Problem 1. Normal Modes in One and Two Dimensions The first configuration is confined to one dimensions, the second and third are confined to two dimen- sions, answer the following questions: 1. How many degrees of freedom are there in total? How many translational, rotational and vibra- tional? 2. Write the Hamiltonian. Assume a nearest neighbor harmonic interaction, with force constants k as indicated on the figures. 3. Define mass weighted coordinates and calculate the Hessian. 4. Diagonalize the Hessian to find the force constants ; and coordinates Qi of the normal modes. Indicate which modes are translational, rotational and vibrational. k m m m k k k k k k m 0.5m m m k m m k m