A system with a single degree of freedom is governed by the Lagrangian \(L=e^{\beta t}\left(\frac{1}{2} m \dot{q}^{2}-\right.\) \(\frac{1}{2} k q^{2}\).

(a) Write down the equation of motion. What sort of motion does it describe?

(b) Show that the explicit time dependence of \(L\) can be removed by introducing a new coordinate variable \(Q=e^{\beta t / 2} q\).

(c) Construct the first integral whose conservation is implied by this property.