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Given the equation $x''+2 \beta x'+ \omega^2 x=0$ for a damped oscillator, I can get to the equation of motion x(t) and deriving it with respect to time: $x'(t)$.
I am asked to plot the phase plane $\{x,x'\}$, considering the 3 cases: $\beta=0$, $\beta << \omega$, $\omega << \beta$.
I have plotted the phase diagram for an undamped harmonic oscillator ($x''+\omega^2 x=0$) which consisted of ellipses but I can´t find a formula linking $x$ and $x'$.$\\$ I have also tried using the Hamiltonian but arrived nowhere.

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  • $\begingroup$ Concerning Lagrangian & Hamiltonian formulations, see e.g. this Phys.SE post. $\endgroup$ – Qmechanic Dec 16 '20 at 17:46
  • $\begingroup$ What about the answer looks problematic to you? $\endgroup$ – Cosmas Zachos Dec 18 '20 at 15:05
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$x(t)$ is a function of time, so is $x'(t)$. You should be able to use time to relate the two.

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