Canonical coordinates

In classical mechanics, canonical coordinates are coordinates ${\displaystyle q^{i}}$ and ${\displaystyle p_{i}}$ in phase space that are used in the Hamiltonian formalism. The canonical coordinates satisfy the fundamental Poisson bracket relations:

${\displaystyle \left\{q^{i},q^{j}\right\}=0\qquad \left\{p_{i},p_{j}\right\}=0\qquad \left\{q^{i},p_{j}\right\}=\delta _{ij}}$

A typical example of canonical coordinates is for ${\displaystyle q^{i}}$ to be the usual Cartesian coordinates, and ${\displaystyle p_{i}}$ to be the components of momentum. Hence in general, the ${\displaystyle p_{i}}$ coordinates are referred to as "conjugate momenta".

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.

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 ${\displaystyle \left(q^{i},p_{j}\right)}$ or ${\displaystyle \left(x^{i},p_{j}\right)}$ 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

${\displaystyle \sum _{i}p_{i}\,\mathrm {d} q^{i}}$

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.

Given a manifold Q, a vector field X on Q (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

${\displaystyle P_{X}:T^{*}Q\to \mathbb {R} }$

such that

${\displaystyle P_{X}(q,p)=p(X_{q})}$

holds for all cotangent vectors p in ${\displaystyle T_{q}^{*}Q}$. Here, ${\displaystyle X_{q}}$ is a vector in ${\displaystyle T_{q}Q}$, the tangent space to the manifold Q at point q. The function ${\displaystyle P_{X}}$ is called the momentum function corresponding to X.

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

${\displaystyle X_{q}=\sum _{i}X^{i}(q){\frac {\partial }{\partial q^{i}}}}$

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

${\displaystyle P_{X}(q,p)=\sum _{i}X^{i}(q)\;p_{i}}$

where the ${\displaystyle p_{i}}$ are defined as the momentum functions corresponding to the vectors ${\displaystyle \partial /\partial q^{i}}$:

${\displaystyle p_{i}=P_{\partial /\partial q^{i}}}$

The ${\displaystyle q^{i}}$ together with the ${\displaystyle p_{j}}$ together form a coordinate system on the cotangent bundle ${\displaystyle T^{*}Q}$; these coordinates are called the canonical coordinates.

In Lagrangian mechanics, a different set of coordinates are used, called the generalized coordinates. These are commonly denoted as ${\displaystyle \left(q^{i},{\dot {q}}^{i}\right)}$ with ${\displaystyle q^{i}}$ called the generalized position and ${\displaystyle {\dot {q}}^{i}}$ 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.

• Goldstein, Herbert; Poole, Charles P., Jr.; Safko, John L. (2002). Classical Mechanics (3rd ed.). San Francisco: Addison Wesley. pp. 347–349. ISBN 0-201-65702-3.{{cite book}}: CS1 maint: multiple names: authors list (link)
• Ralph Abraham and Jerrold E. Marsden, Foundations of Mechanics, (1978) Benjamin-Cummings, London ISBN 0-8053-0102-X See section 3.2.