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:
“The manifestation of poetry in external life is formal perfection. True sentiment grows within, and art must represent internal phenomena externally.”
—Franz Grillparzer (1791–1872)
“Dissonance between family and school, therefore, is not only inevitable in a changing society; it also helps to make children more malleable and responsive to a changing world. By the same token, one could say that absolute homogeneity between family and school would reflect a static, authoritarian society and discourage creative, adaptive development in children.”
—Sara Lawrence Lightfoot (20th century)