In mathematics and classical mechanics, canonical coordinates are sets of coordinates which can be used to describe a physical system at any given point in time (locating the system within phase space). Canonical coordinates are used in the Hamiltonian formulation of classical mechanics. A closely related concept also appears in quantum mechanics; see the Stone–von Neumann theorem and canonical commutation relations for details.
Contents
- Definition in classical mechanics
- Definition on cotangent bundles
- Formal development
- Generalized coordinates
- References
As Hamiltonian mechanics is generalized by symplectic geometry and canonical transformations are generalized by contact transformations, so the 19th century definition of canonical coordinates in classical mechanics may be generalized to a more abstract 20th century definition of coordinates on the cotangent bundle of a manifold.
Definition, in classical mechanics
In classical mechanics, canonical coordinates are coordinates
A typical example of canonical coordinates is for
Canonical coordinates can be obtained from the generalized coordinates of the Lagrangian formalism by a Legendre transformation, or from another set of canonical coordinates by a canonical transformation.
Definition, on cotangent bundles
Canonical coordinates are defined as a special set of coordinates on the cotangent bundle of a manifold. They are usually written as a set of
A common definition of canonical coordinates is any set of coordinates on the cotangent bundle that allow the canonical one-form to be written in the form
up to a total differential. A change of coordinates that preserves this form is a canonical transformation; these are a special case of a symplectomorphism, which are essentially a change of coordinates on a symplectic manifold.
In the following exposition, we assume that the manifolds are real manifolds, so that cotangent vectors acting on tangent vectors produce real numbers.
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
In local coordinates, the vector field X at point q may be written as
where the
where the
The
Generalized coordinates
In Lagrangian mechanics, a different set of coordinates are used, called the generalized coordinates. These are commonly denoted as