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Display information for equation id:math.1410.62 on revision:1410
* Page found: Symmetrien und Erhaltungsgrößen (eq math.1410.62)
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Hash: 22fb6e596eb2116e9355f6da179fdb95
TeX (original user input):
I(\bar{r},\dot{\bar{r}})=\sum\limits_{i=1}^{N}{{}}\frac{\partial L}{\partial {{{\dot{\bar{r}}}}_{i}}}\cdot {{\left( \frac{d{{h}^{s}}}{ds} \right)}_{s=0}}=\sum\limits_{i}{{{m}_{i}}{{{\dot{\bar{r}}}}_{i}}\cdot \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)}=-{{\bar{e}}_{z}}\bar{l}=-{{l}_{z}}
TeX (checked):
I({\bar {r}},{\dot {\bar {r}}})=\sum \limits _{i=1}^{N}{}{\frac {\partial L}{\partial {{\dot {\bar {r}}}_{i}}}}\cdot {{\left({\frac {d{{h}^{s}}}{ds}}\right)}_{s=0}}=\sum \limits _{i}{{{m}_{i}}{{\dot {\bar {r}}}_{i}}\cdot \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)}=-{{\bar {e}}_{z}}{\bar {l}}=-{{l}_{z}}
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MathML (4.083 KB / 501 B) :
<math class="mwe-math-element" xmlns="http://www.w3.org/1998/Math/MathML"><mrow data-mjx-texclass="ORD"><mstyle displaystyle="true" scriptlevel="0"><mi>I</mi><mo stretchy="false">(</mo><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mover><mi>r</mi><mo>¯</mo></mover></mrow></mrow><mo>,</mo><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><mo stretchy="false">)</mo><mo>=</mo><munderover><mo form="prefix" texclass="OP">∑</mo><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mi>i</mi><mo>=</mo><mn>1</mn></mrow></mrow><mrow data-mjx-texclass="ORD"><mi>N</mi></mrow></munderover><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><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></mrow></mfrac></mrow><mo>⋅</mo><msub><mrow data-mjx-texclass="INNER"><mo data-mjx-texclass="OPEN">(</mo><mrow data-mjx-texclass="ORD"><mfrac><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mi>d</mi><msup><mi>h</mi><mrow data-mjx-texclass="ORD"><mi>s</mi></mrow></msup></mrow></mrow><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mi>d</mi><mi>s</mi></mrow></mrow></mfrac></mrow><mo data-mjx-texclass="CLOSE">)</mo></mrow><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mi>s</mi><mo>=</mo><mn>0</mn></mrow></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><mo>⋅</mo><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><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><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><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mover><mi>l</mi><mo>¯</mo></mover></mrow></mrow><mo>=</mo><mo>−</mo><msub><mi>l</mi><mrow data-mjx-texclass="ORD"><mi>z</mi></mrow></msub></mstyle></mrow></math>
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