Home Work Three

Consider the FPUT chain with alternating masses $M>m$, with $m$ being mass of odd particle and $M$ being mass of even particle. Write the Hamiltonian, equations of motion and find the linear dispersion relationship $\omega(k)$:

$$\omega_{\pm}^2(q) = \frac{k(M+m) \pm \sqrt{k^2 (M+m)^2 - 4 k^2 M m \, \sin^2\left(\frac{q a}{2}\right)}}{M m}, \pm$$ (13)