Can $S^4$ be the cotangent bundle of a manifold? I am asking the question because in the classical mechanics book by Arnold, he states that there is a **distinguished** 1-form on $T^*V $. It seems that there is no such distinguished 1-form on a general even-dimensional manifold. So, not every even-dimensional manifold can be the phase space of a system, or not every even-dimensional manifold can be a cotangent bundle? Is $S^2$ a cotangent bundle (intuitively it is not)? How about $S^4$?

No, the total space of a cotangent bundle $T^*M$ is always non-compact, while a sphere $S^{2n}$ is compact.