Showing papers on "Canonical coordinates published in 1996"

Nonconservative lagrangian and Hamiltonian mechanics

Abstract: Traditional Lagrangian and Hamiltonian mechanics cannot be used with nonconservative forces such as friction. A method is proposed that uses a Lagrangian containing derivatives of fractional order. A direct calculation gives an Euler-Lagrange equation of motion for nonconservative forces. Conjugate momenta are defined and Hamilton's equations are derived using generalized classical mechanics with fractional and higher-order derivatives. The method is applied to the case of a classical frictional force proportional to velocity. \textcopyright{} 1996 The American Physical Society.

Separation of Variables in the Quantum Integrable Models Related to the Yangian Y(sl(3))

Abstract: In the absence of a precise definition of the quantum integrability, the separability of variables can serve as its practical substitute. For any quantum integrable model generated by the Yangian Y[sl(3)], the canonical coordinates and the conjugate operators are constructed which satisfy the “quantum characteristic equation” (the quantum counterpart of the spectral algebraic curve for the L-operator). The coordinates constructed provide a local separation of variables. Conditions are listed which are necessary for the separation of variables to take place. Bibliography: 17 titles.

Reactive Scattering with Row-Orthonormal Hyperspherical Coordinates. 2. Transformation Properties and Hamiltonian for Tetraatomic Systems

Abstract: The formalism for tetraatomic reactive scattering using row-orthonormal hyperspherical coordinates is presented. The transformation properties of these coordinates under kinematic rotations and symmetry operations are derived, as are the corresponding Hamiltonian and volume element. Each of the nine operators which contribute to this Hamiltonian is kinematic-rotation invariant. Continuity conditions appropriate for the absence and presence of the geometric phase associated with conical intersections are described. It is concluded that the row-orthonormal hyperspherical coordinates are particularly well suited for calculations of reactive scattering in tetraatomic systems.

Canonical decomposition of steerable functions

Abstract: Steerable functions find application in numerous problems in image processing, computer vision and computer graphics. As such, it is important to develop the appropriate mathematical tools to analyze them. In this paper, we introduce the mathematics of Lie group theory in the context of steerable functions and present a canonical decomposition of these functions under any transformation group. The theory presented in this paper can be applied and extended in various ways.

Symplectic structures and quantum mechanics

Abstract: Canonical coordinates for the Schrodinger equation are introduced, making more transparent its Hamiltonian structure. It is shown that the Schrodinger equation, considered as a classical field theory, shares with Liouville completely integrable field theories the existence of a recursion operator which allows for the infinitely many conserved functionals pairwise commuting with respect to the corresponding Poisson bracket. The approach may provide a good starting point to get a clear interpretation of Quantum Mechanics in the general setting, provided by Stone–von Neumann theorem, of Symplectic Mechanics. It may give new tools to solve in the general case the inverse problem of quantum mechanics whose solution is given up to now only for one-dimensional systems by the Gel’fand-Levitan-Marchenko formula.

Contradirectional propagation and canonical transformations

Abstract: We examine the preservation of Poisson brackets and commutation relations in contradirectional propagation. It is shown that the relation between the input and the output amplitudes is a canonical transformation. This result is translated to the quantum case where preservation of the commutation relations is demonstrated in the case of linear propagation equations for the amplitudes.

Admissible gauges for constrained systems.

Abstract: Gauge-fixing and gaugeless methods for reducing the phase space in generalized Hamiltonian dynamics are compared with the aim to define the class of admissible gauges. In the gaugeless approach, the reduced phase space of a Hamiltonian system with first class constraints is constructed locally, without any gauge fixing, using the following procedure: Abelianization of constraints with a subsequent canonical transformation so that some of the new momenta are equal to the new Abelian constraints. As a result, the corresponding conjugate coordinates are ignorable (nonphysical) while the remaining canonical pairs correspond to the true dynamical variables. This representation of the phase space prompts the definition of the subclass of admissible gauges, canonical gauges, as functions depending only on the ignorable coordinates. A practical method to determine the canonical gauge is proposed. \textcopyright{} 1996 The American Physical Society.

A second invariant for one-degree-of-freedom, time-dependent Hamiltonians given a first invariant

Abstract: An explicit formula for a second invariant of a one‐degree‐of‐freedom time‐dependent Hamiltonian is derived in terms of the Hamiltonian and an assumed first invariant. If the first invariant is expressed as a function of two canonical functions, a transformation to an autonomous Hamiltonian system is possible.

Spectral properties of quantum N body systems versus chaotic properties of their mean field approximations

Abstract: We present numerical evidence that in a system of interacting bosons there exists a correspondence between the spectral properties of the exact quantum Hamiltonian and the dynamical chaos of the associated mean-field evolution. This correspondence, analogous to the usual quantum-classical correspondence, is related to the formal parallel between the second quantization of the mean field, which generates the exact dynamics of the quantum N -body system, and the first quantization of classical canonical coordinates. The limit of infinite density and the thermodynamic limit are then briefly discussed.

Spectral properties of quantum $N$-body systems versus chaotic properties of their mean field approximations

Abstract: We present numerical evidence that in a system of interacting bosons there exists a correspondence between the spectral properties of the exact quantum Hamiltonian and the dynamical chaos of the associated mean field evolution. This correspondence, analogous to the usual quantum-classical correspondence, is related to the formal parallel between the second quantization of the mean field, which generates the exact dynamics of the quantum $N$-body system, and the first quantization of classical canonical coordinates. The limit of infinite density and the thermodynamic limit are then briefly discussed.

Quantum groups and thermo field dynamics

Abstract: The algebraic structure of Thermo Field Dynamics for bosons can be fully incorporated in the q-deformation of the Weyl-Heisenberg algebra hq(1) The doubling of the degrees of freedom, the set of the tilde-conjugation rules, the Bogoliubov transformation and its generator have a direct and simple interpretation in terms of operators and of properties of hq(1) The notion of “thermal degree of freedom” introduced by Umezawa also finds a more specific formalization since the corresponding “thermal conjugate momentum” can be formally introduced, thus providing us with a set of canonical “thermal” variables

Canonical coordinates and Bergman metrics

Abstract: In this paper we will discuss local coordinates canonically corresponding to a Kahler metric We will also discuss and prove the $C^\infty$ convergence of Bergman metrics following Tian's result on $C^2$ convergence of Bergman metrics At the end, we present an interesting characterization of ample line bundle that could be useful in arithmetic geometry

Canonical coordinates and Bergman metrics

Abstract: In this paper we will discuss local coordinates canonically corresponding to a Kahler metric. We will also discuss and prove the $C^\infty$ convergence of Bergman metrics following Tian's result on $C^2$ convergence of Bergman metrics. At the end, we present an interesting characterization of ample line bundle that could be useful in arithmetic geometry.

Hamiltonian reduction of free particle motion on group SL(2, ${\Bbb R}$)

Abstract: The structure of the reduced phase space arising in the Hamiltonian reduction of the phase space corresponding to a free particle motion on the group ${\rm SL}(2, {\Bbb R})$ is investigated. The considered reduction is based on the constraints similar to those used in the Hamiltonian reduction of the Wess--Zumino--Novikov--Witten model to Toda systems. It is shown that the reduced phase space is diffeomorphic either to the union of two two--dimensional planes, or to the cylinder $S^1 \times {\Bbb R}$. Canonical coordinates are constructed for the both cases, and it is shown that in the first case the reduced phase space is symplectomorphic to the union of two cotangent bundles $T^*({\Bbb R})$ endowed with the canonical symplectic structure, while in the second case it is symplectomorphic to the cotangent bundle $T^*(S^1)$ also endowed with the canonical symplectic structure.

A family of separable integrable Hamiltonian systems and their classical dynamical r-matrix Poisson structures

Abstract: We present a Lax representation for a family of finite-dimensional integrable Hamiltonian systems (FDIHS). Their classical Poisson structures are constructed. The associated r-matrices depend not only on the spectral parameters, but also on the dynamical variables. These dynamical r-matrices provide new solutions of the classical Yang - Baxter equations of dynamical type. We show the separability of their Hamilton - Jacobi equation in new canonical coordinates introduced via the Lax representation.

Exotic (Quixotic?) Applications of Bohm Theory

Abstract: IN THIS PAPER I WOULD LIKE TO DISCUSS, in a simple way, applying Bohm’s theory to general relativity and string theory. Ordinarily, I would preface this with the reasons why I think this is worth doing. But Jim Cushing, in his contribution to this volume, explains, better than I ever could, the advantages of Bohm theory as an interpretation of quantum mechanics. So I will just refer you to his paper.

Non-Hermitian quantum canonical variables and the generalized ladder operators

Abstract: Quantum canonical transformations of the second kind and the non-Hermitian realizations of the basic canonical commutation relations are investigated with a special interest in the generalization of the conventional ladder operators. The operator ordering problem is shown to be resolved when the non-Hermitian realizations for the canonical variables that cannot be measured simultaneously with the energy are chosen for the canonical quantizations. Another merit of the non-Hermitian representation is that it naturally allows us to introduce the generalized ladder operators with which one can solve eigenvalue problems quite neatly. \textcopyright{} 1996 The American Physical Society.

Global canonical symmetry in a quantum system

Abstract: Based on the phase-space path integral for a system with a regular or singular Lagrangian the generalized canonical Ward identities under the global symmetry transformation in extended phase space are deduced respectively, thus the relations among Green functions can be found The connection between canonical symmetries and conservation laws at the quantum level is established It is pointed out that this connection in classical theories, in general, is no longer always preserved in quantum theories The advantage of our formulation is that we do not need to carry out the integration over the canonical momenta in phase-space generating functional as usually performed A precise discussion of quantization for a nonlinear sigma model with Hopf and Chern-Simons terms is reexamined The property of fractional spin at quantum level has been clarified

Canonical collective coordinates of electron transfer based on time‐dependent variational principle

Abstract: Electron transfer reactions have been studied as the time development of many‐electron wave functions based on the time‐dependent variational principle (TDVP). The equation of motion (EOM) of the pseudoclassical mechanics, which is described with canonical coordinates, has been derived by introducing a new trial wave function of the TDVP in the polar form. Based on the EOM, it has been shown that the transition state of the electron transfer corresponds to the saddle point at which an unstable broken‐symmetry wave function occurs in the variational method for stationary states. By the maximal decoupling condition, canonical collective coordinates of the electron transfer have been separated in the TDVP–EOM with canonical variables. A simple example has been given for a symmetric electron transfer reaction in H3 by using the time‐dependent cluster expansion of wave function.

Classical and quantum localization and delocalization in the Fermi accelerator, kicked rotor and two‐sided kicked rotor models

Abstract: The phenomena of dynamical localization, both classical and quantum, are studied in the Fermi accelerator model. The model consists of two vertical oscillating walls and a ball bouncing between them. The classical localization boundary is calculated in the case of ‘‘sinusoidal velocity transfer’’ [A. J. Lichtenberg and M. A. Lieberman, Regular and Stochastic Motion (Springer‐Verlag, Berlin, 1983)] on the basis of the analysis of resonances. In the case of the ‘‘sawtooth’’ wall velocity we show that the quantum localization is determined by the analytical properties of the canonical transformations to the action and angle coordinates of the unperturbed Hamiltonian, while the existence of the classical localization is determined by the number of continuous derivatives of the distance between the walls with respect to time.

Canonical and D-transformations in theories with constraints

Abstract: We describe a class of transformations in a super phase space (we call them D-transformations) which play the role of ordinary canonical transformations in theories with second-class constraints. Namely, in such theories they preserve the form invariance of equations of motion, their quantum analogs are unitary transformations, and the measure of integration in the corresponding Hamiltonian path integral is invariant under these transformations.

The parametric-manifold approach to canonical gravity

Abstract: The canonical structure of gravitation in general relativity is investigated. The Einstein-Hilbert action is decomposed with respect to a generic congruence of timelike curves. The non-Riemannian geometry of the curves, considered to be points of a 3 — D differentiate manifold, incorporates time as a parameter in the differential structure of the manifold.

Introduction of local coordinates for a countable system of differential equations in the neighborhood of an invariant manifold

Abstract: We establish conditions of the existence of local coordinates for a countable system of differential equations in the neighborhood of an invariant manifold and present the form of this system in these coordinates.

Quasi-linear coordinates in general relativity

Abstract: An extension of the Jost and Karcher coordinates to the relativistic space-time is studied. The domain and Hessian of these extended coordinates are bounded by functions vanishing if the space-time is flat. The quasi-linearity property is analysed in the class of space-times obtained by Tzanakis which generalize the Robertson-Walker model. The transformation of normal Fermi coordinates into quasi-linear coordinates is obtained. Finally, the approximate position vector fields defined using Jost-Karcher coordinates and normal Fermi coordinates are compared.