Rather than solve the double pendulum problem with two masses in the usual way.
Instead express the coordinates of the second mass, in terms of the coordinates of the mass above it.
$ x2=x_1+\xi = L_1Sin[\theta]Cos[\phi]+L_2Sin[\alpha]Cos[\beta]$\\
$ y2=y_1+ \eta = L_1Sin[\theta]Sin[\phi]+L_2Sin[\alpha]Sin[\beta]$\\
$ z2=z_1-\xi = L_1-L_1Cos[\theta]-L_2Sin[\alpha]Cos[\beta]$
Wouldn't you suspect that the Lagrangian remain invariant? Is there a way to reparameterize these equations?