Cotangent space
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
'R'nxRn. 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.
Referenced By
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