Modern Mathematical Description
In mathematical terms, canonical coordinates are any coordinates on the phase space (cotangent bundle) of the system that allow the canonical one-form to be written as
up to a total differential (exact form). The change of variable between one set of canonical coordinates and another is a canonical transformation. The index of the generalized coordinates is written here as a superscript, not as a subscript as done above . The superscript conveys the contravariant transformation properties of the generalized coordinates, and does not mean that the coordinate is being raised to a power. Further details may be found at the symplectomorphism article.
Read more about this topic: Canonical Transformation
Famous quotes containing the words modern, mathematical and/or description:
“In modern America, anyone who attempts to write satirically about the events of the day finds it difficult to concoct a situation so bizarre that it may not actually come to pass while his article is still on the presses.”
—Calvin Trillin (b. 1940)
“The most distinct and beautiful statement of any truth must take at last the mathematical form.”
—Henry David Thoreau (1817–1862)
“The great object in life is Sensation—to feel that we exist, even though in pain; it is this “craving void” which drives us to gaming, to battle, to travel, to intemperate but keenly felt pursuits of every description whose principal attraction is the agitation inseparable from their accomplishment.”
—George Gordon Noel Byron (1788–1824)