In Hamiltonian mechanics, a canonical transformation is a change of canonical coordinates (q, p) → (Q, P) that preserves the form of Hamilton's equations...

In Hamiltonian mechanics, the linear canonical transformation (LCT) is a family of integral transforms that generalizes many classical transforms. It has...

theorem and canonical commutation relations for details. As Hamiltonian mechanics are generalized by symplectic geometry and canonical transformations are generalized...

homogeneous system. The Bogoliubov transformation is an isomorphism of either the canonical commutation relation algebra or canonical anticommutation relation algebra...

coordinate transformations, called canonical transformations, which map canonical coordinate systems into other canonical coordinate systems. A "canonical coordinate...

which acts as a bridge between two sets of canonical variables when performing a canonical transformation. There are four basic generating functions,...

{\displaystyle Q=Q(q,p),P=P(q,p)} is a canonical transformation, then the Lagrange bracket is an invariant of the transformation, in the sense that [ u , v ] q...

tightened in the linear canonical transformation, discussed below. From the point of view of the linear canonical transformation, Fresnel diffraction can...

published, based upon a stationary phase approximation. Linear canonical transformations are the linear transforms of the time–frequency representation...

treating S {\displaystyle S} as the generating function for a canonical transformation of the classical Hamiltonian H = H ( q 1 , q 2 , … , q N ; p 1...

that they become Fourier transform duals Canonical transformation, in Hamiltonian mechanics Grand canonical ensemble, a probability distribution of microscopic...

such as {x, p} = 1. All transformations of variables which preserve these brackets are allowed as canonical transformations in classical mechanics. Motion...

the time–frequency domain: from the point of view of the linear canonical transformation, the Fourier transform is rotation by 90° in the time–frequency...

picture, where Ehrenfest discovered it. In classical mechanics, a canonical transformation of phase space coordinates is one which preserves the structure...

mirrors, etc. A ray transfer matrix can be regarded as a linear canonical transformation. According to the eigenvalues of the optical system, the system...

In computer science, canonicalization (sometimes standardization or normalization) is a process for converting data that has more than one possible representation...

represents a transformation of phase space that is volume-preserving and preserves the symplectic structure of phase space, and is called a canonical transformation...

momenta, conjugate momenta, and canonical momenta). For a time instant t , {\displaystyle t,} the Legendre transformation of L {\displaystyle {\mathcal...

operations, and can also be further generalized into the linear canonical transformation (LCT). An early definition of the FRFT was introduced by Condon...

of studying differential equations (e.g. the Legendre transformation or canonical transformation) and describing the 'change of space element', familiar...

called canonical coordinates; transformations between different canonical coordinate systems are known as canonical transformations. The canonical symplectic...

every choice leads to a canonical transformation. One simple criterion for a transformation q → Q and p → P to be canonical is the Poisson bracket be...

Fourier Transform, the Laplace transform, and linear canonical transformations. These transformations are used in signal processing, optics, and quantum...

level sets of all first integrals are compact, then there exists a canonical transformation to action-angle coordinates in which the transformed Hamiltonian...

are similar if and only if they have the same rational canonical form. The rational canonical form is determined by the elementary divisors of A; these...

the subclass of geometric integrators which, by definition, are canonical transformations. They are widely used in nonlinear dynamics, molecular dynamics...

derived from a variational argument, could be obtained using a canonical transformation of the electronic Hamiltonian. In 1959, Lev Gor'kov showed that...

can only be among the free modes. To obtain the free modes, a canonical transformation can be applied that removes all modes that are not free (that do...

of a canonical transform that has now come to be known as the Foldy–Wouthuysen transformation. A brief account of the history of the transformation is to...

Time-frequency analysis – for processing non-stationary signals Linear canonical transformation Spectral estimation – for determining the spectral content (i.e...