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Zusammenhang der beiden Prinzipien
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{\displaystyle {\frac {d({\frac {\partial T}{\partial q_{i}^{\prime }}})}{dt}}=2\sum _{j=1}^{k}{\frac {da_{ij}}{dt}}\cdot q_{j}^{\prime }+2\sum _{j=1}^{k}a_{ij}\cdot q_{j}^{\prime \prime }.\,}
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{\displaystyle q_{i}^{\prime }}
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{\displaystyle \ 1,2,...\,k}
(3)
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{\displaystyle {\begin{aligned}\sum _{i=1}^{k}{\frac {d({\frac {\partial T}{\partial q_{i}^{\prime }}})}{dt}}\cdot q_{i}^{\prime }&=2\sum _{i=1}^{k}\sum _{j=1}^{k}{\frac {da_{ij}}{dt}}\cdot q_{i}^{\prime }\cdot q_{j}^{\prime }\\&+2\sum _{i=1}^{k}\sum _{j=1}^{k}a_{ij}\cdot q_{i}^{\prime }\cdot q_{j}^{\prime \prime }\\\end{aligned}}}
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{\displaystyle {\frac {da_{ij}}{dt}}=\sum _{h=1}^{k}{\frac {\partial a_{ij}}{\partial q_{h}}}\cdot q_{h}^{\prime }}
(4)
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{\displaystyle \sum _{i=1}^{k}\sum _{j=1}^{k}{\frac {da_{ij}}{dt}}\cdot q_{i}^{\prime }\cdot q_{j}^{\prime }=\sum _{h=1}^{k}q_{h}^{\prime }\cdot \left\lbrace \sum _{i=1}^{k}\sum _{j=1}^{k}{\frac {\partial a_{ij}}{\partial q_{h}}}q_{i}^{\prime }\cdot q_{j}^{\prime }\right\rbrace }
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{\displaystyle =\sum _{h=1}^{k}{\frac {\partial T}{\partial q_{h}}}\cdot q_{h}^{\prime },}
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{\displaystyle \sum _{i=1}^{k}q_{i}^{\prime \prime }\cdot \left\{2\sum _{j=1}^{k}a_{ij}\cdot q_{j}^{\prime }\right\},}
(5)
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{\displaystyle 2\sum _{i=1}^{k}\sum _{j=1}^{k}a_{ij}\cdot q_{i}^{\prime }\cdot q_{j}^{\prime \prime }=\sum _{i=1}^{k}{\frac {\partial T}{\partial q_{i}^{\prime }}}\cdot q_{i}^{\prime \prime }.}
(6)
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{\displaystyle \sum _{i=1}^{k}{\frac {d({\frac {\partial T}{\partial q_{i}^{\prime }}})}{dt}}\cdot q_{i}^{\prime }=2\sum _{i=1}^{k}{\frac {\partial T}{\partial q_{i}}}\cdot q_{i}^{\prime }+\sum _{i=1}^{k}{\frac {\partial T}{\partial q_{i}^{\prime }}}\cdot q_{i}^{\prime \prime }}
