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Display information for equation id:math.1515.7 on revision:1515

* Page found: 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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L=TV=12(j,kTjkq˙jq˙kj,kVjkqjqk)Lq˙l=12j,kTjkq˙l(q˙jq˙k)=12j,kTjk(δjlq˙k+δklq˙j)=12j,kTlkq˙k+Tljq˙j=kTlkq˙kmitTjl=Tljddt(Lq˙l)=kTlkq¨kLql=kVlkqk
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Calculated based on the variables occurring on the entire Normalschwingungen page

Identifiers

  • L
  • T
  • V
  • j
  • k
  • Tjk
  • q˙j
  • q˙k
  • j
  • k
  • Vjk
  • qj
  • qk
  • L
  • q˙l
  • j
  • k
  • Tjk
  • q˙l
  • q˙j
  • q˙k
  • j
  • k
  • Tjk
  • δjl
  • q˙k
  • δkl
  • q˙j
  • j
  • k
  • Tlk
  • q˙k
  • Tlj
  • q˙j
  • k
  • Tlk
  • q˙k
  • m
  • i
  • t
  • Tjl
  • Tlj
  • d
  • d
  • t
  • L
  • q˙l
  • k
  • Tlk
  • q¨k
  • L
  • ql
  • k
  • Vlk
  • qk

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