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* Page found: Normalschwingungen (eq math.1515.7)

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Normalschwingungen Eq: math.1515.7

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TeX (original user input):

\begin{align}
  & L=T-V=\frac{1}{2}\left( \sum\limits_{j,k}{{{T}_{jk}}}{{{\dot{q}}}_{j}}{{{\dot{q}}}_{k}}-\sum\limits_{j,k}{{{V}_{jk}}}{{q}_{j}}{{q}_{k}} \right) \\
 & \frac{\partial L}{\partial {{{\dot{q}}}_{l}}}=\frac{1}{2}\sum\limits_{j,k}{{{T}_{jk}}}\frac{\partial }{\partial {{{\dot{q}}}_{l}}}\left( {{{\dot{q}}}_{j}}{{{\dot{q}}}_{k}} \right)=\frac{1}{2}\sum\limits_{j,k}{{{T}_{jk}}}\left( {{\delta }_{jl}}{{{\dot{q}}}_{k}}+{{\delta }_{kl}}{{{\dot{q}}}_{j}} \right)=\frac{1}{2}\sum\limits_{j,k}{{{T}_{lk}}}{{{\dot{q}}}_{k}}+{{T}_{lj}}{{{\dot{q}}}_{j}}=\sum\limits_{k}{{{T}_{lk}}}{{{\dot{q}}}_{k}}\quad mit\ {{T}_{jl}}={{T}_{lj}} \\
 & \frac{d}{dt}\left( \frac{\partial L}{\partial {{{\dot{q}}}_{l}}} \right)=\sum\limits_{k}{{{T}_{lk}}}{{{\ddot{q}}}_{k}} \\
 & \frac{\partial L}{\partial {{q}_{l}}}=-\sum\limits_{k}{{{V}_{lk}}{{q}_{k}}} \\
\end{align}

TeX (checked):

{\begin{aligned}&L=T-V={\frac {1}{2}}\left(\sum \limits _{j,k}{{T}_{jk}}{{\dot {q}}_{j}}{{\dot {q}}_{k}}-\sum \limits _{j,k}{{V}_{jk}}{{q}_{j}}{{q}_{k}}\right)\\&{\frac {\partial L}{\partial {{\dot {q}}_{l}}}}={\frac {1}{2}}\sum \limits _{j,k}{{T}_{jk}}{\frac {\partial }{\partial {{\dot {q}}_{l}}}}\left({{\dot {q}}_{j}}{{\dot {q}}_{k}}\right)={\frac {1}{2}}\sum \limits _{j,k}{{T}_{jk}}\left({{\delta }_{jl}}{{\dot {q}}_{k}}+{{\delta }_{kl}}{{\dot {q}}_{j}}\right)={\frac {1}{2}}\sum \limits _{j,k}{{T}_{lk}}{{\dot {q}}_{k}}+{{T}_{lj}}{{\dot {q}}_{j}}=\sum \limits _{k}{{T}_{lk}}{{\dot {q}}_{k}}\quad mit\ {{T}_{jl}}={{T}_{lj}}\\&{\frac {d}{dt}}\left({\frac {\partial L}{\partial {{\dot {q}}_{l}}}}\right)=\sum \limits _{k}{{T}_{lk}}{{\ddot {q}}_{k}}\\&{\frac {\partial L}{\partial {{q}_{l}}}}=-\sum \limits _{k}{{{V}_{lk}}{{q}_{k}}}\\\end{aligned}}

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\begin{aligned} L = T - V = \frac{1}{2} (\sum_{j, k} T_{j k} {\dot{q}}_{j} {\dot{q}}_{k} - \sum_{j, k} V_{j k} q_{j} q_{k}) \\ \frac{\partial L}{\partial {\dot{q}}_{l}} = \frac{1}{2} \sum_{j, k} T_{j k} \frac{\partial}{\partial {\dot{q}}_{l}} ({\dot{q}}_{j} {\dot{q}}_{k}) = \frac{1}{2} \sum_{j, k} T_{j k} (δ_{j l} {\dot{q}}_{k} + δ_{k l} {\dot{q}}_{j}) = \frac{1}{2} \sum_{j, k} T_{l k} {\dot{q}}_{k} + T_{l j} {\dot{q}}_{j} = \sum_{k} T_{l k} {\dot{q}}_{k} m i t T_{j l} = T_{l j} \\ \frac{d}{d t} (\frac{\partial L}{\partial {\dot{q}}_{l}}) = \sum_{k} T_{l k} {\ddot{q}}_{k} \\ \frac{\partial L}{\partial q_{l}} = - \sum_{k} V_{l k} q_{k} \end{aligned}

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Identifiers

$L$

$T$

$V$

$j$

$k$

$T_{j k}$

${\dot{q}}_{j}$

${\dot{q}}_{k}$

$j$

$k$

$V_{j k}$

$q_{j}$

$q_{k}$

$L$

${\dot{q}}_{l}$

$j$

$k$

$T_{j k}$

${\dot{q}}_{l}$

${\dot{q}}_{j}$

${\dot{q}}_{k}$

$j$

$k$

$T_{j k}$

$δ_{j l}$

${\dot{q}}_{k}$

$δ_{k l}$

${\dot{q}}_{j}$

$j$

$k$

$T_{l k}$

${\dot{q}}_{k}$

$T_{l j}$

${\dot{q}}_{j}$

$k$

$T_{l k}$

${\dot{q}}_{k}$

$m$

$i$

$t$

$T_{j l}$

$T_{l j}$

$d$

$d$

$t$

$L$

${\dot{q}}_{l}$

$k$

$T_{l k}$

${\ddot{q}}_{k}$

$L$

$q_{l}$

$k$

$V_{l k}$

$q_{k}$

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