Consider a system with Hamiltonian:

$$H(p, q)=\frac{1}{2}p^2+1-\cos q$$

The corresponding system of equations is $\dot{q}=p, \dot{p}=-\sin q$. For this system, the critical point $(0,0)$ is a stable center but not asymptotically stable since the Hamiltonian is a constant. Now, we add energy dissipation so that the system becomes

$\dot{q}=p$,

$\dot{p}=-p-\sin q$.

We expect that $(0,0)$ becomes asymptotically stable. Which one of the following statements is correct for showing that $(0,0)$ is asymptotically stable?