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Display information for equation id:math.1515.6 on revision:1515

* Page found: Normalschwingungen (eq math.1515.6)

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\begin{align}
  & {{{\vec{v}}}_{i}}=\sum\limits_{j}{{}}\left( \frac{\partial {{{\vec{r}}}_{i}}}{\partial {{q}_{j}}} \right){{{\dot{q}}}_{j}} \\
 & T=\frac{1}{2}\sum\limits_{i}{{{m}_{i}}}\left( \sum\limits_{j,k}{\left( \frac{\partial {{{\vec{r}}}_{i}}}{\partial {{q}_{j}}} \right)\left( \frac{\partial {{{\vec{r}}}_{i}}}{\partial {{q}_{j}}} \right)}{{{\dot{q}}}_{j}}{{{\dot{q}}}_{k}} \right)\ge 0 \\
 & T=\frac{1}{2}\sum\limits_{j,k}{{{T}_{jk}}}{{{\dot{q}}}_{j}}{{{\dot{q}}}_{k}} \\
 & {{T}_{jk}}={{T}_{kj}}\approx \sum\limits_{i}{{{m}_{i}}{{\left( \frac{\partial {{{\vec{r}}}_{i}}}{\partial {{q}_{j}}} \right)}_{0}}{{\left( \frac{\partial {{{\vec{r}}}_{i}}}{\partial {{q}_{j}}} \right)}_{0}}} \\
\end{align}

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v → i = ∑ j ( ∂ ⁡ r → i ∂ ⁡ q j ) ⁢ q ˙ j T = 1 2 ⁢ ∑ i m i ⁢ ( ∑ j , k ( ∂ ⁡ r → i ∂ ⁡ q j ) ⁢ ( ∂ ⁡ r → i ∂ ⁡ q j ) ⁢ q ˙ j ⁢ q ˙ k ) ≥ 0 T = 1 2 ⁢ ∑ j , k T j ⁢ k ⁢ q ˙ j ⁢ q ˙ k T j ⁢ k = T k ⁢ j ≈ ∑ i m i ⁢ ( ∂ ⁡ r → i ∂ ⁡ q j ) 0 ⁢ ( ∂ ⁡ r → i ∂ ⁡ q j ) 0 missing-subexpression subscript → 𝑣 𝑖 subscript 𝑗 subscript → 𝑟 𝑖 subscript 𝑞 𝑗 subscript ˙ 𝑞 𝑗 missing-subexpression 𝑇 1 2 subscript 𝑖 subscript 𝑚 𝑖 subscript 𝑗 𝑘 subscript → 𝑟 𝑖 subscript 𝑞 𝑗 subscript → 𝑟 𝑖 subscript 𝑞 𝑗 subscript ˙ 𝑞 𝑗 subscript ˙ 𝑞 𝑘 0 missing-subexpression 𝑇 1 2 subscript 𝑗 𝑘 subscript 𝑇 𝑗 𝑘 subscript ˙ 𝑞 𝑗 subscript ˙ 𝑞 𝑘 missing-subexpression subscript 𝑇 𝑗 𝑘 subscript 𝑇 𝑘 𝑗 subscript 𝑖 subscript 𝑚 𝑖 subscript subscript → 𝑟 𝑖 subscript 𝑞 𝑗 0 subscript subscript → 𝑟 𝑖 subscript 𝑞 𝑗 0 {\displaystyle{\displaystyle\begin{aligned} &\displaystyle{{{\vec{v}}}_{i}}=% \sum\limits_{j}{{}}\left(\frac{\partial{{{\vec{r}}}_{i}}}{\partial{{q}_{j}}}% \right){{{\dot{q}}}_{j}}\\ &\displaystyle T=\frac{1}{2}\sum\limits_{i}{{{m}_{i}}}\left(\sum\limits_{j,k}{% \left(\frac{\partial{{{\vec{r}}}_{i}}}{\partial{{q}_{j}}}\right)\left(\frac{% \partial{{{\vec{r}}}_{i}}}{\partial{{q}_{j}}}\right)}{{{\dot{q}}}_{j}}{{{\dot{% q}}}_{k}}\right)\geq 0\\ &\displaystyle T=\frac{1}{2}\sum\limits_{j,k}{{{T}_{jk}}}{{{\dot{q}}}_{j}}{{{% \dot{q}}}_{k}}\\ &\displaystyle{{T}_{jk}}={{T}_{kj}}\approx\sum\limits_{i}{{{m}_{i}}{{\left(% \frac{\partial{{{\vec{r}}}_{i}}}{\partial{{q}_{j}}}\right)}_{0}}{{\left(\frac{% \partial{{{\vec{r}}}_{i}}}{\partial{{q}_{j}}}\right)}_{0}}}\\ \end{aligned}}}

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v→i=∑j(∂r→i∂qj)q˙jT=12∑imi(∑j,k(∂r→i∂qj)(∂r→i∂qj)q˙jq˙k)≥0T=12∑j,kTjkq˙jq˙kTjk=Tkj≈∑imi(∂r→i∂qj)0(∂r→i∂qj)0

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