Canonical coordinates
Encyclopedia
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 transformation
s 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 relation
s for details.
, canonical coordinates are coordinates
and 
in phase space
that are used in the Hamiltonian
formalism. The canonical coordinates satisfy the fundamental Poisson bracket
relations:
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
.
of a manifold
. They are usually written as a set of
or 
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
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.
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
such that
holds for all cotangent vectors p in
. Here, 
is a vector in 
, the tangent space to the manifold Q at point q. The function 
is called the momentum function corresponding to X.
In local coordinates
, the vector field X at point q may be written as
where the
are the coordinate frame on TQ. The conjugate momentum then has the expression
where the
are defined as the momentum functions corresponding to the vectors 
:
The
together with the 
together form a coordinate system on the cotangent bundle 
; these coordinates are called the canonical coordinates.
, a different set of coordinates are used, called the generalized coordinates
. These are commonly denoted as
with 
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 equation
s.
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
and classical mechanics
Classical mechanics
In physics, classical mechanics is one of the two major sub-fields of mechanics, which is concerned with the set of physical laws describing the motion of bodies under the action of a system of forces...
, canonical coordinates are particular sets of coordinates on the phase space
Phase space
In mathematics and physics, a phase space, introduced by Willard Gibbs in 1901, is a space in which all possible states of a system are represented, with each possible state of the system corresponding to one unique point in the phase space...
, or equivalently, on the cotangent manifold of a manifold
Manifold
In mathematics , a manifold is a topological space that on a small enough scale resembles the Euclidean space of a specific dimension, called the dimension of the manifold....
. Canonical coordinates arise naturally in physics
Physics
Physics is a natural science that involves the study of matter and its motion through spacetime, along with related concepts such as energy and force. More broadly, it is the general analysis of nature, conducted in order to understand how the universe behaves.Physics is one of the oldest academic...
in the study of Hamiltonian mechanics
Hamiltonian mechanics
Hamiltonian mechanics is a reformulation of classical mechanics that was introduced in 1833 by Irish mathematician William Rowan Hamilton.It arose from Lagrangian mechanics, a previous reformulation of classical mechanics introduced by Joseph Louis Lagrange in 1788, but can be formulated without...
. As Hamiltonian mechanics is generalized by symplectic geometry and canonical transformation
Canonical transformation
In Hamiltonian mechanics, a canonical transformation is a change of canonical coordinates → that preserves the form of Hamilton's equations , although it...
s 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
Classical mechanics
In physics, classical mechanics is one of the two major sub-fields of mechanics, which is concerned with the set of physical laws describing the motion of bodies under the action of a system of forces...
. A closely related concept also appears in quantum mechanics
Quantum mechanics
Quantum mechanics, also known as quantum physics or quantum theory, is a branch of physics providing a mathematical description of much of the dual particle-like and wave-like behavior and interactions of energy and matter. It departs from classical mechanics primarily at the atomic and subatomic...
; see the Stone-von Neumann theorem and canonical commutation relation
Canonical commutation relation
In physics, the canonical commutation relation is the relation between canonical conjugate quantities , for example:[x,p_x] = i\hbar...
s for details.
Definition, in classical mechanics
In classical mechanicsClassical mechanics
In physics, classical mechanics is one of the two major sub-fields of mechanics, which is concerned with the set of physical laws describing the motion of bodies under the action of a system of forces...
, canonical coordinates are coordinates
and in phase spacePhase space
In mathematics and physics, a phase space, introduced by Willard Gibbs in 1901, is a space in which all possible states of a system are represented, with each possible state of the system corresponding to one unique point in the phase space...
that are used in the Hamiltonian
Hamiltonian mechanics
Hamiltonian mechanics is a reformulation of classical mechanics that was introduced in 1833 by Irish mathematician William Rowan Hamilton.It arose from Lagrangian mechanics, a previous reformulation of classical mechanics introduced by Joseph Louis Lagrange in 1788, but can be formulated without...
formalism. The canonical coordinates satisfy the fundamental Poisson bracket
Poisson bracket
In mathematics and classical mechanics, the Poisson bracket is an important binary operation in Hamiltonian mechanics, playing a central role in Hamilton's equations of motion, which govern the time-evolution of a Hamiltonian dynamical system...
relations:
Canonical coordinates can be obtained from the generalized coordinates of the Lagrangian
Lagrangian mechanics
Lagrangian mechanics is a re-formulation of classical mechanics that combines conservation of momentum with conservation of energy. It was introduced by the Italian-French mathematician Joseph-Louis Lagrange in 1788....
formalism by a Legendre transformation
Legendre transformation
In mathematics, the Legendre transformation or Legendre transform, named after Adrien-Marie Legendre, is an operation that transforms one real-valued function of a real variable into another...
, or from another set of canonical coordinates by a canonical transformation
Canonical transformation
In Hamiltonian mechanics, a canonical transformation is a change of canonical coordinates → that preserves the form of Hamilton's equations , although it...
.
Definition, on cotangent bundles
Canonical coordinates are defined as a special set of coordinates on the cotangent bundleCotangent bundle
In mathematics, especially differential geometry, the cotangent bundle of a smooth manifold is the vector bundle of all the cotangent spaces at every point in the manifold...
of a manifold
Manifold
In mathematics , a manifold is a topological space that on a small enough scale resembles the Euclidean space of a specific dimension, called the dimension of the manifold....
. They are usually written as a set of
or 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
up to a total differential. A change of coordinates that preserves this form is a canonical transformation
Canonical transformation
In Hamiltonian mechanics, a canonical transformation is a change of canonical coordinates → that preserves the form of Hamilton's equations , although it...
; these are a special case of a symplectomorphism
Symplectomorphism
In mathematics, a symplectomorphism is an isomorphism in the category of symplectic manifolds.-Formal definition:A diffeomorphism between two symplectic manifolds f: \rightarrow is called symplectomorphism, iff^*\omega'=\omega,...
, which are essentially a change of coordinates on a symplectic manifold
Symplectic manifold
In mathematics, a symplectic manifold is a smooth manifold, M, equipped with a closed nondegenerate differential 2-form, ω, called the symplectic form. The study of symplectic manifolds is called symplectic geometry or symplectic topology...
.
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 fieldVector field
In vector calculus, a vector field is an assignmentof a vector to each point in a subset of Euclidean space. A vector field in the plane for instance can be visualized as an arrow, with a given magnitude and direction, attached to each point in the plane...
X on Q (or equivalently, a section
Section (fiber bundle)
In the mathematical field of topology, a section of a fiber bundle π is a continuous right inverse of the function π...
of the tangent bundle
Tangent bundle
In differential geometry, the tangent bundle of a differentiable manifold M is the disjoint unionThe disjoint union assures that for any two points x1 and x2 of manifold M the tangent spaces T1 and T2 have no common vector...
TQ) can be thought of as a function acting on the cotangent bundle
Cotangent bundle
In mathematics, especially differential geometry, the cotangent bundle of a smooth manifold is the vector bundle of all the cotangent spaces at every point in the manifold...
, by the duality between the tangent and cotangent spaces. That is, define a function
such that
holds for all cotangent vectors p in
. Here, is a vector in , the tangent space to the manifold Q at point q. The function is called the momentum function corresponding to X.In local coordinates
Atlas (topology)
In mathematics, particularly topology, one describesa manifold using an atlas. An atlas consists of individualcharts that, roughly speaking, describe individual regionsof the manifold. If the manifold is the surface of the Earth,...
, the vector field X at point q may be written as
where the
are the coordinate frame on TQ. The conjugate momentum then has the expressionwhere the
are defined as the momentum functions corresponding to the vectors :The
together with the together form a coordinate system on the cotangent bundle ; these coordinates are called the canonical coordinates.Generalized coordinates
In Lagrangian mechanicsLagrangian mechanics
Lagrangian mechanics is a re-formulation of classical mechanics that combines conservation of momentum with conservation of energy. It was introduced by the Italian-French mathematician Joseph-Louis Lagrange in 1788....
, a different set of coordinates are used, called the generalized coordinates
Generalized coordinates
In the study of multibody systems, generalized coordinates are a set of coordinates used to describe the configuration of a system relative to some reference configuration....
. These are commonly denoted as
with 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 equationHamilton–Jacobi equation
In mathematics, the Hamilton–Jacobi equation is a necessary condition describing extremal geometry in generalizations of problems from the calculus of variations. In physics, the Hamilton–Jacobi equation is a reformulation of classical mechanics and, thus, equivalent to other formulations such as...
s.
See also
- Linear discriminant analysisLinear discriminant analysisLinear discriminant analysis and the related Fisher's linear discriminant are methods used in statistics, pattern recognition and machine learning to find a linear combination of features which characterizes or separates two or more classes of objects or events...
- symplectic manifoldSymplectic manifoldIn mathematics, a symplectic manifold is a smooth manifold, M, equipped with a closed nondegenerate differential 2-form, ω, called the symplectic form. The study of symplectic manifolds is called symplectic geometry or symplectic topology...
- symplectic vector field
- symplectomorphismSymplectomorphismIn mathematics, a symplectomorphism is an isomorphism in the category of symplectic manifolds.-Formal definition:A diffeomorphism between two symplectic manifolds f: \rightarrow is called symplectomorphism, iff^*\omega'=\omega,...
- Kinetic momentumKinetic momentumIn physics, in particular electromagnetism, the kinetic momentum is a nonstandard term for the momentum of a charged particle due to its inertia. When a charged particle interacts with an electromagnetic field , there are two momenta: due to its inertia and due to the field...