 Solution

The quadratic total mechanical energy isFor this frictionless conservative double pendulum, the Euler-Lagrange equations for $L=K-U$ giveor $8{\ddot{\theta }}_1+2{\ddot{\theta }}_2+(4g/l){\theta }_1=0$ and $2{\ddot{\theta }}_1+{\ddot{\theta }}_2+(g/l){\theta }_2=0$. Substitution immediately verifies energy conservation.

Logical qualification. Energy conservation does not uniquely determine equations of motion: the scalar equation $dE/dt=0$ alone does not determine both equations. For example, replacing their right side by $\mathbf{G}\dot{\mathbf{\theta }}$ with any nonzero antisymmetric matrix $\mathbf{G}$ preserves the same energy because ${\dot{\mathbf{\theta }}}^{\mathsf{T}}\mathbf{G}\dot{\mathbf{\theta }}=0$, while changing the dynamics. Such a gyroscopic force is absent from the specified pendulum. The conservative-force structure, supplied here by the Lagrangian, is the additional information needed to justify the displayed equations rather than infer two independent equations from one scalar identity.