Simple pendulum ($\ddot{\theta }+(g/\ell )\sin \theta =0$)

A point mass constrained to a fixed-length massless arm under uniform gravitational acceleration is a simple pendulum. With no drag its angular equation is $\ddot{\theta }+(g/\ell )\sin \theta =0$. The small-angle approximation yields simple harmonic motion, whereas the full restoring force is nonlinear. Adding drag gives a damped pendulum.

In units where the small-amplitude natural frequency is one, linear velocity drag gives $\ddot{\theta }+c\dot{\theta }+\sin \theta =0$ with $c>0$. The even multiples of $\pi$ are attracting equilibria; the odd multiples are saddle equilibria. Small displacements obey a damped harmonic oscillator; the energy dissipation of a damped pendulum governs the global phase portrait.

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.