Canonical coordinates

Canonical coordinates

In mathematics and classical mechanics, canonical coordinates are particular sets of coordinates on the phase space, or equivalently, on the cotangent manifold of a manifold. Canonical coordinates arise naturally in physics in the study of Hamiltonian mechanics. 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 in terms of cotangent bundles.

This article defines the canonical coordinates as they appear in 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

In classical mechanics, canonical coordinates are coordinates q_i\, and p_i\, in phase space that are used in the Hamiltonian formalism. The canonical coordinates satisfy the fundamental Poisson bracket relations:

\{q_i, q_j\} = 0 \qquad \{p_i, p_j\} = 0 \qquad \{q_i, p_j\} = \delta_{ij}

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 (qi,pj) or (xi,pj) 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

\sum_i p_i\,\mathrm{d}q^i

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

P_X:T^*Q\to \mathbb{R}

such that

PX(q,p) = p(Xq)

holds for all cotangent vectors p in T_q^*Q. Here, Xq is a vector in TqQ, the tangent space to the manifold Q at point q. The function PX is called the momentum function corresponding to X.

In local coordinates, the vector field X at point q may be written as

X_q=\sum_i X^i(q) \frac{\partial}{\partial q^i}

where the \partial /\partial q^i are the coordinate frame on TQ. The conjugate momentum then has the expression

P_X(q,p)=\sum_i X^i(q) \;p_i

where the pi are defined as the momentum functions corresponding to the vectors \partial /\partial q^i:

p_i = P_{\partial /\partial q^i}

The qi together with the pj together form a coordinate system on the cotangent bundle T * Q; 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 (q^i,\dot{q}^i) with qi called the generalized position and \dot{q}^i the generalized velocity. When a Hamiltonian is defined on the cotangent bundle, then the generalized coordinates are related to the canonical coordinates by means of the Hamilton–Jacobi equations.

See also

References

see H. Goldstein

External links


Wikimedia Foundation. 2010.

Игры ⚽ Поможем сделать НИР

Look at other dictionaries:

  • canonical coordinates — kanoninės koordinatės statusas T sritis fizika atitikmenys: angl. canonical coordinates vok. kanonische Koordinaten, f rus. канонические координаты, f pranc. coordonnées canoniques, f …   Fizikos terminų žodynas

  • Canonical transformation — In Hamiltonian mechanics, a canonical transformation is a change of canonical coordinates (mathbf{q}, mathbf{p}, t) ightarrow (mathbf{Q}, mathbf{P}, t) that preserves the form of Hamilton s equations, although it might not preserve the… …   Wikipedia

  • Canonical commutation relation — In physics, the canonical commutation relation is the relation between canonical conjugate quantities (quantities which are related by definition such that one is the Fourier transform of another), for example: between the position x and momentum …   Wikipedia

  • Canonical ensemble — A canonical ensemble in statistical mechanics is a statistical ensemble representing a probability distribution of microscopic states of the system. The probability distribution is characterised by the proportion pi of members of the ensemble… …   Wikipedia

  • Canonical — is an adjective derived from . Canon comes from the Greek word kanon , rule (perhaps originally from kanna reed , cognate to cane ), and is used in various meanings. Basic, canonic, canonical : reduced to the simplest and most significant form… …   Wikipedia

  • Canonical general relativity — In physics, canonical quantum gravity is an attempt to quantize the canonical formulation of general relativity (or canonical gravity). It is a Hamiltonian formulation of Einstein s general theory of relativity. The basic theory was outlined by… …   Wikipedia

  • Canonical bundle — In mathematics, the canonical bundle of a non singular algebraic variety V of dimension n is the line bundle which is the nth exterior power of the cotangent bundle Ω on V. Over the complex numbers, it is the determinant bundle of holomorphic n… …   Wikipedia

  • Canonical analysis — In statistics, canonical analysis (from Gk. κανων bar, measuring rod, ruler) belongs to the family of regression methods for data analysis. Regression analysis quantifies a relationship between a predictor variable and a criterion variable by the …   Wikipedia

  • Canonical quantization — In physics, canonical quantization is one of many procedures for quantizing a classical theory. Historically, this was the earliest method to be used to build quantum mechanics. When applied to a classical field theory it is also called second… …   Wikipedia

  • Canonical correlation — In statistics, canonical correlation analysis, introduced by Harold Hotelling, is a way of making sense of cross covariance matrices.DefinitionGiven two column vectors X = (x 1, dots, x n) and Y = (y 1, dots, y m) of random variables with finite… …   Wikipedia

Share the article and excerpts

Direct link
Do a right-click on the link above
and select “Copy Link”