= Oscillator energy dissipation
{title2=$\dot E=-b\dot x^2$}
For the <damped harmonic oscillator>, the <mechanical energy> is the sum of <kinetic energy> and <spring> <potential energy>:
$$
E=\frac12m\dot x^2+\frac12kx^2,\qquad
\frac{dE}{dt}=\dot x(m\ddot x+kx)=-b\dot x^2\leq0.
$$
Thus $b\dot x^2$ is the <power> transferred out of the oscillator's <mechanical energy>. For $b>0$, the rate vanishes instantaneously at a turning point, despite nonzero stored <potential energy>; it is not a constant loss per unit <time>. In <eddy-current damping> the loss becomes <Joule heating>. <Oscillations and Waves by Richard Fitzpatrick/Damped Harmonic Oscillation>{ref}
For an <underdamped> response, advancing one <oscillation period> multiplies both <displacement> and <velocity> by $e^{-\zeta\omega_0T_{\mathrm d}}$, so $E(t+T_{\mathrm d})=e^{-2\zeta\omega_0T_{\mathrm d}}E(t)$. This exact cycle relation does not imply that instantaneous <energy> is a pure <exponential function> throughout each cycle.
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