Canonical transformation

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

Canonical formulation for the geometrical optics of concave gratings

Abstract: Dragt presented a method for characterizing optical systems that uses the concept of Lie transformations [ J. Opt. Soc. Am.72, 372 ( 1982)]. This method is applied to the geometrical optics of concave gratings that have plane symmetry, and the Lie transformation is derived for holographic gratings and for ruled gratings. This transformation corresponds to the canonical transformation in analytical mechanics. Unlike the systems treated by Dragt, concave grating systems do not have axial symmetry, and consequently odd-order Lie transformations are required. This formulation enables us to derive analytical expressions for the image coordinates. Applications of this formulation to compound optical systems are given.

Hamiltonian normalization in the restricted many-body problem by computer algebra methods

Abstract: A symbolic algorithm for construction of a real canonical transformation that reduces the Hamiltonian determining motion of an autonomous two-degree-of-freedom system in a neighborhood of an equilibrium state to the normal form is discussed. The application of the algorithm to the restricted planar circular three-body problem is demonstrated. The expressions obtained for the coefficients of the Hamiltonian normal form substantiate results derived earlier by A. Deprit. Symbolic computations are performed in the computer algebra system Mathematica.

On the Canonical Formalism for a Higher-Curvature Gravity

Abstract: Following the method of Buchbinder and Lyahovich, we carry out a canonical formalism for a higher-curvature gravity in which the Lagrangian density ${\cal L}$ is given in terms of a function of the salar curvature $R$ as ${\cal L}=\sqrt{-\det g_{\mu u}}f(R)$. The local Hamiltonian is obtained by a canonical transformation which interchanges a pair of the generalized coordinate and its canonical momentum coming from the higher derivative of the metric.

Integral Representations for the Class of Generalized Metaplectic Operators

Abstract: This article gives explicit integral formulas for the so-called generalized metaplectic operators, i.e. Fourier integral operators of Schrodinger type, having a symplectic matrix as their canonical transformation. These integrals are over specific linear subspaces of $$\mathbb {R}^d$$ , related to the $$d\times d$$ upper left-hand side submatrix of the underlying $$2d\times 2d$$ symplectic matrix. The arguments use the integral representations for the classical metaplectic operators obtained by Morsche and Oonincx in a previous paper, algebraic properties of symplectic matrices and time-frequency tools. As an application, we give a specific integral representation for solutions of the Cauchy problem of Schrodinger equations with bounded perturbations for every instant time $$t\in \mathbb {R}$$ , even at the (so-called) caustic points.