= Energy decay of a viscously damped harmonic oscillator
{title2=$\dot E=-m\gamma\dot y^2,\quad\overline E\simeq E_0e^{-\gamma t}$}
For a <damped harmonic oscillator> $\ddot y+\gamma\dot y+\omega_0^2y=0$, differentiation of the <mechanical energy> $E=m(\dot y^2+\omega_0^2y^2)/2$ gives $\dot E=-m\gamma\dot y^2$. This identity is exact. In the weak-damping regime $\gamma\ll\omega_0$, the leading <period average> has equal <kinetic energy> and <potential energy>, so $\overline{\dot y^2}\simeq\overline E/m$ and $\dot{\overline E}\simeq-\gamma\overline E$. Thus the averaged <energy> has <exponential decay>, while the instantaneous <energy> has a small oscillatory correction. Its derivative vanishes at every turning point, ruling out an exact pointwise nonconstant exponential. The <quality factor> is $Q\simeq\omega_0/\gamma$ in this approximation.
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