[QUE/CM-03006]

For a simple harmonic oscillator with Lagrangian $$ L = {1\over 2} m \dot{q}^2 - {1\over 2} m \omega^2 q^2  $$ show that the transformation $$ q \rightarrow q^\prime = q+ \epsilon \sin \omega t $$ is a symmetry transformation. Find the conserved quantity associated with this symmetry transformation. Using equations of motion verify explicitly that this quantity is indeed conserved.