A linear molecule is modelled as four equal masses connected by three equal springs. Using the Cartesian coordinates $x_{1}, x_{2}, x_{3}, x_{4}$ of the centres of the four masses, and neglecting any forces other than those due to the springs, write down the Lagrangian of the system describing longitudinal motions of the molecule.

Rewrite and simplify the Lagrangian in terms of the generalized coordinates

$$q_{1}=\frac{x_{1}+x_{4}}{2}, \quad q_{2}=\frac{x_{2}+x_{3}}{2}, \quad q_{3}=\frac{x_{1}-x_{4}}{2}, \quad q_{4}=\frac{x_{2}-x_{3}}{2}$$

Deduce Lagrange's equations for $q_{1}, q_{2}, q_{3}, q_{4}$. Hence find the normal modes of the system and their angular frequencies, treating separately the symmetric and antisymmetric modes of oscillation.