Formal Development
Given a manifold Q, a vector field X on Q (or equivalently, a section of the tangent bundle TQ) can be thought of as a function acting on the cotangent bundle, by the duality between the tangent and cotangent spaces. That is, define a function
such that
holds for all cotangent vectors p in . Here, is a vector in, the tangent space to the manifold Q at point q. The function is called the momentum function corresponding to X.
In local coordinates, the vector field X at point q may be written as
where the are the coordinate frame on TQ. The conjugate momentum then has the expression
where the are defined as the momentum functions corresponding to the vectors :
The together with the together form a coordinate system on the cotangent bundle ; these coordinates are called the canonical coordinates.
Read more about this topic: Canonical Coordinates
Famous quotes containing the words formal and/or development:
“That anger can be expressed through words and non-destructive activities; that promises are intended to be kept; that cleanliness and good eating habits are aspects of self-esteem; that compassion is an attribute to be prized—all these lessons are ones children can learn far more readily through the living example of their parents than they ever can through formal instruction.”
—Fred Rogers (20th century)
“There are two things which cannot be attacked in front: ignorance and narrow-mindedness. They can only be shaken by the simple development of the contrary qualities. They will not bear discussion.”
—John Emerich Edward Dalberg, 1st Baron Acton (1834–1902)