A system of three particles of equal mass $m$ moves along the $x$ axis with $x_{i}$ denoting the $x$ coordinate of particle $i$. There is an equilibrium configuration for which $x_{1}=0$, $x_{2}=a$ and $x_{3}=2 a$.

Particles 1 and 2, and particles 2 and 3, are connected by springs with spring constant $\mu$ that provide restoring forces when the respective particle separations deviate from their equilibrium values. In addition, particle 1 is connected to the origin by a spring with spring constant $16 \mu / 3$. The Lagrangian for the system is

$$L=\frac{m}{2}\left(\dot{x}_{1}^{2}+\dot{\eta}_{1}^{2}+\dot{\eta}_{2}^{2}\right)-\frac{\mu}{2}\left(\frac{16}{3} x_{1}^{2}+\left(\eta_{1}-x_{1}\right)^{2}+\left(\eta_{2}-\eta_{1}\right)^{2}\right)$$

where the generalized coordinates are $x_{1}, \eta_{1}=x_{2}-a$ and $\eta_{2}=x_{3}-2 a$.

Write down the equations of motion. Show that the generalized coordinates can oscillate with a period $P=2 \pi / \omega$, where

$$\omega^{2}=\frac{\mu}{3 m}$$

and find the form of the corresponding normal mode in this case.