Past exam of the mathematical methods II course of the University of Oxford / 2013 / 8 / d / Solution

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.