i$

$\bar{V}$

$i$

$n$

$m_{i}$

$\bar{x}$

$i$

$\bar{l}$

$i$

$n$

$m_{i}$

${\bar{r}}_{S}$

$\bar{V}$

$i$

$m_{i}$

$\bar{x}$

$i$

$\bar{ω}$

$\bar{x}$

$i$

$M$

${\bar{r}}_{S}$

$\bar{V}$

$i$

$m_{i}$

$\bar{x}$

$i$

$\bar{ω}$

$\bar{x}$

$i$

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