Why is the function $H(I, \varphi) = -\epsilon \varphi$ not globally defined on $\mathbb{R} \times \mathbb{S}^{1}$ ?

Question

I am reading a research paper concerning Hamiltonian dynamics. It is remarked that the Hamiltonian " $H(I, \varphi) = -\epsilon \varphi$ is not globally defined on $\mathbb{R} \times \mathbb{S}^{1}$ ." Here it is meant that $I \in \mathbb{R}$, $\varphi \in \mathbb{S}^{1}$ and $\epsilon$ is a small positive parameter (but that's not important, just treat it as some constant).

I don't understand why the function $H$ is not globally defined. This implies it should be multivalued on the cylinder, but how does the periodicity in $\varphi$ give rise to the multiple possible values of $H$?

Answer 1

You should have, for example, $H(I,0)=H(I,2\pi)$, but this doesn't happen.