Cotangent bundle

In differential geometry, the cotangent bundle of a manifold is the union of all the cotangent spaces at each point of the manifold.

One-shapes

The differentiable sections of the cotangent bundle are differential one-forms, also called Pfaff forms or Pfaffian forms.

Cotangent bundle as phase space

Symplectic form

The cotangent bundle has a canonical symplectic 2-form in it, as an exterior derivative of the canonical one-form.

The one-form maps a vector onto the tangent bundle from the cotangent bundle by mapping the element onto the cotangent bundle (a linear functional) to the projection of the vector onto the tangent bundle (the differential of the projection of the cotangent bundle to the original variety). One can prove that the exterior derivative of this form is symplectic by noting that being symplectic is a local property: since the cotangent bundle is locally trivial, this definition need only be checked on Rⁿ × Rⁿ. But there the one definite form is the sum of y_idx_i, and the differential is the canonical symplectic form, the sum of dy_i∧dx_i.

Phase space

If the manifold M represents the set of possible positions in a dynamical system, then the cotangent bundle of T^*M can be thought of as the set of possible positions and moments. For example, this is an easy way to describe the (non-trivial) phase space of a three-dimensional spherical pendulum: a massive ball constrained to move along a 2-sphere. The above symplectic construction, together with an appropriate energy function, gives a complete determination of the physics of the system. See Hamiltonian mechanics for more information.

Related Issues

• tangent fiber