I have a one-dimensional chain of identical atoms with mass m. There are "springs" between all pairs of atoms. Thus, the elastic energy is
$$ U = \frac{1}{2}\sum_j\sum_{l > 0} K_l (u_j - u_{j+l})^2, $$
where $K_j$ is spring constant and $u_j$ is the displacement of $j$-th atom. I need to find the equation of motion
$$ m\ddot{u}_j = -\frac{\partial U}{\partial u_j}. $$
So far I managed to get the following equation
$$ m\ddot{u}_j = \sum_{n = 1}^{j - 1} K_n(u_n - u_j) -K_j \sum_{l>0} (u_j - u_{j + l}), $$ but this does not seem right because later I need to find a dispersion relation and it looks like it will depend on $j$.
Edit: To simplify thing, let's assume periodic boundary conditions $u_{N +1} = u_1$
So, as @Sal mentioned, the right solution is
$$ m\ddot{u}_j = \sum_{l > 0}K_l(u_{j-l} - 2u_{j} + u_{j+l}) $$