Exercise 5.5: three coupled oscillators

1. Run the program with $N=3$, $k_{c}=0.2$ and $k=1$ and arbitrary but nonzero initial displacements. Describe the time-dependence of the motion of the particles.

2. Consider the following initial conditions: ${u_{1}=u_{2}=u_{3}=0.5}$, ${u_{1}=0.5,u_{2}=-0.5,u_{3}=0.5}$, ${u_{1}=0.5,u_{2}=0,u_{3}=-0.5}$ (all the velocities are initially zero) If these initial conditions correspond to normal modes, determine the normal modes frequencies.

3. Add an external driving force to particle 1 and determine the normal mode frequencies. Compare your results with the frequencies you obtained in part 2. How many normal modes are there?