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Display information for equation id:math.1965.21 on revision:1965

* Page found: Symplektische Struktur des Phasenraums (eq math.1965.21)

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TeX (original user input): 

\begin{align}
  & J{{\left( J{{M}^{-1}} \right)}^{T}}=\left( \begin{matrix}
   0 & 1  \\
   -1 & 0  \\
\end{matrix} \right){{\left[ \left( \begin{matrix}
   0 & 1  \\
   -1 & 0  \\
\end{matrix} \right)\left( \begin{matrix}
   \frac{\partial Q}{\partial q} & \frac{\partial Q}{\partial p}  \\
   \frac{\partial P}{\partial q} & \frac{\partial P}{\partial p}  \\
\end{matrix} \right) \right]}^{T}}=\left( \begin{matrix}
   0 & 1  \\
   -1 & 0  \\
\end{matrix} \right){{\left( \begin{matrix}
   \frac{\partial P}{\partial q} & \frac{\partial P}{\partial p}  \\
   -\frac{\partial Q}{\partial q} & -\frac{\partial Q}{\partial p}  \\
\end{matrix} \right)}^{T}} \\ 
 & =\left( \begin{matrix}
   0 & 1  \\
   -1 & 0  \\
\end{matrix} \right){{\left( \begin{matrix}
   {{\left( \frac{\partial P}{\partial q} \right)}^{T}} & -{{\left( \frac{\partial Q}{\partial q} \right)}^{T}}  \\
   {{\left( \frac{\partial P}{\partial p} \right)}^{T}} & -{{\left( \frac{\partial Q}{\partial p} \right)}^{T}}  \\
\end{matrix} \right)}^{{}}}=\left( \begin{matrix}
   {{\left( \frac{\partial P}{\partial p} \right)}^{T}} & -{{\left( \frac{\partial Q}{\partial p} \right)}^{T}}  \\
   -{{\left( \frac{\partial P}{\partial q} \right)}^{T}} & {{\left( \frac{\partial Q}{\partial q} \right)}^{T}}  \\
\end{matrix} \right) \\ 
\end{align}

TeX (checked): 

{\begin{aligned}&J{{\left(J{{M}^{-1}}\right)}^{T}}=\left({\begin{matrix}0&1\\-1&0\\\end{matrix}}\right){{\left[\left({\begin{matrix}0&1\\-1&0\\\end{matrix}}\right)\left({\begin{matrix}{\frac {\partial Q}{\partial q}}&{\frac {\partial Q}{\partial p}}\\{\frac {\partial P}{\partial q}}&{\frac {\partial P}{\partial p}}\\\end{matrix}}\right)\right]}^{T}}=\left({\begin{matrix}0&1\\-1&0\\\end{matrix}}\right){{\left({\begin{matrix}{\frac {\partial P}{\partial q}}&{\frac {\partial P}{\partial p}}\\-{\frac {\partial Q}{\partial q}}&-{\frac {\partial Q}{\partial p}}\\\end{matrix}}\right)}^{T}}\\&=\left({\begin{matrix}0&1\\-1&0\\\end{matrix}}\right){{\left({\begin{matrix}{{\left({\frac {\partial P}{\partial q}}\right)}^{T}}&-{{\left({\frac {\partial Q}{\partial q}}\right)}^{T}}\\{{\left({\frac {\partial P}{\partial p}}\right)}^{T}}&-{{\left({\frac {\partial Q}{\partial p}}\right)}^{T}}\\\end{matrix}}\right)}^{}}=\left({\begin{matrix}{{\left({\frac {\partial P}{\partial p}}\right)}^{T}}&-{{\left({\frac {\partial Q}{\partial p}}\right)}^{T}}\\-{{\left({\frac {\partial P}{\partial q}}\right)}^{T}}&{{\left({\frac {\partial Q}{\partial q}}\right)}^{T}}\\\end{matrix}}\right)\\\end{aligned}}

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MathML (75.194 KB / 6.188 KB) :

J ⁢ ( J ⁢ M - 1 ) T = ( 0 1 - 1 0 ) ⁢ [ ( 0 1 - 1 0 ) ⁢ ( ∂ ⁡ Q ∂ ⁡ q ∂ ⁡ Q ∂ ⁡ p ∂ ⁡ P ∂ ⁡ q ∂ ⁡ P ∂ ⁡ p ) ] T = ( 0 1 - 1 0 ) ⁢ ( ∂ ⁡ P ∂ ⁡ q ∂ ⁡ P ∂ ⁡ p - ∂ ⁡ Q ∂ ⁡ q - ∂ ⁡ Q ∂ ⁡ p ) T = ( 0 1 - 1 0 ) ⁢ ( ( ∂ ⁡ P ∂ ⁡ q ) T - ( ∂ ⁡ Q ∂ ⁡ q ) T ( ∂ ⁡ P ∂ ⁡ p ) T - ( ∂ ⁡ Q ∂ ⁡ p ) T ) = ( ( ∂ ⁡ P ∂ ⁡ p ) T - ( ∂ ⁡ Q ∂ ⁡ p ) T - ( ∂ ⁡ P ∂ ⁡ q ) T ( ∂ ⁡ Q ∂ ⁡ q ) T ) missing-subexpression 𝐽 superscript 𝐽 superscript 𝑀 1 𝑇 0 1 1 0 superscript delimited-[] 0 1 1 0 𝑄 𝑞 𝑄 𝑝 𝑃 𝑞 𝑃 𝑝 𝑇 0 1 1 0 superscript 𝑃 𝑞 𝑃 𝑝 𝑄 𝑞 𝑄 𝑝 𝑇 missing-subexpression absent 0 1 1 0 superscript 𝑃 𝑞 𝑇 superscript 𝑄 𝑞 𝑇 superscript 𝑃 𝑝 𝑇 superscript 𝑄 𝑝 𝑇 superscript 𝑃 𝑝 𝑇 superscript 𝑄 𝑝 𝑇 superscript 𝑃 𝑞 𝑇 superscript 𝑄 𝑞 𝑇 {\displaystyle{\displaystyle\begin{aligned} &\displaystyle J{{\left(J{{M}^{-1}% }\right)}^{T}}=\left(\begin{matrix}0&1\\ -1&0\\ \end{matrix}\right){{\left[\left(\begin{matrix}0&1\\ -1&0\\ \end{matrix}\right)\left(\begin{matrix}\frac{\partial Q}{\partial q}&\frac{% \partial Q}{\partial p}\\ \frac{\partial P}{\partial q}&\frac{\partial P}{\partial p}\\ \end{matrix}\right)\right]}^{T}}=\left(\begin{matrix}0&1\\ -1&0\\ \end{matrix}\right){{\left(\begin{matrix}\frac{\partial P}{\partial q}&\frac{% \partial P}{\partial p}\\ -\frac{\partial Q}{\partial q}&-\frac{\partial Q}{\partial p}\\ \end{matrix}\right)}^{T}}\\ &\displaystyle=\left(\begin{matrix}0&1\\ -1&0\\ \end{matrix}\right){{\left(\begin{matrix}{{\left(\frac{\partial P}{\partial q}% \right)}^{T}}&-{{\left(\frac{\partial Q}{\partial q}\right)}^{T}}\\ {{\left(\frac{\partial P}{\partial p}\right)}^{T}}&-{{\left(\frac{\partial Q}{% \partial p}\right)}^{T}}\\ \end{matrix}\right)}}=\left(\begin{matrix}{{\left(\frac{\partial P}{\partial p% }\right)}^{T}}&-{{\left(\frac{\partial Q}{\partial p}\right)}^{T}}\\ -{{\left(\frac{\partial P}{\partial q}\right)}^{T}}&{{\left(\frac{\partial Q}{% \partial q}\right)}^{T}}\\ \end{matrix}\right)\\ \end{aligned}}}

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J(JM−1)T=(01−10)[(01−10)(∂Q∂q∂Q∂p∂P∂q∂P∂p)]T=(01−10)(∂P∂q∂P∂p−∂Q∂q−∂Q∂p)T=(01−10)((∂P∂q)T−(∂Q∂q)T(∂P∂p)T−(∂Q∂p)T)=((∂P∂p)T−(∂Q∂p)T−(∂P∂q)T(∂Q∂q)T)

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