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* Page found: Normalschwingungen (eq math.1515.20)

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\begin{align}
  & \sum\limits_{k}{({{V}_{lk}}-{{\omega }^{2}}{{T}_{lk}}){{A}_{k}}=0}\left| \cdot \sum\limits_{l}{{{A}_{l}}^{*}} \right. \\
 & \sum\limits_{l,k}{{{V}_{lk}}{{A}_{l}}^{*}{{A}_{k}}-}{{\omega }^{2}}\sum\limits_{l,k}{{{T}_{lk}}{{A}_{l}}^{*}{{A}_{k}}}=0 \\
 & {{\omega }^{2}}=\frac{\sum\limits_{l,k}{{{V}_{lk}}{{A}_{l}}^{*}{{A}_{k}}}}{\sum\limits_{l,k}{{{T}_{lk}}{{A}_{l}}^{*}{{A}_{k}}}} \\
 & \sum\limits_{l,k}{{{V}_{lk}}{{A}_{l}}^{*}{{A}_{k}}}=\frac{1}{2}\sum\limits_{l,k}{{{V}_{lk}}{{A}_{l}}^{*}{{A}_{k}}}+\frac{1}{2}\sum\limits_{l,k}{{{V}_{kl}}{{A}_{k}}^{*}{{A}_{l}}=}\frac{1}{2}\sum\limits_{l,k}{{{V}_{lk}}\left( {{A}_{l}}^{*}{{A}_{k}}+{{A}_{k}}^{*}{{A}_{l}} \right)=}\frac{1}{2}\sum\limits_{l,k}{{{V}_{lk}}2\cdot \operatorname{Re}\left( {{A}_{l}}^{*}{{A}_{k}} \right)} \\
\end{align}

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∑ k ( V l ⁢ k - ω 2 T l ⁢ k ) A k = 0 | ⋅ ∑ l A l * ∑ l , k V l ⁢ k ⁢ A l * ⁢ A k - ω 2 ⁢ ∑ l , k T l ⁢ k ⁢ A l * ⁢ A k = 0 ω 2 = ∑ l , k V l ⁢ k ⁢ A l * ⁢ A k ∑ l , k T l ⁢ k ⁢ A l * ⁢ A k ∑ l , k V l ⁢ k ⁢ A l * ⁢ A k = 1 2 ⁢ ∑ l , k V l ⁢ k ⁢ A l * ⁢ A k + 1 2 ⁢ ∑ l , k V k ⁢ l ⁢ A k * ⁢ A l = 1 2 ⁢ ∑ l , k V l ⁢ k ⁢ ( A l * ⁢ A k + A k * ⁢ A l ) = 1 2 ⁢ ∑ l , k V l ⁢ k ⁢ 2 ⋅ Re ⁡ ( A l * ⁢ A k ) missing-subexpression fragments subscript 𝑘 fragments ( subscript 𝑉 𝑙 𝑘 superscript 𝜔 2 subscript 𝑇 𝑙 𝑘 ) subscript 𝐴 𝑘 0 | ⋅ subscript 𝑙 superscript subscript 𝐴 𝑙 missing-subexpression subscript 𝑙 𝑘 subscript 𝑉 𝑙 𝑘 superscript subscript 𝐴 𝑙 subscript 𝐴 𝑘 superscript 𝜔 2 subscript 𝑙 𝑘 subscript 𝑇 𝑙 𝑘 superscript subscript 𝐴 𝑙 subscript 𝐴 𝑘 0 missing-subexpression superscript 𝜔 2 subscript 𝑙 𝑘 subscript 𝑉 𝑙 𝑘 superscript subscript 𝐴 𝑙 subscript 𝐴 𝑘 subscript 𝑙 𝑘 subscript 𝑇 𝑙 𝑘 superscript subscript 𝐴 𝑙 subscript 𝐴 𝑘 missing-subexpression subscript 𝑙 𝑘 subscript 𝑉 𝑙 𝑘 superscript subscript 𝐴 𝑙 subscript 𝐴 𝑘 1 2 subscript 𝑙 𝑘 subscript 𝑉 𝑙 𝑘 superscript subscript 𝐴 𝑙 subscript 𝐴 𝑘 1 2 subscript 𝑙 𝑘 subscript 𝑉 𝑘 𝑙 superscript subscript 𝐴 𝑘 subscript 𝐴 𝑙 1 2 subscript 𝑙 𝑘 subscript 𝑉 𝑙 𝑘 superscript subscript 𝐴 𝑙 subscript 𝐴 𝑘 superscript subscript 𝐴 𝑘 subscript 𝐴 𝑙 1 2 subscript 𝑙 𝑘 ⋅ subscript 𝑉 𝑙 𝑘 2 Re superscript subscript 𝐴 𝑙 subscript 𝐴 𝑘 {\displaystyle{\displaystyle\begin{aligned} &\displaystyle\sum\limits_{k}{({{V% }_{lk}}-{{\omega}^{2}}{{T}_{lk}}){{A}_{k}}=0}\left|\cdot\sum\limits_{l}{{{A}_{% l}}^{*}}\right.\\ &\displaystyle\sum\limits_{l,k}{{{V}_{lk}}{{A}_{l}}^{*}{{A}_{k}}-}{{\omega}^{2% }}\sum\limits_{l,k}{{{T}_{lk}}{{A}_{l}}^{*}{{A}_{k}}}=0\\ &\displaystyle{{\omega}^{2}}=\frac{\sum\limits_{l,k}{{{V}_{lk}}{{A}_{l}}^{*}{{% A}_{k}}}}{\sum\limits_{l,k}{{{T}_{lk}}{{A}_{l}}^{*}{{A}_{k}}}}\\ &\displaystyle\sum\limits_{l,k}{{{V}_{lk}}{{A}_{l}}^{*}{{A}_{k}}}=\frac{1}{2}% \sum\limits_{l,k}{{{V}_{lk}}{{A}_{l}}^{*}{{A}_{k}}}+\frac{1}{2}\sum\limits_{l,% k}{{{V}_{kl}}{{A}_{k}}^{*}{{A}_{l}}=}\frac{1}{2}\sum\limits_{l,k}{{{V}_{lk}}% \left({{A}_{l}}^{*}{{A}_{k}}+{{A}_{k}}^{*}{{A}_{l}}\right)=}\frac{1}{2}\sum% \limits_{l,k}{{{V}_{lk}}2\cdot\operatorname{Re}\left({{A}_{l}}^{*}{{A}_{k}}% \right)}\\ \end{aligned}}}

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∑k(Vlk−ω2Tlk)Ak=0|⋅∑lAl*∑l,kVlkAl*Ak−ω2∑l,kTlkAl*Ak=0ω2=∑l,kVlkAl*Ak∑l,kTlkAl*Ak∑l,kVlkAl*Ak=12∑l,kVlkAl*Ak+12∑l,kVklAk*Al=12∑l,kVlk(Al*Ak+Ak*Al)=12∑l,kVlk2⋅ℜ⁡(Al*Ak)

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