Symplectic vector field

Definition

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. The equivalence follows by differentiating the pullback of $\omega$ along the flow, as developed in McDuff and Salamon, Chapter 10.

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.

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