Sets of coordinates on phase space which can be used to describe a physical system
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In mathematics and classical mechanics, canonical coordinates are sets of coordinates on phase space which can be used to describe a physical system at any given point in time. Canonical coordinates are used in the Hamiltonian formulation of classical mechanics. A closely related concept also appears in quantum mechanics; see the Stone–von Neumann theorem and canonical commutation relations for details.
As Hamiltonian mechanics are generalized by symplectic geometry and canonical transformations 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 of coordinates on the cotangent bundle of a manifold (the mathematical notion of phase space).
and 26 Related for: Canonical coordinates information
canonicalcoordinates are sets of coordinates on phase space which can be used to describe a physical system at any given point in time. Canonical coordinates...
notation for some object Canonical basis – Basis of a type of algebraic structure Canonicalcoordinates, sets of coordinates that can be used to describe...
In Hamiltonian mechanics, a canonical transformation is a change of canonicalcoordinates (q, p) → (Q, P) that preserves the form of Hamilton's equations...
{\boldsymbol {q}})} is called phase space coordinates. (Also canonicalcoordinates). In phase space coordinates ( p , q ) {\displaystyle ({\boldsymbol {p}}...
In quantum mechanics, the canonical commutation relation is the fundamental relation between canonical conjugate quantities (quantities which are related...
( q , p , t ) {\displaystyle H=H(q,p,t)} as one of the new canonical momentum coordinates. In a more general sense, the Poisson bracket is used to define...
system. Generalized coordinates are paired with generalized momenta to provide canonicalcoordinates on phase space. Generalized coordinates are usually selected...
(x,y)} be the standard Cartesian coordinates, and ( r , θ ) {\displaystyle (r,\theta )} the standard polar coordinates. x = r cos θ y = r sin θ ∂ (...
In statistical mechanics, a canonical ensemble is the statistical ensemble that represents the possible states of a mechanical system in thermal equilibrium...
other external coordinates are kept the same in all possible states of the system. The thermodynamic variables of the grand canonical ensemble are chemical...
foliation are tori, and the natural linear coordinates on these are called "angle" variables. The cycles of the canonical 1 {\displaystyle 1} -form are called...
vector bundle). A special set of coordinates can be defined on the cotangent bundle; these are called the canonicalcoordinates. Because cotangent bundles can...
Astronomy portal Canonicalcoordinates Fundamental lemma of the calculus of variations Functional derivative Generalized coordinates Hamiltonian mechanics...
group Sp(2n, R) comes up in classical physics as the symmetries of canonicalcoordinates preserving the Poisson bracket. Consider a system of n particles...
the conjugate variable of the density (or probability density). Canonicalcoordinates "Heisenberg – Quantum Mechanics, 1925–1927: The Uncertainty Relations"...
the three coordinates (r, θ, φ), known as a 3-tuple, provide a coordinate system on a sphere, typically called the spherical polar coordinates. Nota bene:...
defined directly from the action and does not require the canonicalcoordinates and their canonical momenta to be defined in advance.[clarification needed]...
momentum coordinates ( q , p ) {\displaystyle (q,p)} are called canonicalcoordinates. (See Hamiltonian mechanics for more background.) The time evolution...
In mathematics, the canonical bundle of a non-singular algebraic variety V {\displaystyle V} of dimension n {\displaystyle n} over a field is the line...
2n-dimensional symplectic manifold. Then locally, one may choose canonicalcoordinates (q1, ..., qn, p1, ..., pn) on M, in which the symplectic form is...
evolves in time. These are the metric canonicalcoordinates. In 1986 Abhay Ashtekar introduced a new set of canonical variables, Ashtekar (new) variables...
mechanics, a trajectory is defined by Hamiltonian mechanics via canonicalcoordinates; hence, a complete trajectory is defined by position and momentum...
from the fact that the Lagrangian depends on fewer coordinates than there are canonicalcoordinates (which correspond to the initial parameters needed...
mechanics through the Wigner transform of Heisenberg operators of canonicalcoordinates and momenta. These trajectories obey the Hamilton equations in quantum...
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