Three particles, each of mass $m$, move along a straight line. Their positions on the line containing the origin, $O$, are $x_{1}, x_{2}$ and $x_{3}$. They are subject to forces derived from the potential energy function

$$V=\frac{1}{2} m \Omega^{2}\left[\left(x_{1}-x_{2}\right)^{2}+\left(x_{2}-x_{3}\right)^{2}+\left(x_{3}-x_{1}\right)^{2}+x_{1}^{2}+x_{2}^{2}+x_{3}^{2}\right]$$

Obtain Lagrange's equations for the system, and show that the frequency, $\omega$, of a normal mode satisfies

$$f^{3}-9 f^{2}+24 f-16=0$$

where $f=\left(\omega^{2} / \Omega^{2}\right)$. Find a complete set of normal modes for the system, and draw a diagram indicating the nature of the corresponding motions.