= Solution
For the constant small-oscillation matrices, the total <mechanical energy> satisfies
$$
E=\tfrac12\dot{\boldsymbol\theta}^{\mathsf T}\mathbf T\dot{\boldsymbol\theta}+\tfrac12\boldsymbol\theta^{\mathsf T}\mathbf V\boldsymbol\theta,\qquad \dot E=\dot{\boldsymbol\theta}^{\mathsf T}(\mathbf T\ddot{\boldsymbol\theta}+\mathbf V\boldsymbol\theta).
$$
The specified conservative pendulum has <Lagrangian> $L=K-U$, whose <Euler-Lagrange equations> give
$$
\boxed{\mathbf T\ddot{\boldsymbol\theta}+\mathbf V\boldsymbol\theta=0,}
$$
or $8\ddot\theta_1+2\ddot\theta_2+(4g/l)\theta_1=0$ and $2\ddot\theta_1+\ddot\theta_2+(g/l)\theta_2=0$. These equations make $\dot E=0$.
The scalar energy law alone is insufficient to infer two unique equations: <energy conservation does not uniquely determine equations of motion>. In particular, adding $\mathbf G\dot{\boldsymbol\theta}$ on the right with antisymmetric $\mathbf G$ changes the dynamics but contributes zero power. The conservative-force structure of the given pendulum excludes that extra gyroscopic force and supplies the information needed for the displayed matrix equations.
Back to article page
