Past exam of the mathematical methods I course of the University of Oxford / 2017 resit / 7 / c / Solution

The initial conditions in the underdamped solution give $A=x_0$ and $B=\gamma x_0/\Omega$. ConsequentlyThe exact mechanical energy is thereforeFor weak damping, $\gamma \ll {\omega }_0$, the bracket differs from one by a bounded relative correction of order $\gamma /{\omega }_0$. To leading order the energy and its half-life areThe exact identity $\dot{E}=-2m\gamma {\dot{x}}^2\le 0$ confirms dissipation. The small rapid variation of the exponential-normalized energy does not mean that total energy increases; the half-life formula is a weak-damping approximation, not an exact equality for arbitrary $\gamma$.