there we have the EOM: \begin{align*} \alpha q_{2} + \lambda - \ddot{q}_1=0 \\ \alpha q_{1} + \lambda - \ddot{q}_2=0 \end{align*}
and $q_{i}$ is the canonical coordinates. Can I use the Fourier transform to solve it? and how?
I want to solve the EOM like the coupled Hamonic oscillators by using Fourier transform like
$$q_{i} (t) =\frac{1}{\sqrt{N}}\int \tilde{q}_{i}(\omega)e^{i \omega t}d \omega $$ and $$ -\frac{1}{\sqrt{N}}\int \omega^{2}\tilde{q}_{1}(\omega)e^{-i \omega t}d \omega = \alpha \frac{1}{\sqrt{N}}\int \tilde{q}_{2}(\omega')e^{i \omega^{'} t}d \omega^{'} + \lambda.$$ $$ -\frac{1}{\sqrt{N}}\int {\omega^{\prime}}^{2} \tilde{q}_{2}(\omega^{'})e^{-i \omega^{\prime} t}d \omega^{\prime} = \alpha \frac{1}{\sqrt{N}}\int \tilde{q}_{1}(\omega)e^{i \omega t}d \omega + \lambda.$$ How to solve this equation?
