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Display information for equation id:math.1325.183 on revision:1325
* Page found: Der Hamiltonsche kanonische Formalismus (eq math.1325.183)
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TeX (original user input):
\begin{align}
& {{{\dot{y}}}_{i}}=\sum\limits_{k}^{{}}{\frac{\partial {{y}_{i}}}{\partial {{x}_{k}}}{{{\dot{x}}}_{k}}}\Leftrightarrow \dot{\bar{y}}={{M}^{-1}}\dot{\bar{x}}=\left( {{J}^{-1}}{{M}^{T}}J \right)J{{{\bar{H}}}_{,x}} \\
& \frac{\partial \bar{H}}{\partial {{y}_{i}}}=\sum\limits_{k}^{{}}{\frac{\partial \bar{H}}{\partial {{x}_{k}}}\frac{\partial {{x}_{k}}}{\partial {{y}_{i}}}\Leftrightarrow {{{\bar{H}}}_{,y}}={{M}^{T}}{{{\bar{H}}}_{,x}}} \\
& \Rightarrow \dot{\bar{y}}=\left( {{J}^{-1}}{{M}^{T}}J \right)J{{\left( {{M}^{T}} \right)}^{-1}}{{{\bar{H}}}_{,y}}=-J\left( -1 \right){{M}^{T}}{{\left( {{M}^{T}} \right)}^{-1}}{{{\bar{H}}}_{,y}}=J{{{\bar{H}}}_{,y}} \\
\end{align}
TeX (checked):
{\begin{aligned}&{{\dot {y}}_{i}}=\sum \limits _{k}^{}{{\frac {\partial {{y}_{i}}}{\partial {{x}_{k}}}}{{\dot {x}}_{k}}}\Leftrightarrow {\dot {\bar {y}}}={{M}^{-1}}{\dot {\bar {x}}}=\left({{J}^{-1}}{{M}^{T}}J\right)J{{\bar {H}}_{,x}}\\&{\frac {\partial {\bar {H}}}{\partial {{y}_{i}}}}=\sum \limits _{k}^{}{{\frac {\partial {\bar {H}}}{\partial {{x}_{k}}}}{\frac {\partial {{x}_{k}}}{\partial {{y}_{i}}}}\Leftrightarrow {{\bar {H}}_{,y}}={{M}^{T}}{{\bar {H}}_{,x}}}\\&\Rightarrow {\dot {\bar {y}}}=\left({{J}^{-1}}{{M}^{T}}J\right)J{{\left({{M}^{T}}\right)}^{-1}}{{\bar {H}}_{,y}}=-J\left(-1\right){{M}^{T}}{{\left({{M}^{T}}\right)}^{-1}}{{\bar {H}}_{,y}}=J{{\bar {H}}_{,y}}\\\end{aligned}}
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data-mjx-texclass="OPEN">(</mo><msup><mi>M</mi><mrow data-mjx-texclass="ORD"><mi>T</mi></mrow></msup><mo data-mjx-texclass="CLOSE">)</mo></mrow><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mo>−</mo><mn>1</mn></mrow></mrow></msup><msub><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mover><mi>H</mi><mo>¯</mo></mover></mrow></mrow><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mo>,</mo><mi>y</mi></mrow></mrow></msub><mo>=</mo><mo>−</mo><mi>J</mi><mrow data-mjx-texclass="INNER"><mo data-mjx-texclass="OPEN">(</mo><mo>−</mo><mn>1</mn><mo data-mjx-texclass="CLOSE">)</mo></mrow><msup><mi>M</mi><mrow data-mjx-texclass="ORD"><mi>T</mi></mrow></msup><msup><mrow data-mjx-texclass="INNER"><mo data-mjx-texclass="OPEN">(</mo><msup><mi>M</mi><mrow data-mjx-texclass="ORD"><mi>T</mi></mrow></msup><mo data-mjx-texclass="CLOSE">)</mo></mrow><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mo>−</mo><mn>1</mn></mrow></mrow></msup><msub><mrow 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