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Display information for equation id:math.2035.47 on revision:2035

* Page found: Stabilität und Langzeitverhalten (eq math.2035.47)

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Stabilität und Langzeitverhalten Eq: math.2045.47

Hash: 22a95a8d342847f2225a4e7ec1c15708

TeX (original user input):

\begin{align}
  & \bar{\varpi }{{*}^{(1)}}:{{\varpi }_{1}}=\varpi ,{{\varpi }_{2}}=0,{{\varpi }_{3}}=0 \\
 & 0=\det (A-\lambda 1)=\left| \begin{matrix}
   -\lambda  & 0 & 0  \\
   0 & -\lambda  & {{k}_{2}}{{\omega }_{{}}}  \\
   0 & -{{k}_{3}}\omega  & -\lambda   \\
\end{matrix} \right|=-\lambda \left( {{\lambda }^{2}}+{{k}_{2}}{{k}_{3}}{{\omega }^{2}} \right) \\
 & \Rightarrow {{\lambda }_{1}}^{(1)}=0,{{\lambda }_{2/3}}^{(1)}=\pm i\omega \sqrt{{{k}_{2}}{{k}_{3}}} \\
\end{align}

TeX (checked):

{\begin{aligned}&{\bar {\varpi }}{{*}^{(1)}}:{{\varpi }_{1}}=\varpi ,{{\varpi }_{2}}=0,{{\varpi }_{3}}=0\\&0=\det(A-\lambda 1)=\left|{\begin{matrix}-\lambda &0&0\\0&-\lambda &{{k}_{2}}{{\omega }_{}}\\0&-{{k}_{3}}\omega &-\lambda \\\end{matrix}}\right|=-\lambda \left({{\lambda }^{2}}+{{k}_{2}}{{k}_{3}}{{\omega }^{2}}\right)\\&\Rightarrow {{\lambda }_{1}}^{(1)}=0,{{\lambda }_{2/3}}^{(1)}=\pm i\omega {\sqrt {{{k}_{2}}{{k}_{3}}}}\\\end{aligned}}

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\begin{aligned} \bar{ϖ} *^{(1)} : ϖ_{1} = ϖ, ϖ_{2} = 0, ϖ_{3} = 0 \\ 0 = \det (A - λ 1) = | \begin{matrix} - λ & 0 & 0 \\ 0 & - λ & k_{2} ω_{} \\ 0 & - k_{3} ω & - λ \end{matrix} | = - λ (λ^{2} + k_{2} k_{3} ω^{2}) \\ \Rightarrow {λ_{1}}^{(1)} = 0, {λ_{2 / 3}}^{(1)} = \pm i ω \sqrt{k_{2} k_{3}} \end{aligned}

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$\bar{ϖ}$

$ϖ_{1}$

$ϖ$

$ϖ_{2}$

$ϖ_{3}$

$A$

$λ$

$λ$

$λ$

$k_{2}$

$ω$

$k_{3}$

$ω$

$λ$

$λ$

$λ$

$k_{2}$

$k_{3}$

$ω$

$λ_{1}$

$λ$

$i$

$ω$

$k_{2}$

$k_{3}$

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