= Turning points of an underdamped release from rest
{title2=$t_n=n\pi/\omega_d,\quad x(t_n)=(-1)^n x_0e^{-\beta n\pi/\omega_d}$}
For $\ddot x+2\beta\dot x+\omega_0^2x=0$ with $0<\beta<\omega_0$, put $\omega_d=\sqrt{\omega_0^2-\beta^2}$. The <initial conditions> $x(0)=x_0$ and $\dot x(0)=0$ give
$$
x(t)=x_0e^{-\beta t}\left[\cos(\omega_dt)+\frac\beta{\omega_d}\sin(\omega_dt)\right],\qquad
\dot x(t)=-\frac{x_0\omega_0^2}{\omega_d}e^{-\beta t}\sin(\omega_dt).
$$
Thus the turning points have the times and displacements in the title. For release from $x_0<0$, the first positive maximum is $|x_0|e^{-\beta\pi/\omega_d}$, sufficient to test whether the body reaches a fixed positive stop before that first maximum. Every later positive maximum is smaller.
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