3. Consider a system with $T = \frac{1}{2}(\dot{q}_1^2 + \dot{q}_2^2)$, $V = f(q_1 - q_2)$, where $f$ is yet unspecified. Choose a new set of generalized coordinates, $q_1$, $q_2$, to reduce the equations of motion to an integral involving $f$. Find $q_1(t)$, $q_2(t)$ for general initial conditions if $f(x) = x^2$. (Hint: Choose, say, $q_2$, so that $\frac{\partial \mathcal{L}(q_i, \dot{q}_i)}{\partial q_2} = 0$.)
