Are Hamiltonians proper maps?

Question

In the context of classical mechanics:

Which assumption on a Hamiltonian function $H : T^* \mathcal M \to \mathbb R$ is most reasonable to assure its phase flow $\varphi_t : T^* \mathcal M \to T^* \mathcal M$ maps compact sets to bounded sets?

I want to find a condition which holds for most reasonable systems.

Details about why proper might do the trick:

1. If $H$ is proper and continuous and $K \subseteq T^* \mathcal M$ a compact subset of the phase space. Then $H(K)$ is compact (by continuity), in particular bounded.
2. Since the energy is conserved, we know $H( \varphi_t(K) )$ bounded as well.
3. Therefore $\overline{H( \varphi_t(K))}$ is compact (Hopf–Rinow theorem) and hence $\varphi_t(K) \subseteq H^{-1}(\overline{H( \varphi_t(K))})$ is a subset of a compact set (by properness of $H$) and therefore bounded.

(Maybe this is too complicated. As seen here, alternatively it would work to assume that preimages of bounded sets are bounded, but I don't know anything the strength of this condition.)