# Derivation of the dynamical matrix

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.

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