Canonical coordinates

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: :\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 q^i to be the usual Cartesian coordinates, and p_i to be the components of momentum. Hence in general, the 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. ==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 \left(q^i, p_j\right) or \left(x^i, p_j\right) with the xs or qs denoting the coordinates on the underlying manifold and the ps 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 , a vector field on (a section of the tangent bundle ) 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 :P_X(q, p) = p(X_q) holds for all cotangent vectors in T_q^*Q. Here, X_q is a vector in T_qQ, the tangent space to the manifold at point . The function P_X is called the momentum function corresponding to . In local coordinates, the vector field at point 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 . The conjugate momentum then has the expression :P_X(q, p) = \sum_i X^i(q)\; p_i where the p_i are defined as the momentum functions corresponding to the vectors \partial /\partial q^i: :p_i = P_{\partial /\partial q^i} The q^i together with the p_j 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 \left(q^i, \dot{q}^i\right) with q^i 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==