Normal modes of a double pendulum with a two-to-one length ratio (${\omega }_{\pm }^2=g(3\pm \sqrt{5})/(2l)$)

For equal point masses and upper/lower massless rods of length $2l,l$, angular coordinates from the downward vertical give mass matrix $m{l}^2\left(\begin{array}{cc}8& 2\\ 2& 1\end{array}\right)$ and stiffness matrix $mgl\mathrm{diag}(4,1)$. The generalized eigenvalue problem reduces to $s^2-3s+1=0$, where $s={\omega }^{2}l/g$. The slow mode has angular-amplitude ratio ${\theta }_2/{\theta }_1=\sqrt{5}-1$, and the fast mode has ratio $-1-\sqrt{5}$.