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Display information for equation id:math.1411.69 on revision:1411
* Page found: Symmetrien und Erhaltungsgrößen (eq math.1411.69)
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Hash: 58ed4ffc43577d88e59cce86bccb7f45
TeX (original user input):
{{p}_{1}}=\frac{\partial L}{\partial {{{\dot{q}}}_{1}}}=\frac{\partial L}{\partial {{{\dot{q}}}_{1}}}=\frac{1}{2}\sum\limits_{i}{{{m}_{i}}\frac{\partial }{\partial {{{\dot{q}}}_{1}}}{{{\dot{\bar{r}}}}_{i}}^{2}=}\sum\limits_{i}{{{m}_{i}}{{{\dot{\bar{r}}}}_{i}}\frac{\partial }{\partial {{{\dot{q}}}_{1}}}{{{\dot{\bar{r}}}}_{i}}=\sum\limits_{i}{{{m}_{i}}{{{\dot{\bar{r}}}}_{i}}\left( {{{\bar{r}}}_{i}}\times {{{\bar{e}}}_{z}} \right)}}=-{{\bar{e}}_{z}}\sum\limits_{i}{\left( {{{\bar{r}}}_{i}}\times {{m}_{i}}{{{\dot{\bar{r}}}}_{i}} \right)}=-{{l}_{z}}
TeX (checked):
{{p}_{1}}={\frac {\partial L}{\partial {{\dot {q}}_{1}}}}={\frac {\partial L}{\partial {{\dot {q}}_{1}}}}={\frac {1}{2}}\sum \limits _{i}{{{m}_{i}}{\frac {\partial }{\partial {{\dot {q}}_{1}}}}{{\dot {\bar {r}}}_{i}}^{2}=}\sum \limits _{i}{{{m}_{i}}{{\dot {\bar {r}}}_{i}}{\frac {\partial }{\partial {{\dot {q}}_{1}}}}{{\dot {\bar {r}}}_{i}}=\sum \limits _{i}{{{m}_{i}}{{\dot {\bar {r}}}_{i}}\left({{\bar {r}}_{i}}\times {{\bar {e}}_{z}}\right)}}=-{{\bar {e}}_{z}}\sum \limits _{i}{\left({{\bar {r}}_{i}}\times {{m}_{i}}{{\dot {\bar {r}}}_{i}}\right)}=-{{l}_{z}}
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<math class="mwe-math-element" xmlns="http://www.w3.org/1998/Math/MathML"><mrow data-mjx-texclass="ORD"><mstyle displaystyle="true" scriptlevel="0"><msub><mi>p</mi><mrow data-mjx-texclass="ORD"><mn>1</mn></mrow></msub></mstyle><mo>=</mo><mrow data-mjx-texclass="ORD"><mfrac><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mi>∂</mi><mi>L</mi></mrow></mrow><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mi>∂</mi><msub><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mover><mi>q</mi><mo>˙</mo></mover></mrow></mrow><mrow data-mjx-texclass="ORD"><mn>1</mn></mrow></msub></mrow></mrow></mfrac></mrow><mo>=</mo><mrow data-mjx-texclass="ORD"><mfrac><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mi>∂</mi><mi>L</mi></mrow></mrow><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mi>∂</mi><msub><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mover><mi>q</mi><mo>˙</mo></mover></mrow></mrow><mrow data-mjx-texclass="ORD"><mn>1</mn></mrow></msub></mrow></mrow></mfrac></mrow><mo>=</mo><mrow data-mjx-texclass="ORD"><mfrac><mrow data-mjx-texclass="ORD"><mn>1</mn></mrow><mrow data-mjx-texclass="ORD"><mn>2</mn></mrow></mfrac></mrow><munder><mo form="prefix" texclass="OP">∑</mo><mrow data-mjx-texclass="ORD"><mi>i</mi></mrow></munder><mrow data-mjx-texclass="ORD"><msub><mi>m</mi><mrow data-mjx-texclass="ORD"><mi>i</mi></mrow></msub><mrow data-mjx-texclass="ORD"><mfrac><mrow data-mjx-texclass="ORD"><mi>∂</mi></mrow><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mi>∂</mi><msub><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mover><mi>q</mi><mo>˙</mo></mover></mrow></mrow><mrow data-mjx-texclass="ORD"><mn>1</mn></mrow></msub></mrow></mrow></mfrac></mrow><msup><msub><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mover><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mover><mi>r</mi><mo>¯</mo></mover></mrow></mrow><mo>˙</mo></mover></mrow></mrow><mrow data-mjx-texclass="ORD"><mi>i</mi></mrow></msub><mrow data-mjx-texclass="ORD"><mn>2</mn></mrow></msup><mo>=</mo></mrow><munder><mo form="prefix" texclass="OP">∑</mo><mrow data-mjx-texclass="ORD"><mi>i</mi></mrow></munder><mrow data-mjx-texclass="ORD"><msub><mi>m</mi><mrow data-mjx-texclass="ORD"><mi>i</mi></mrow></msub><msub><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mover><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mover><mi>r</mi><mo>¯</mo></mover></mrow></mrow><mo>˙</mo></mover></mrow></mrow><mrow data-mjx-texclass="ORD"><mi>i</mi></mrow></msub><mrow data-mjx-texclass="ORD"><mfrac><mrow data-mjx-texclass="ORD"><mi>∂</mi></mrow><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mi>∂</mi><msub><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mover><mi>q</mi><mo>˙</mo></mover></mrow></mrow><mrow data-mjx-texclass="ORD"><mn>1</mn></mrow></msub></mrow></mrow></mfrac></mrow><msub><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mover><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mover><mi>r</mi><mo>¯</mo></mover></mrow></mrow><mo>˙</mo></mover></mrow></mrow><mrow data-mjx-texclass="ORD"><mi>i</mi></mrow></msub><mo>=</mo><munder><mo form="prefix" texclass="OP">∑</mo><mrow data-mjx-texclass="ORD"><mi>i</mi></mrow></munder><mrow data-mjx-texclass="ORD"><msub><mi>m</mi><mrow data-mjx-texclass="ORD"><mi>i</mi></mrow></msub><msub><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mover><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mover><mi>r</mi><mo>¯</mo></mover></mrow></mrow><mo>˙</mo></mover></mrow></mrow><mrow data-mjx-texclass="ORD"><mi>i</mi></mrow></msub><mrow data-mjx-texclass="INNER"><mo data-mjx-texclass="OPEN">(</mo><msub><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mover><mi>r</mi><mo>¯</mo></mover></mrow></mrow><mrow data-mjx-texclass="ORD"><mi>i</mi></mrow></msub><mo>×</mo><msub><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mover><mi>e</mi><mo>¯</mo></mover></mrow></mrow><mrow data-mjx-texclass="ORD"><mi>z</mi></mrow></msub><mo data-mjx-texclass="CLOSE">)</mo></mrow></mrow></mrow><mo>=</mo><mo>−</mo><msub><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mover><mi>e</mi><mo>¯</mo></mover></mrow></mrow><mrow data-mjx-texclass="ORD"><mi>z</mi></mrow></msub><munder><mo form="prefix" texclass="OP">∑</mo><mrow data-mjx-texclass="ORD"><mi>i</mi></mrow></munder><mrow data-mjx-texclass="INNER"><mo data-mjx-texclass="OPEN">(</mo><msub><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mover><mi>r</mi><mo>¯</mo></mover></mrow></mrow><mrow data-mjx-texclass="ORD"><mi>i</mi></mrow></msub><mo>×</mo><msub><mi>m</mi><mrow data-mjx-texclass="ORD"><mi>i</mi></mrow></msub><msub><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mover><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mover><mi>r</mi><mo>¯</mo></mover></mrow></mrow><mo>˙</mo></mover></mrow></mrow><mrow data-mjx-texclass="ORD"><mi>i</mi></mrow></msub><mo data-mjx-texclass="CLOSE">)</mo></mrow><mo>=</mo><mo>−</mo><msub><mi>l</mi><mrow data-mjx-texclass="ORD"><mi>z</mi></mrow></msub></mrow></math>
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