View source for Physics:Canonical coordinates

{{Short description|Sets of coordinates on phase space which can be used to describe a physical system}}
{{Classical mechanics}}
In [[Mathematics|mathematics]] and [[Physics:Classical mechanics|classical mechanics]], '''canonical coordinates''' are sets of coordinates on [[Physics:Phase space|phase space]] which can be used to describe a physical system at any given point in time. Canonical coordinates are used in the [[Physics:Hamiltonian mechanics|Hamiltonian formulation]] of [[Physics:Classical mechanics|classical mechanics]]. A closely related concept also appears in [[Physics:Quantum mechanics|quantum mechanics]]; see the [[Stone–von Neumann theorem]] and [[Physics:Canonical commutation relation|canonical commutation relation]]s for details.

As Hamiltonian mechanics are generalized by [[Symplectic geometry|symplectic geometry]] and [[Physics:Canonical transformation|canonical transformation]]s 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|cotangent bundle]] of a [[Manifold|manifold]] (the mathematical notion of phase space).

==Definition in classical mechanics==
In [[Physics:Classical mechanics|classical mechanics]], '''canonical coordinates''' are coordinates <math>q^i</math> and <math>p_i</math> in [[Physics:Phase space|phase space]] that are used in the [[Physics:Hamiltonian mechanics|Hamiltonian]] formalism. The canonical coordinates satisfy the fundamental [[Physics:Poisson bracket|Poisson bracket]] relations:

:<math>\left\{q^i, q^j\right\} = 0 \qquad \left\{p_i, p_j\right\} = 0 \qquad \left\{q^i, p_j\right\} = \delta_{ij}</math>

A typical example of canonical coordinates is for <math>q^i</math> to be the usual Cartesian coordinates, and <math>p_i</math> to be the components of [[Physics:Momentum|momentum]]. Hence in general, the <math>p_i</math> coordinates are referred to as "conjugate momenta".

Canonical coordinates can be obtained from the [[Generalized coordinates|generalized coordinates]] of the [[Lagrangian mechanics|Lagrangian]] formalism by a [[Legendre transformation]], or from another set of canonical coordinates by a [[Physics:Canonical transformation|canonical transformation]].

==Definition on cotangent bundles==
Canonical coordinates are defined as a special set of coordinates on the [[Cotangent bundle|cotangent bundle]] of a [[Manifold|manifold]]. They are usually written as a set of <math>\left(q^i, p_j\right)</math> or <math>\left(x^i, p_j\right)</math> with the ''x''{{'}}s or ''q''{{'}}s denoting the coordinates on the underlying manifold and the ''p''{{'}}s denoting the '''conjugate momentum''', which are 1-forms in the cotangent bundle at point ''q'' in the manifold.

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
:<math>\sum_i p_i\,\mathrm{d}q^i</math>

up to a total differential. A change of coordinates that preserves this form is a [[Physics:Canonical transformation|canonical transformation]]; these are a special case of a [[Symplectomorphism|symplectomorphism]], which are essentially a change of coordinates on a [[Physics:Symplectic manifold|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 {{mvar|Q}}, a [[Vector field|vector field]] {{mvar|X}} on {{mvar|Q}} (a [[Section (fiber bundle)|section]] of the [[Tangent bundle|tangent bundle]] {{math|''TQ''}}) can be thought of as a function acting on the [[Cotangent bundle|cotangent bundle]], by the duality between the tangent and cotangent spaces.  That is, define a function

:<math>P_X: T^*Q \to \mathbb{R}</math>

such that

:<math>P_X(q, p) = p(X_q)</math>

holds for all cotangent vectors {{mvar|p}} in <math>T_q^*Q</math>. Here, <math>X_q</math> is a vector in <math>T_qQ</math>, the tangent space to the manifold {{mvar|Q}} at point {{mvar|q}}.  The function <math>P_X</math> is called the ''momentum function'' corresponding to {{mvar|X}}.

In [[Atlas (topology)|local coordinates]], the vector field {{mvar|X}} at point {{mvar|q}} may be written as

:<math>X_q = \sum_i X^i(q) \frac{\partial}{\partial q^i}</math>

where the <math>\partial /\partial q^i</math> are the coordinate frame on {{mvar|TQ}}. The conjugate momentum then has the expression

:<math>P_X(q, p) = \sum_i X^i(q)\; p_i</math>

where the <math>p_i</math> are defined as the momentum functions corresponding to the vectors <math>\partial /\partial q^i</math>:

:<math>p_i = P_{\partial /\partial q^i}</math>

The <math>q^i</math> together with the <math>p_j</math> together form a coordinate system on the cotangent bundle <math>T^*Q</math>; these coordinates are called the ''canonical coordinates''.

==Generalized coordinates==
In [[Lagrangian mechanics]], a different set of coordinates are used, called the [[Generalized coordinates|generalized coordinates]].  These are commonly denoted as <math>\left(q^i, \dot{q}^i\right)</math> with <math>q^i</math> called the '''generalized position''' and <math>\dot{q}^i</math> the '''generalized velocity'''.  When a [[Symplectic vector field|Hamiltonian]] is defined  on the cotangent bundle, then the generalized coordinates are related to the canonical coordinates by means of the [[Physics:Hamilton–Jacobi equation|Hamilton–Jacobi equation]]s.

==See also==
{{div col|colwidth=20em}}
* [[Linear discriminant analysis]]
* [[Physics:Symplectic manifold|Symplectic manifold]]
* [[Symplectic vector field]]
* [[Symplectomorphism]]
* Kinetic momentum
* [[Physics:Complementarity|Complementarity]]
* [[Physics:Canonical quantization|Canonical quantization]]
* [[Physics:Canonical quantum gravity|Canonical quantum gravity]]
{{div col end}}

==References==
<references/>
*{{cite book |last1=Goldstein |first1=Herbert |last2=Poole | first2=Charles P. Jr. |last3=Safko |first3=John L. |title=Classical Mechanics |edition=3rd |year=2002 |url=http://www.pearsonhighered.com/educator/product/Classical-Mechanics/9780201657029.page |isbn=0-201-65702-3 |publisher=Addison Wesley |location=San Francisco|pages=347–349}}
* [[Biography:Ralph Abraham (mathematician)|Ralph Abraham]] and Jerrold E. Marsden, ''Foundations of Mechanics'', (1978) Benjamin-Cummings, London {{isbn|0-8053-0102-X}} ''See section 3.2''.

[[Category:Differential topology]]
[[Category:Symplectic geometry]]
[[Category:Hamiltonian mechanics]]
[[Category:Lagrangian mechanics]]
[[Category:Coordinate systems]]

{{Sourceattribution|Canonical coordinates}}