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
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