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[QUE/CM-03010]

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 Consider $N$- particle system with Lagrangian $$ L = \sum_{i=1}^N {1\over 2} m_i \dot{\vec{r}}^{\,2} - {1\over 2} \sum_{i,j} V_{ij}(|\vec{r}_i - \vec{r}_j|)  $$ Show that the Lagrangian is invariant $$  \vec{r} \rightarrow \vec{r}^{\,\prime}=\vec{r}-(\delta \vec{v})\,t$$ up to a total time derivative. Use the results of the previous problem to derive a conservation law and hence show that the center of mass of the  system moves like a free particle.


 

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