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...

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

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

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

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

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...

{\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...

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

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

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

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...

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...

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...

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

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...

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...

This is a list of some of the most commonly used coordinate transformations. Let ( x , y ) {\displaystyle (x,y)} be the standard Cartesian coordinates...

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...

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

In quantum mechanics, the canonical commutation relation is the fundamental relation between canonical conjugate quantities (quantities which are related...

dispersion relation, and have no resonant interactions. In this case, canonical transformations are applied, with the goal of eliminating terms that are non-interacting...

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

ISBN 978-0-521-76360-8 Janssen, P. A. E. M. (2009). "On some consequences of the canonical transformation in the hamiltonian theory of water waves". J. Fluid Mech. 637:...