These are word's examples related to cotangent. Click on any word to go to its word's detail page. Or, go to the definition of cotangent.
Similarly, the second of equations (30) yields equation (32) and the inequality (33)with the tangent replaced by the minus cotangent.
The cotangent bundle of a manifold is equipped with a canonical symplectic form and a canonical section (the vertical section).
The metric g determines the isomorphism of the tangent and cotangent bundles.
This condition is particularly significant, being verified in all natural mechanical systems on cotangent bundles.
One of them is closely related to propagation of trajectories for symplectomorphisms of cotangent bundles.
This transformation has an invariant form that comes from the symplectic form in the cotangent bundle of the billiard table.
They may be geometrically interpreted as the forward propagation of tangents or the backward propagation of normals or cotangents.
But why should forces take values in the cotangent bundle?
This is more or less what is done when a tangent linear code is first developed and then the partials are used in the cotangent linear, or adjoint code.
Typically, the cotangent space is defined as the dual space of the tangent space at "x", although there are more direct definitions (see below).
For mechanical systems, the phase space usually consists of all possible values of position and momentum variables (i.e. the cotangent space of configuration space).
Likewise, cotangent space is a contravariant functor, essentially the composition of the tangent space with the dual space above.
Important examples of vector bundles include the tangent bundle and cotangent bundle of a smooth manifold.
Abstractly, it is a second order operator on each exterior power of the cotangent bundle.
Readers familiar with more advanced mathematics such as cotangent bundles, exterior derivatives and symplectic manifolds should read the related symplectomorphism article.
