cotangent bundle

Definition

The cotangent bundle is the vector bundle of all the cotangent spaces.

As a smooth manifold

Let be a smooth manifold, and be a chart on the open set .

Given this chart, we have a basis for the tangent space at given by

Then, using the dual basis, we get the differentials

which form a basis for . Then, for any , we have

As we move the point smoothly, then we can think of how are smooth functions. This gives a map

which is a coordinate chart on called the cotangent coordinates which makes the cotangent bundle into a smooth manifold.

As a symplectic manifold

The cotangent bundle for a smooth manifold is a symplectic manifold using the canonical symplectic form

is the exterior derivative of the tautological 1-form.

Locally, with coordinates we have that

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References

• @dasilva2001Lectures