You are asked to recreate and then go beyond Figure 1 of the FPU article by writing and running your own simulation code. Your work is to be written up and submitted. This project has two parts. First, you generate the eigenvalues and eigenvectors for the linear system. This process is simple, because the exact solution is known. All you need to do is to normalize the eigenvectors. Second, you are asked to simulate the motion of the chain with a quadratic term in the restoring force - Eq (1) of the FPU article [but please note a misprint: the lhs of the equation should read ]. This exercise involves solving a coupled system of ordinary differential equations, which can be done using the ``leapfrog'' method and periodically measuring the energy in a few of the lowest normal modes, which you constructed by solving the linear system. In the process you determine the mode energies of the system, i.e the contributions to the total energy attributed to each of the eigenmodes of the linear problem. The result of this simulation will be a set of graphs of mode energy vs simulation time.