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fluidistic
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Homework Statement
Basically I found the following system of DE's:
[itex]\frac{dx}{dt}=y[/itex]
[itex]\frac{dy}{dt}=-\frac{g}{l} \sin x - \frac{cy}{ml}[/itex]. (Damped pendulum)
I'm asked to analize the stability of the critical points [itex]x=0[/itex], [itex]y=0[/itex] and [itex]x=\pi[/itex], [itex]y=0[/itex].
Using intuition the first point is asymptotically stable while the second point is unstable.
Homework Equations
I tried to put the system under matrix form and then check out the eigenvalues of a matrix but I have some problems.
The Attempt at a Solution
[itex]\begin {bmatrix} \frac{dx}{dt} \\ \frac{dy}{dt} \end{bmatrix} =A \begin {bmatrix} x \\ y \end{bmatrix}[/itex] where [itex]A= \begin {bmatrix} a& b \\ c & d \end{bmatrix}[/itex].
I tried to determine the elements of the matrix A but was unable to perform this without making the assumption that [itex]\sin x \approx x[/itex] (for small x or x close to pi I guess?). Making that assumption I reached [itex]A\approx \begin {bmatrix} 0 &1 \\ -\frac{g}{l} & -\frac{c}{ml} \end{bmatrix}[/itex]. I found the eigenvalues to be [itex]\lambda _1 =- \frac{c}{2ml} + \sqrt {\frac{c^2}{ml^2} - \frac{4g}{l}}[/itex] and [itex]\lambda _2 =- \frac{c}{2ml} - \sqrt {\frac{c^2}{ml^2} - \frac{4g}{l}}[/itex].
So that I have a bunch of possible cases which doesn't look good at all to me. Furthermore I never used to information that the critical points are [itex](0,0)[/itex] and [itex](\pi , 0 )[/itex] yet.
Can someone make some comments so far? Thanks in advance.