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.