I am following a derivation of the dynamical matrix given at http://physik.uni-graz.at/~pep/Lehre/PP/DynMat.pdf. Here T and W are kinetic and potential energies.

$T=\sum_{n\alpha i}\frac{M_{\alpha}}{2}\dot{s}_{n\alpha i}^{2}$

$W = \frac{1}{2}\sum_{n\alpha i}\sum_{n'\alpha' i'} \Phi_{n\alpha i}^{n'\alpha' i'} s_{n\alpha i} s_{n'\alpha' i'}$

What I get after plugging T and W to the Euler-Lagrange equation ( $\frac{d}{dt}\frac{\partial L}{\partial \dot{s}_{n\alpha i}} -\frac{\partial L}{\partial s_{n\alpha i}} = 0$ ) is,

$\sum_{n\alpha i}M_{\alpha}\ddot{s}_{n\alpha i} = -\sum_{n'\alpha' i'} \Phi_{n\alpha i}^{n'\alpha' i'} s_{n\alpha i} $

But in the derivation, he has got, $M_{\alpha}\ddot{s}_{n\alpha i} = -\sum_{n'\alpha' i'} \Phi_{n\alpha i}^{n'\alpha' i'} s_{n\alpha i} $.

I don't understand how he got rid of the summation sign on the left hand side.


The sum vanishes when you take the derivative, because only the term where the indices match the ones of the derivative remain. It becomes clear when you use different names for the variables.

$$\frac{\partial}{\partial \dot{s}_{n'\alpha 'i'}} \sum_{n\alpha i}\frac{M_{\alpha}}{2}\dot{s}_{n\alpha i}^{2} = M_{\alpha '}\dot{s}_{n'\alpha 'i'}$$


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