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For the constant small-oscillation matrices, the total mechanical energy satisfiesThe specified conservative pendulum has Lagrangian $L=K-U$, whose Euler-Lagrange equations giveor $8{\ddot{\theta }}_1+2{\ddot{\theta }}_2+(4g/l){\theta }_1=0$ and $2{\ddot{\theta }}_1+{\ddot{\theta }}_2+(g/l){\theta }_2=0$. 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 $\mathbf{G}\dot{\mathbf{\theta }}$ 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.

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.