In mathematics and physics, the canonical coordinates are a special set of coordinates on the cotangent bundle of a manifold. They are usually written as a set of (qⁱ,p_j) or (xⁱ,p_j) 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 contangent bundle at point q in the manifold. This article defines the canonical coordinates as they appear in classical physics. A closely related concept also appears in quantum mechanics; see the Stone-von Neumann theorem and canonical commutation relations for details. In the following exposition, we assume that the manifolds are real manifolds, so that cotangent vectors acting on tengent vectors produce real numbers. Mathematics, often abbreviated maths in Commonwealth English and math in American English, is the study of abstraction. ... Physics (from the Greek, φυσικός (phusikos), natural, and φύσις (phusis), nature) is the science of nature in the broadest sense. ... See Cartesian coordinate system or Coordinates (elementary mathematics) for a more elementary introduction to this topic. ... In differential geometry, the cotangent bundle of a manifold is the vector bundle of all the cotangent spaces at every point in the manifold. ... In mathematics, a manifold M is a type of space, characterized in one of two equivalent ways: near every point of the space, we have a coordinate system; or near every point, the environment is like that in Euclidean space of a given dimension. ... (Redirected from 1-form) A differential form is a mathematical concept in the fields of multivariate calculus, differential topology and tensors. ... Classical physics is physics based on principles developed before the rise of quantum theory. ... Fig. ... In mathematics and in theoretical physics, the Stone-von Neumann theorem is any one of a number of different formulations of the uniqueness of the canonical commutation relations between position and momentum operators. ... In physics, the canonical commutation relation is the relation among the position and momentum of a point particle in one dimension, where is the so-called commutator of and , is the imaginary unit and is the reduced Plancks constant. ...

Definition

Given a manifold Q, a vector field X on 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 Vector field given by vectors of the form (-y, x) In mathematics a vector field is a construction in vector calculus which associates a vector to every point in Euclidean space. ... In mathematics, the tangent bundle of a manifold is a vector bundle which as a set is the disjoint union of all the tangent spaces at every point in the manifold with natural topology and smooth structure. ... In differential geometry, the cotangent bundle of a manifold is the vector bundle of all the cotangent spaces at every point in the manifold. ...

such that

P_X(q,p) = p(X_q)

holds for all cotangent vectors p in . Here, X_q is the vector in T_qQ, the tangent space to the manifold Q at point q. The function P_X is called the momentum function corresponding to X.

In local coordinates, the vector field X at point q may be written as In topology, an atlas describes how a complicated space is glued together from simpler pieces. ...

where the are the coordinate frame on TQ. The conjugate momentum then has the expression

where the p_i are defined as the momentum functions corresponding to the vectors :

The qⁱ 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 with qⁱ called the generalized position and 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. Lagrangian mechanics is a re-formulation of classical mechanics introduced by Joseph Louis Lagrange in 1788. ... Lagrangian mechanics is a re-formulation of classical mechanics introduced by Joseph Louis Lagrange in 1788. ... In mathematics and physics, the symplectic vector field, also known as the Hamiltonian vector field, is a vector field induced on a symplectic manifold by an energy function or Hamiltonian. ... ...

See also

• symplectic manifold
• symplectic vector field
• symplectomorphism