= Energy ratio between equal-phase damped crossings
{title2=$E(t+T)/E(t)=e^{-4\pi\gamma/\omega_d}$}
A linearly damped oscillator $m\ddot x+b\dot x+kx=0$ has $x=Ae^{-\gamma t}\cos(\omega_dt+\phi)$, with $\gamma=b/(2m)$ and $\omega_d^2=k/m-\gamma^2>0$. After one damped period $T=2\pi/\omega_d$, both displacement and <velocity> are multiplied by $e^{-\gamma T}$. Their quadratic <mechanical energy> is therefore multiplied exactly by $e^{-2\gamma T}$, regardless of the starting <phase>. In particular successive same-direction equilibrium crossings have this ratio even though <potential energy> is zero at each. A cosine <phase> maximum is not exactly a displacement turning point when <damping> is nonzero.
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