A system of $N$ particles $i=1,2,3, \ldots, N$, with mass $m_{i}$, moves around a circle of radius $a$. The angle between the radius to particle $i$ and a fixed reference radius is $\theta_{i}$. The interaction potential for the system is

$$V=\frac{1}{2} k \sum_{j=1}^{N}\left(\theta_{j+1}-\theta_{j}\right)^{2}$$

where $k$ is a constant and $\theta_{N+1}=\theta_{1}+2 \pi$.

The Lagrangian for the system is

$$L=\frac{1}{2} a^{2} \sum_{j=1}^{N} m_{j} \dot{\theta}_{j}^{2}-V$$

Write down the equation of motion for particle $i$ and show that the system is in equilibrium when the particles are equally spaced around the circle.

Show further that the system always has a normal mode of oscillation with zero frequency. What is the form of the motion associated with this?

Find all the frequencies and modes of oscillation when $N=2, m_{1}=k m / a^{2}$ and $m_{2}=2 \mathrm{~km} / \mathrm{a}^{2}$, where $m$ is a constant.