A Riemannian metric on the manifold provides a (non-canonical) isomorphism between the cotangent space and the tangent space. Thus, they have the same smoothness properties. However, many definitions are more natural on the cotangent bundle.

For example, the cotangent bundle has a
canonical symplectic two-form on it, as an exterior derivative of a one-form. The one-form assigns to a vector in the tangent bundle to the cotangent
bundle the application of the element in the cotangent bundle (a linear
functional) to the projection of the vector into the tangent bundle
(the differential of the projection of the cotangent bundle to the
original manifold). Proving this form is, indeed, symplectic can be done
by noting that being symplectic is a local property: since the cotangent
bundle is locally trivial, this definition need only be checked on
**R**^{n}x**R**^{n}. But there the one form defined is the
sum of y_{i}dx_{i}, and the differential is the
canonical symplectic form, the sum of dy_{i}dx_{i}.

The above symplectic construction, along with an appropriate energy function, gives a complete determination of the physics of systems, such as the pendulum example cited above.