differential geometry - Is there a physical interpretation of symplectic manifolds which are not cotangent bundles?

The inspiration for symplectic geometry was from Hamiltonian mechanics. However, I am wondering how close the ties are between arbitrary symplectic manifolds and real physical systems.

In particular, it is my understanding that that the symplectic manifolds usually of interest to physicists are cotangent bundles of configuration spaces (but I am a mathematician so please correct me here if this is not correct). However, not every symplectic manifold is a cotangent bundle (for instance, the 2-sphere or the torus).

Question: For any given symplectic manifold $(M,\omega)$, is there a classical mechanical system which has $(M,\omega)$ as its phase space?

Particular examples of non cotangent bundle symplectic manifolds corresonding to real mechanical systems would also be useful.