Symplectic vector field

Let $(M,\omega)$ be a symplectic manifold. A symplectic vector field is a smooth vector field $X$ satisfying

\mathcal L_X\omega=0,

where $\mathcal L_X\omega$ is the Lie derivative of the differential form. Since $d\omega=0$ , Cartan’s formula gives $\mathcal L_X\omega=d(\iota_X\omega)$ . Thus $X$ is symplectic exactly when the one-form $\iota_X\omega$ is closed. It is Hamiltonian under the convention $\iota_X\omega=dH$ when that one-form is globally exact, a strictly stronger condition in general.

Flow characterization

A vector field is symplectic exactly when each map in its local flow preserves $\omega$ wherever defined. If $X$ is complete, this yields a one-parameter group of symplectomorphisms.

Closed versus exact

Every symplectic vector field is locally Hamiltonian by the Poincaré lemma. It is globally Hamiltonian precisely when the de Rham cohomology class $[\iota_X\omega]$ vanishes. In particular, if the first de Rham cohomology group of $M$ vanishes, every symplectic vector field is Hamiltonian.

On the standard symplectic two-torus, a translation field gives a closed contraction one-form that need not be exact, hence a symplectic but non-Hamiltonian field.

Conventions and scope

References

Dusa McDuff and Dietmar Salamon, Introduction to Symplectic Topology, 3rd ed., Oxford University Press, 2017. Publisher record. Relevant: Chapter 10, symplectic and Hamiltonian vector fields.
Ana Cannas da Silva, Lectures on Symplectic Geometry, Lecture Notes in Mathematics 1764, Springer, 2001. DOI record. Relevant: Hamiltonian vector fields and flows.