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[QUE/CM-03004] Symmetry Transformation and Runge Lenz Vector

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Show that the transformation 
$$ \vec{r} \rightarrow \vec{r}^\prime = \vec{r} - m [ 2\dot{\vec{r}}        (\vec{\epsilon}\cdot\vec{r}) - \vec{r}(\vec{\epsilon}\cdot\dot{\vec{r}})  (\vec{r}.\dot{\vec{r}}) \vec{\epsilon} ]  $$

are symmetry transformations for a particle of mass $m$, moving in Coulomb potential $V(r)= -k/r$. Show that this symmetry implies conservation of  Runge Lenz vector $$ \vec{N} = \vec{p}\times \vec{L} - k\, m\, {\vec{r}\over r}  $$


 

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4727:Diamond Point

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