 Solution

Retain terms quadratic in the small displacements and velocities. The approximations $1-\cos \theta \simeq {\theta }^2/2$ and $\cos ({\theta }_1-{\theta }_2)\simeq 1$ yieldThus the correct quadratic forms areSource correction. The printed kinetic-energy matrix formula omits both time-derivative dots. It cannot hold as a function of angles alone: at $\mathbf{\theta }=0$, ${\dot{\theta }}_1\ne 0$, ${\dot{\theta }}_2=0$, the actual kinetic energy is $4m{l}^2{\dot{\theta }}_1^2>0$, whereas the printed undifferentiated quadratic form vanishes.