Canonical coordinates

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 ${\displaystyle \left(q^i,p_j\right)}$ 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.

Formal development

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

${\displaystyle P_X:T^{\ast }Q\to \mathbb{R}}$

such that

${\displaystyle P_X(q,p)=p(X_q)}$

holds for all cotangent vectors p in ${\displaystyle T_q^{\ast }Q}$. 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

${\displaystyle X_q=\sum _i{X}^i(q)\frac{\mathrm{\partial }}{\mathrm{\partial }{q}^i}}$

where the ${\displaystyle \mathrm{\partial }/\mathrm{\partial }{q}^i}$ 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 ${\displaystyle \mathrm{\partial }/\mathrm{\partial }{q}^i}$:

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

The ${\displaystyle q^i}$ together with the ${\displaystyle p_j}$ together form a coordinate system on the cotangent bundle ${\displaystyle T^{\ast }Q}$; 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 ${\displaystyle \left(q^i,{\dot{q}}^i\right)}$ 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.