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Canonical transformation

About: Canonical transformation is a research topic. Over the lifetime, 1854 publications have been published within this topic receiving 38019 citations.


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Journal ArticleDOI
TL;DR: In this paper, a general implicit solution for determining volume-preserving transformations in the n -dimensional Euclidean space is obtained in terms of a set of 2 n generating functions in mixed coordinates.
Abstract: A general implicit solution for determining volume-preserving transformations in the n -dimensional Euclidean space is obtained in terms of a set of 2 n generating functions in mixed coordinates. For n =2, the proposed representation corresponds to the classical definition of a potential stream function in a canonical transformation. For n =3, the given solution defines a more general class of isochoric transformations, when compared to existing methods based on multiple potentials. Illustrative examples are discussed both in rectangular and in cylindrical coordinates for applications in mechanical problems of incompressible continua. Solving exactly the incompressibility constraint, the proposed representation method is suitable for determining three-dimensional isochoric perturbations to be used in bifurcation theory. Applications in non-linear elasticity are envisaged for determining the occurrence of complex instability patterns for soft elastic materials.

14 citations

Journal ArticleDOI
TL;DR: It is shown that the expectationvalue of a dynamical variable can be written in terms of its vacuum expectation value of the canonically transformed variable, and parallel-axis theorems are established for the photon number and its variant.
Abstract: It is possible to calculate expectation values and transition probabilities from the Wigner phase-space distribution function. Based on the canonical transformation properties of the Wigner function, an algorithm is developed for calculating these quantities in quantum optics for coherent and squeezed states. It is shown that the expectation value of a dynamical variable can be written in terms of its vacuum expectation value of the canonically transformed variable. Parallel-axis theorems are established for the photon number and its variant. It is also shown that the transition probability between two squeezed states can be reduced to that of the transition from one squeezed state to vacuum.

14 citations

Journal ArticleDOI
TL;DR: In this article, the authors derived the canonical transformation that takes the Hamiltonian of the Coulomb problem (in the Fock-Bargmann formulation) into that of the harmonic oscillator, while transforming the angular momenta of both problems into each other.
Abstract: The present paper can be viewed from two standpoints. The first is that it derives the canonical transformation that takes the Hamiltonian of the Coulomb problem (in the Fock–Bargmann formulation) into that of the harmonic oscillator, while transforming the angular momenta of both problems into each other. The second is the one in which the solution of the previous problem is required if we wish to find the canonical transformation relating microscopic and macroscopic collective models, where the former is derived from a system of A particles moving in two dimensions and interacting through harmonic oscillator forces. The canonical transformation shows the existence of a U(3) symmetry group in the microscopic collective model corresponding to that of the three‐dimensional oscillator which is the Hamiltonian of the macroscopic collective model. The importance of this result rests on the fact that had the motion of the particles taken place in the physical three‐dimensional space, rather than the hypothetical two‐dimensional one discussed here, the symmetry group would have been U(6) rather than U(3). Thus, the group theoretical structure of an s‐d boson picture or, equivalently, of a generalized Bohr–Mottelson approach, is present implicitly in an A‐body system interacting through harmonic oscillator forces.

14 citations

Journal ArticleDOI
TL;DR: In this article, the authors evaluate the quality of the approximation of periodic orbits in the logarithmic potential constructed using perturbation theory based on Hamiltonian normal forms, and show that with a normal form truncated at the lowest order incorporating the relevant resonance, it is possible to construct quite accurate solutions both for normal modes and periodic orbit in general position.
Abstract: Analytic methods to investigate periodic orbits in galactic potentials. To evaluate the quality of the approximation of periodic orbits in the logarithmic potential constructed using perturbation theory based on Hamiltonian normal forms. The solutions of the equations of motion corresponding to periodic orbits are obtained as series expansions computed by inverting the normalizing canonical transformation. To improve the convergence of the series a resummation based on a continued fraction may be performed. This method is analogous to that looking for approximate rational solutions (Prendergast method). It is shown that with a normal form truncated at the lowest order incorporating the relevant resonance it is possible to construct quite accurate solutions both for normal modes and periodic orbits in general position.

14 citations

Book ChapterDOI
01 Jan 2016
TL;DR: In this paper, the authors summarize the construction of the LCT integral transforms, detailing their Lie-algebraic relation with second-order differential operators, which is the origin of the metaplectic phase.
Abstract: Linear canonical transformations (LCTs) were introduced almost simultaneously during the early 1970s by Stuart A. Collins Jr. in paraxial optics, and independently by Marcos Moshinsky and Christiane Quesne in quantum mechanics, to understand the conservation of information and of uncertainty under linear maps of phase space. Only in the 1990s did both sources begin to be referred jointly in the growing literature, which has expanded into a field common to applied optics, mathematical physics, and analogic and digital signal analysis. In this introductory chapter we recapitulate the construction of the LCT integral transforms, detailing their Lie-algebraic relation with second-order differential operators, which is the origin of the metaplectic phase. Radial and hyperbolic LCTs are reviewed as unitary integral representations of the two-dimensional symplectic group, with complex extension to a semigroup for systems with loss or gain. Some of the more recent developments on discrete and finite analogues of LCTs are commented with their concomitant problems, whose solutions and alternatives are contained the body of this book.

14 citations


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Performance
Metrics
No. of papers in the topic in previous years
YearPapers
20237
202218
202158
202042
201932
201829