In the text we saw an example involving a non-diagonal mass matrix arising in the case of a single particle. In this problem, we will look at a similar scenario for two particles. Consider two interacting particles of mass \(m_{1}\) and \(m_{2}\) constrained to move in one dimension described by the Lagrangian

\[L=\frac{1}{2} m_{1} \dot{q}_{1}^{2}+\frac{1}{2} m_{2} \dot{q}_{2}^{2}-U\left(q_{1}+q_{2}\right)\]

The coordinates of the two particles are represented by \(q_{1}\) and \(q_{2}\) and the potential energy function is given by \(U(Q)=\alpha Q^{2} / 2\) for some constant \(\alpha\). The novelty here is that the potential between the two particles is not translationally invariant; it does not depend on the distance between the particles, \(q \equiv q_{1}-q_{2}\). Instead, the potential depends on the sum of the two coordinates \(Q \equiv q_{1}+q_{2}\). As a result, the usual coordinate transformation from \(q_{1}\) and \(q_{2}\) to the center of mass coordinate \(Q_{c m}=\left(m_{1} q_{1}+m_{2} q_{2}\right) /\left(m_{1}+m_{2}\right)\) and the relative distance \(q=q_{1}-q_{2}\) is not very useful. Instead, we want to transform to \(Q=q_{1}+q_{2}\) and \(q=q_{1}-q_{2}\).

(a) Show that the Lagrangian in the \(Q\) and \(q\) coordinates takes the form

\[L=\frac{1}{8} M \dot{Q}^{2}+\frac{1}{8} M \dot{q}^{2}+\frac{1}{4} m \dot{Q} \dot{q}-U(Q)\]

where \(M \equiv m_{1}+m_{2}\) and \(m \equiv m_{1}-m_{2}\).

(b) Find the eigenvalues and eigenvectors of small oscillations. Elaborate briefly on the meaning of each normal mode.