Exercise 5.6: $N$ coupled oscillators

1. Choose $k_{c}=k=1$ and $N=10$. Find the normal modes by applying an external force to one of the particles and determine the resonant frequencies. Drive the system for several periods of the external force and compute the steady state amplitude of the displacement of each particle for each value of $\omega$. Try values of $\omega$ in the range $0.2(k/m)^{1/2}$ to $3(k/m)^{1/2}$. If you think that you are close to a resonance, use several other values of $\omega$ to obtain a better estimate. How many normal modes are there?

2. Compare your results in part 1 with the analytic result
   
   $$\omega _{n}^{2}=\frac{4k}{m}\sin ^{2}\frac{n\pi }{2(N+1)},$$
   
   
   where $N$ is the number of particles and the mode index is $n=1,2,...,N$.