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{{Classical mechanics}}
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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 mechanics|Hamiltonian formulation]] of [[classical mechanics]]. A closely related concept also appears in [[quantum mechanics]]; see the [[Stone-von Neumann theorem]] and [[canonical commutation relation]]s for details.


As Hamiltonian mechanics is generalized by [[symplectic geometry]] and [[canonical transformation]]s are generalized by [[contact transformation]]s, 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]].
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==Definition, in classical mechanics==
In [[classical mechanics]], '''canonical coordinates''' are coordinates <math>q_i\,</math> and <math>p_i\,</math> in [[phase space]] that are used in the [[Hamiltonian mechanics|Hamiltonian]] formalism. The canonical coordinates satisfy the fundamental [[Poisson bracket]] relations:
 
:<math>\{q_i, q_j\} = 0 \qquad \{p_i, p_j\} = 0 \qquad \{q_i, p_j\} = \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 [[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]] of the [[Lagrangian mechanics|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 <math>(q^i,p_j)</math> or <math>(x^i,p_j)</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-form]]s 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 [[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 (fiber bundle)|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
:<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 ''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 ''Q'' at point ''q''.  The function <math>P_X</math> is called the '''momentum function''' corresponding to ''X''.  
 
In [[atlas (topology)|local coordinates]], the vector field ''X'' at point ''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 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]]. These are commonly denoted as <math>(q^i,\dot{q}^i)</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 [[Hamilton–Jacobi equation]]s.
 
==See also==
* [[Linear discriminant analysis]]
* [[symplectic manifold]]
* [[symplectic vector field]]
* [[symplectomorphism]]
* [[Kinetic momentum]]
 
==References==
<references/>
*{{cite book |last1=Goldstein |first1=Herbert |authorlink1=Herbert Goldstein |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, CA |pages=347–349}}
 
==External links==
 
[[Category:Differential topology]]
[[Category:Symplectic geometry]]
[[Category:Hamiltonian mechanics]]
[[Category:Lagrangian mechanics]]
[[Category:Coordinate systems]]

Latest revision as of 18:10, 5 January 2015

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