= Normal modes of a double pendulum with a two-to-one length ratio
{title2=$\omega_\pm^2=g(3\pm\sqrt5)/(2l)$}
For equal point masses and upper/lower massless rods of length $2l,l$, angular coordinates from the downward vertical give mass matrix $ml^2\begin{pmatrix}8&2\\2&1\end{pmatrix}$ and stiffness matrix $mgl\operatorname{diag}(4,1)$. The <generalized eigenvalue problem> reduces to $s^2-3s+1=0$, where $s=\omega^2l/g$. The slow mode has angular-amplitude ratio $\theta_2/\theta_1=\sqrt5-1$, and the fast mode has ratio $-1-\sqrt5$.
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