Damped simple harmonic oscillator, phase space

Question

I want to calculate and draw the phase space trajectory of this damped harmonic oscillator:

$$\ddot{x}+\gamma\,\dot{x}+\omega^2x=0$$

for the two cases $\gamma=2\omega$ and $\gamma=\omega$.

I'm really stuck with this and have no idea what to do... I've found stuff on google about it but am struggling to follow.

Answer 1

A quick orientation can be gotten by using a WolframAlpha query.

In your case:

solve x'' + 2 w x' + w^2 x = 0 (link)

and

solve x'' + w x' + w^2 x = 0 (link)

It will give you algebraic solutions, which you could use for your $(x, \dot{x})$ plot with other software (mathematica example) (octave example).

And it provides some sample plots for some initial conditions it chooses which might give you an idea what is going on.

Answer 2

The standard approach to these problems is to introduce $y = \dot{x}$ and kinda regard $y$ as an independent variable. Upon introduction of $y$, you acquire the following system

\begin{align*} \frac{\mathrm{d}}{\mathrm{d} t} \begin{pmatrix} x \\ y \end{pmatrix} = \begin{pmatrix} 0 & 1 \\ - \omega^2 & - \gamma \end{pmatrix} \begin{pmatrix} x \\ y \end{pmatrix} \end{align*}

You solve this by finding the eigenvalues and eigenvectors of the matrix. Assuming that we can find $2$ eigenvectors $v_1,v_2$ with eigenvalues $\lambda_1,\lambda_2$, the solution is given by

\begin{align*} \begin{pmatrix} x \\ y \end{pmatrix} = c_1 v_1 \mathrm{e}^{\lambda_1 t} + c_2 v_2 \mathrm{e}^{\lambda_2 t} \end{align*}

where $c_1$ and $c_2$ follow from the initial condition(s). To draw the phase space trajectory, simply plot $y$ vs $x$ for all $t$.

Now, the problem gets more complicated when the matrix does not have $2$ eigenvectors. If it has only one eigenvector $v_1$ with eigenvalue $\lambda_1$, you look for a second solution of the form $c_2(t) v_1 \mathrm{e}^{\lambda_1 t}$ with $c_2$ a function of $t$.