The cotangent space at a point P on a smooth manifold M is formally defined as a quotient space of two vector spaces: it is the vector space of all infinitely differentiable functions which have the value 0 at P, divided by the subspace of all functions which also have derivative 0 at this point. As P varies, the cotangent spaces make up the cotangent bundle of M. If M represents the set of possible positions in a dynamical system, then the cotangent bundle can be thought of as the set of possible ''positions and speeds''. For example, this is an easy way to describe the (non-trivial) phase space of a three dimensional pendulum: a weighted ball able to move along a sphere.
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 RnxRn. But there the one form defined is the sum of yidxi, and the differential is the canonical symplectic form, the sum of dyidxi.
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.
