symplectic vector field

Symplectic vector fields

For a symplectic manifold , a symplectic vector field is a smooth vector field with flow such that for each that is defined, then is a symplectomorphism. In other words,

Closed vector fields

For a vector field , the following are equivalent:

• is symplectic
• (i.e. is closed)

Proof

() Assume is symplectic, thus . Then for every , , we have

() For this part we can use Cartan’s magic formula which says:

is a symplectic form and is thus closed, so . Thus, we see

Therefore, .

It is pretty simple to follow the logic back up to show that both conditions 2 and 3 imply 1.