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A symplectic vector field satisfies ${\mathcal{L}}_X\omega =0$, so its local flow consists of symplectomorphisms. By Cartan's magic formula and $d\omega =0$, this is equivalent to the differential one-form ${\iota }_X\omega$ being closed. In the convention fixed above, a Hamiltonian vector field satisfies ${\iota }_X\omega =dH$ for a globally defined smooth function $H$. It is therefore a symplectic vector field, but the converse requires this closed one-form to be exact.

On the torus with $\omega =dx\wedge dy$,Thus ${\partial }_x$ is a symplectic vector field. The closed differential one-form $dy$ is not exact on the torus: its integral on the loop $\gamma (s)=[(0,s)]$, $0\le s\le 1$, equals one, while the integral of an exact one-form around any closed loop is zero. The vector field is symplectic but not Hamiltonian. The local candidate $H=y$ does not descend to a single-valued function on ${\mathbb{R}}^2/{\mathbb{Z}}^2$.