cotangent lift of vector field

Definition

Let be a smooth manifold and be a smooth vector field. The cotangent lift of sometimes denoted is the vector field on whose local flow is the cotangent lift of the flow of .

That is,

Relation to projection pullback

The cotangent lift is the unique vector field such that

where is the projection map and is the tautological 1-form.

Proof

Since we know that cotangent lifts of smooth maps preserves the tautological 1-form, then we have that . Thus, .

To see why is unique, consider for some vector field such that . Then we have

Note that by construction .

Using Cartan’s magic formula, we have

Thus, since the symplectic form is non-degenerate, then .

Hamiltonian vector field

The cotangent lift of a vector field is a Hamiltonian vector field for Hamilton function .

Proof

This follows from Cartan’s Magic formula: