Energy dissipation of a damped pendulum ($\dot{E}=-c{\dot{\theta }}^2$)

For the normalized damped pendulum, $E={\dot{\theta }}^2/2+1-\cos \theta$ obeys $\dot{E}=-c{\dot{\theta }}^2\le 0$. Its mechanical energy is constant only along an equilibrium trajectory, and strictly decreases over every positive-length interval on a nonstationary trajectory. At an individual turning point its derivative vanishes. This excludes nonconstant periodic trajectories despite oscillatory approach to a stable focus.