This article has multiple issues. Please helpimprove it or discuss these issues on thetalk page.(Learn how and when to remove these messages) This article includes alist of references,related reading, orexternal links,but its sources remain unclear because it lacksinline citations. Please helpimprove this article byintroducing more precise citations.(November 2018) (Learn how and when to remove this message) This articleneeds additional citations forverification. Please helpimprove this article byadding citations to reliable sources. Unsourced material may be challenged and removed. Find sources: "Canonical coordinates" – news ·newspapers ·books ·scholar ·JSTOR(November 2018) (Learn how and when to remove this message) (Learn how and when to remove this message)

Part of a series on
Classical mechanics
${\displaystyle \mathbf{\text{F}}=\frac{d\mathbf{p}}{dt}}$ Second law of motion
History Timeline Textbooks
Branches Applied Celestial Continuum Dynamics Field theory Kinematics Kinetics Statics Statistical mechanics
Fundamentals Acceleration Angular momentum Couple D'Alembert's principle Energy kinetic potential Force Frame of reference Inertial frame of reference Impulse Inertia / Moment of inertia Mass Mechanical power Mechanical work Moment Momentum Space Speed Time Torque Velocity Virtual work
Formulations Newton's laws of motion Analytical mechanics Lagrangian mechanics Hamiltonian mechanics Routhian mechanics Hamilton–Jacobi equation Appell's equation of motion Koopman–von Neumann mechanics
Core topics Damping Displacement Equations of motion Euler's laws of motion Fictitious force Friction Harmonic oscillator Inertial / Non-inertial reference frame Motion (linear) Newton's law of universal gravitation Newton's laws of motion Relative velocity Rigid body dynamics Euler's equations Simple harmonic motion Vibration
Rotation Circular motion Rotating reference frame Centripetal force Centrifugal force reactive Coriolis force Pendulum Tangential speed Rotational frequency Angular acceleration / displacement / frequency / velocity
Scientists Kepler Galileo Huygens Newton Horrocks Halley Maupertuis Daniel Bernoulli Johann Bernoulli Euler d'Alembert Clairaut Lagrange Laplace Poisson Hamilton Jacobi Cauchy Routh Liouville Appell Gibbs Koopman von Neumann
Physics portal  Category
v t e

Inmathematics andclassical mechanics,canonical coordinates are sets ofcoordinates onphase space which can be used to describe a physical system at any given point in time. Canonical coordinates are used in theHamiltonian formulation ofclassical mechanics. A closely related concept also appears inquantum mechanics; see theStone–von Neumann theorem andcanonical commutation relations for details.

As Hamiltonian mechanics are generalized bysymplectic geometry andcanonical transformations are generalized bycontact transformations, so the 19th century definition of canonical coordinates in classical mechanics may be generalized to a more abstract 20th century definition of coordinates on thecotangent bundle of amanifold (the mathematical notion of phase space).

Inclassical mechanics,canonical coordinates are coordinates${\displaystyle q^i}$ and${\displaystyle p_i}$ inphase space that are used in theHamiltonian formalism. The canonical coordinates satisfy the fundamentalPoisson bracket relations:

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

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

Canonical coordinates can be obtained from thegeneralized coordinates of theLagrangian formalism by aLegendre transformation, or from another set of canonical coordinates by acanonical transformation.

Canonical coordinates are defined as a special set ofcoordinates on thecotangent bundle of amanifold. They are usually written as a set of${\displaystyle \left(q^i,p_j\right)}$ or${\displaystyle \left(x^i,p_j\right)}$ with thex's orq's denoting the coordinates on the underlying manifold and thep's denoting theconjugate momentum, which are1-forms in the cotangent bundle at pointq in the manifold.

A common definition of canonical coordinates is any set of coordinates on the cotangent bundle that allow thecanonical 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 acanonical transformation; these are a special case of asymplectomorphism, which are essentially a change of coordinates on asymplectic 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 manifoldQ, avector fieldX onQ (asection of thetangent bundleTQ) can be thought of as a function acting on thecotangent 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 vectorsp in${\displaystyle T_q^{\ast }Q}$. Here,${\displaystyle X_q}$ is a vector in${\displaystyle T_{q}Q}$, the tangent space to the manifoldQ at pointq. The function${\displaystyle P_X}$ is called themomentum function corresponding toX.

Inlocal coordinates, the vector fieldX at pointq 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 onTQ. 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 thecanonical coordinates.

InLagrangian mechanics, a different set of coordinates are used, called thegeneralized coordinates. These are commonly denoted as${\displaystyle \left(q^i,{\dot{q}}^i\right)}$ with${\displaystyle q^i}$ called thegeneralized position and${\displaystyle {\dot{q}}^i}$ thegeneralized velocity. When aHamiltonian is defined on the cotangent bundle, then the generalized coordinates are related to the canonical coordinates by means of theHamilton–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.
• Ralph Abraham andJerrold E. Marsden,Foundations of Mechanics, (1978) Benjamin-Cummings, LondonISBN 0-8053-0102-XSee section 3.2.

• Differential topology
• Symplectic geometry
• Hamiltonian mechanics
• Lagrangian mechanics
• Coordinate systems
• Moment (physics)

• Articles with short description
• Short description is different from Wikidata
• Articles lacking in-text citations from November 2018
• All articles lacking in-text citations
• Articles needing additional references from November 2018
• All articles needing additional references
• Articles with multiple maintenance issues