"Appropriate" Hamiltonian function of simple pendulum

Question

Consider a simple pendulum of length $\ell$ and mass $m$, where the only force is gravity. If $\theta$ is the angle between the rod and the vertical direction, and $\xi$ is the coordinate along fibers of $T^*S^1$, then consider $H:T^*S^1\to\mathbb R$ where $$H(\theta,\xi)=\frac{\xi^2}{2m\ell^2}+m\ell(1-\cos\theta).$$ I'm trying to show that $H$ is "an appropriate Hamiltonian function to describe the spherical pendulum." My question is, what does this mean? I guess the corresponding Hamiltonian vector field would be $$X_H=\frac{\xi}{m\ell^2}\frac{\partial}{\partial\theta}+m\ell\sin\theta\frac{\partial}{\partial\xi},$$ but I have no idea what that would mean physicially. I searched for this online, but all the explanations seemed to assume that I know a lot more physics than I do (I don't know what a Lagrangian is, for example). This is from da Silva's book on symplectic geometry, by the way.