symplectic potential

Definition

Let be a symplectic manifold. A symplectic potential is a 1-form such that

Examples

The main example of a symplectic potential is the cotangent bundle of a smooth manifold . In this case, the tautological 1-form is a symplectic potential for .

Commuting with flows

Let ) be a symplectic manifold and be a symplectic potential. Since is non-degenerate, then there is a unique vector field such that

If is a symplectomorphism that preserves (that is, ) then commutes with the one parameter group of diffeomorphisms generated by (the flow of at time ). Written another way

Proof

It suffices to show that
We can differentiate at which gives

Where the last line follows since

since is the unique vector such that then .

Example

For a smooth manifold and cotangent bundle , when is the tautological 1-form, then the unique vector field from above is

and thus for every point then