= Symplectic vector field
{title2=$\mathcal L_X\omega=0$}
A <vector field> on a <symplectic manifold> is symplectic when its flow preserves the <symplectic form>. Infinitesimally this is $\mathcal L_X\omega=0$. By <Cartan's magic formula> it is equivalent to $d(\iota_X\omega)=0$. A <Hamiltonian vector field> has an exact contraction with the form, so is symplectic. On the standard two-dimensional <torus>, $\partial_x$ contracts to the closed but nonexact form $dy$, providing a symplectic non-Hamiltonian example.
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