Home Work Three

Consider the $\alpha$-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 $\omega(k)$:
  $$\omega_{\pm}^2(q) = \frac{k(M+m) \pm \sqrt{k^2 (M+m)^2 - 4 k^2 M m \, \sin^2\left(\frac{q a}{2}\right)}}{M m}$$
  , with $\pm$ for optical/acoustic branches.
• Translate the Hamiltonian to the Fourier space
• Introduce Normal Variables $a_k$ and $b_k$ as we did in class and write the Hamiltonian in terms of $a_k$, $b_k$. Quadratic part of the Hamiltonian should be $\sum\omega_k \vert a_k\vert^2$ and $\sum \Omega_k \vert b_k\vert^2$. The nonlinear part of the Hamiltonian should have cubic products of $a$ and $b$.

Contact me if you have questions