Canonical transformation

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

Quantization of the Liénard II equation and Jacobi’s last multiplier

Abstract: In this paper, the role of Jacobi’s last multiplier in mechanical systems with a position-dependent mass is unveiled. In particular, we map the Lienard II equation to a position-dependent mass system. The quantization of the Lienard II equation is then carried out using the point canonical transformation method together with the Von Roos ordering technique. Finally, we show how their eigenfunctions and eigenspectrum can be obtained in terms of associated Laguerre and exceptional Laguerre functions.

On the canonical reduction of spherically symmetric gravity

Abstract: In a thorough paper Kuchar has examined the canonical reduction of the most general action functional describing the geometrodynamics of the maximally extended Schwarzschild geometry. This reduction yields the true degrees of freedom for (vacuum) spherically symmetric general relativity (SSGR). The essential technical ingredient in Kuchar's analysis is a canonical transformation to a certain chart on the gravitational phase space which features the Schwarzschild mass parameter , expressed in terms of what are essentially Arnowitt - Deser - Misner variables, as a canonical coordinate. (Kuchar's paper complements earlier work by Kastrup and Thiemann, based mostly on Ashtekar variables, which has also explicitly isolated the true degrees of freedom for vacuum SSGR.) In this paper we discuss the geometric interpretation of Kuchar's canonical transformation in terms of the theory of quasilocal energy - momentum in general relativity given by Brown and York. We find Kuchar's transformation to be a `sphere-dependent boost to the rest frame', where the `rest frame' is defined by vanishing quasilocal momentum. Furthermore, our formalism is general enough to cover the case of (vacuum) two-dimensional dilaton gravity. Therefore, besides reviewing Kuchar's original work for Schwarzschild black holes from the framework of hyperbolic geometry, we present new results concerning the canonical reduction of Witten black-hole geometrodynamics. Finally, addressing a recent work of Louko and Whiting, we discuss some delicate points concerning the canonical reduction of the `thermodynamical action', which is of central importance in the path-integral formulation of gravitational thermodynamics.

Generalized Langevin theory for many‐body problems in chemical dynamics: The method of partial clamping and formulation of the solute equations of motion in generalized coordinates

Abstract: A formulation of the analytical dynamics of an n‐atom solute system present at infinite dilution in a monatomic solvent is presented. This treatment, which will aid in the understanding of liquid state spectroscopic, energy‐transfer, and chemical reaction processes, is made by combining the methods of classical dynamics with recently developed generalized Langevin equation techniques [S. A. Adelman, Adv. Chem. Phys. 5 3, 611 (1983)]. The Hamiltonian of the solute system is formulated in generalized coordinates which are related to the underlying Cartesian coordinates by a point canonical transformation. The 3n generalized coordinates are partitioned into a set of p explicit coordinates whose dynamics are of direct interest and a set of q=3n−p i m p l i c i t coordinates whose motion is of lesser interest. A generalized Langevin equation of motion for the explicit coordinates is formulated by computing the reaction force exerted by the solvent on the explicit coordinates in response to small displacements of these coordinates. This generalized Langevin equation will provide a realistic description of explicit coordinate dynamics if the liquid state motion of these coordinates is oscillatory on subpicosecond time scales. Solvent effects appearing in the generalized Langevin equation may be discussed in terms of the fluctuation spectrum which describes atomic motion in the solvation shells. A rigorous statistical mechanical method for assessing the influence of the implicit coordinate motion on this fluctuation spectrum is presented. The formulas for the liquid state quantities appearing in the generalized Langevin equation may be exactly evaluated for particular solute–solvent systems via molecular dynamics (MD) simulation of the motion of the implicit and solvent coordinates given that the solute is partially clamped, i.e., given that the explicit coordinates are fixed. The method of partial clamping provides an improvement of the method of full solute clamping developed earlier. This improvement should permit one to realistically treat many liquid state processes via generalized Langevin equation techniques for which the solute/solvent mass ratio is less than unity.

Effective mass schrödinger equation for exactly solvable class of one-dimensional potentials

Abstract: We deal with the exact solutions of Schrodinger equation characterized by position-dependent effective mass via point canonical transformations. The Morse, Poschl-Teller, and Hulthen type potentials are considered, respectively. With the choice of position-dependent mass forms, exactly solvable target potentials are constructed. Their energy of the bound states and corresponding wavefunctions are determined exactly.

Consistent discretization and canonical classical and quantum Regge calculus

Abstract: We apply the ``consistent discretization'' technique to the Regge action for (Euclidean and Lorentzian) general relativity in arbitrary number of dimensions. The result is a well defined canonical theory that is free of constraints and where the dynamics is implemented as a canonical transformation. This provides a framework for the discussion of topology change in canonical quantum gravity. In the Lorentzian case, the framework appears to be naturally free of the ``spikes'' that plague traditional formulations. It also provides a well defined recipe for determining the measure of the path integral.