Showing papers on "Canonical transformation published in 1986"
TL;DR: In this paper, the authors studied the birational canonical transformations of the Painleve system ℋ, that is, the Hamiltonian system associated with the painleve differential equations.
Abstract: In this series of papers, we study birational canonical transformations of the Painleve system ℋ, that is, the Hamiltonian system associated with the Painleve differential equations. We consider also τ -function related to ℋ and particular solutions of ℋ. The present article concerns the sixth Painleve equation. By giving the explicit forms of the canonical transformations of ℋ associated with the affine transformations of the space of parameters of ℋ, we obtain the non-linear representation: G→G*, of the affine Weyl group of the exceptional root system of the type F4 A canonical transformation of G* can extend to the correspondence of the τ -functions related to ℋ. We show the certain sequence of τ -functions satisfies the equation of the Toda lattice. Solutions of ℋ, which can be written by the use of the hypergeometric functions, are studied in details.
404 citations
TL;DR: The simplest two-body relativistic system with direct interaction, described by two first-class constraints, is investigated and the initial data problem has been solved by means of a quantum canonical transformation, which transforms the integro-differential equations into differential equations.
Abstract: The simplest two-body relativistic system with direct interaction, described by two first-class constraints, is investigated. After a description of the multitime approach (canonical quantization without gauge fixings), the two coupled integro-differential wave equations are solved. The elementary solutions for the bound states are found and are shown to transform as irreducible representations of the Poincare group. Invariant scalar products are introduced assuring the unitarity of the representations and some of the associated conserved currents are discussed. The initial data problem has been solved by means of a quantum canonical transformation, which transforms the integro-differential equations into differential equations.
67 citations
TL;DR: In this article, it was shown that a variational principle in the Langrange form yields the same two-fluid equations as the Lagrange form in the Euler form.
Abstract: In a recent paper the two-phase flow equations for a bubbly liquid/gas mixture were derived by variational methods. Starting point was an Euler form of Hamilton's extended principle of least action. The effect of the virtual mass of the gas bubbles was included. It is demonstrated now that a variational principle in the Langrange form yields the same two-fluid equations. In addition it is shown how the Lagrange form is trandformed in the Euler form by means of a canonical transformation. With regard to a recent discussion in the literature the material frame indifference or objectivity of the virtual-mass terms is investigated. The mutual force between the two-phases which is associated with the virtual mass of the gas bubbles turns out to be objective. In the limit of low bubble concentrations the results of the one-bubble theory are recovered. A corrected value is derived for the lift coefficient of a gas bubble in a rotational flow. It is indicated how a scheme of iterative solutions yields higher order approximations in which the mutual interaction of the gas bubbles is taken into account.
63 citations
TL;DR: By the use of a canonical transformation the problem of the harmonic oscillator with a time-dependent mass has been transformed to that of an oscillatorWith-a-time-dependent frequency.
Abstract: By the use of a canonical transformation the problem of the harmonic oscillator with a time-dependent mass has been transformed to that of an oscillator with a time-dependent frequency. Pseudostationary and quasicoherent states are discussed.
52 citations
TL;DR: In this article, a canonical transformation was proposed to reduce the averaged planar planetary problem near resonance to a one degree of freedom problem when the perturbation is truncated at the first order in the eccentricities.
Abstract: We propose a canonical transformation reducing the averaged planar planetary problem near resonance to a one degree of freedom problem when the perturbation is truncated at the first order in the eccentricities. This reducing transformation leads to a very simple explanation of the puzzling behaviour of the Apocentric Librators, a class of asteroids identified by Franklinet al. (1975). An exploration of the phase space of the average problem with the use of the mapping technique shows that the alternation of two libration mechanism is a common feature for initial conditions near, but not inside, the deep resonance region.
51 citations
TL;DR: In this article, Sessin and Ferraz-Mello (Celes. Mech.32, 307,332) simplified the Hori auxiliary system for the motion of two planets with periods nearly commensurate in the ratio 2∶1 by the introduction of canonical variables.
Abstract: The solution by Sessin and Ferraz-Mello (Celes. Mech.32, 307–332) of the Hori auxiliary system for the motion of two planets with periods nearly commensurate in the ratio 2∶1 is considerably simplified by the introduction of canonical variables. An analogous canonical transformation simplifies the elliptic restricted problem.
49 citations
TL;DR: In this article, an algebraic semiclassical approach to the calculation of vibrational transition probabilities in inelastic collisions between molecules is presented, which leads to a set of linear differential equations for the parameters of the coherent state, coupled to the classical Hamilton equations.
Abstract: An algebraic semiclassical approach to the calculation of vibrational transition probabilities in inelastic collisions between molecules is presented. Translational motion is treated classically, while vibrational motion is described quantum mechanically using the generalized coherent state of a proper Lie algebra. This leads to a set of linear differential equations for the parameters of the coherent state, coupled to the classical Hamilton equations. Use is also made of a time dependent canonical transformation to simplify the algebraic structure. Two examples are treated explicitly: colinear collision of an atom and a diatom and a diatom–diatom collision. Good agreement with the exact quantum results is found.
34 citations
TL;DR: In this paper, the authors present explicit results for aberration order nine for a spherical surface with symmetry algebra, which allows recursive computation of its aberration coefficients to arbitrarily high order.
28 citations
TL;DR: In this article, the Hamiltonian, action-angle equations of motion, and necessary coordinate transformations are presented for both rigid and vibrating rotor models of the diatom, and a quantitatively realistic interatomic diatom potential model U(r) is presented, which allows exact solution of diatom energy "eigenvalues" E(V,J), and facilitates analytical inversion of the transformation equations to obtain virtually exact expressions for r(ψV;V, J) without resort to any dynamical approximations.
Abstract: Application of action‐angle variables for the physical description and direct numerical integration of the exact Hamilton’s classical equations of motion for 3D nonreactive atom–diatom collision systems is discussed in detail. The Hamiltonian, action‐angle equations of motion, and necessary coordinate transformations are presented for both rigid and vibrating rotor models of the diatom. Generalization to the vibrating‐rotor model of the diatom is done via the introduction of internuclear position and momentum variables (r,pr), and subsequent canonical transformation (r,pr) →(V,ψV) to vibrational action‐angle variables. We present a quantitatively realistic interatomic diatom potential model U(r) which allows exact solution of diatom energy ‘‘eigenvalues’’ E(V,J), and which facilitates analytical inversion of the transformation equations to obtain virtually exact expressions for r(ψV;V,J) without resort to any dynamical approximations. Use of rotational and especially vibrational action‐angle variables (J,...
27 citations
TL;DR: In this paper, the transformation of time-varying multivariable systems into canonical structures was studied using differential matrix operators, and it was shown that canonical forms derived by other authors are special cases of the canonical form of this paper.
Abstract: This paper concerns the transformation of time-varying multivariable systems into canonical structures. The study employs differential matrix operators. It leads to a systematic and straightforward technique for developing canonical forms for time-varying multivariable systems that are uniformly observable and «lexicography-fixed’. It shows that canonical forms derived by other authors are special cases of the canonical form of this paper. The derived canonical forms are not unique. However, their structures are controlled by the designer
23 citations
TL;DR: In this paper, the Hamiltonian of a positively charged ion in a homogeneous magnetic field is transformed into the sum of three commuting operators and a small term coupling the centre of mass and internal motions.
Abstract: With a canonical transformation, the Hamiltonian of a positively charged ion in a homogeneous magnetic field is transformed into the sum of three commuting operators and of a small term coupling the centre-of-mass and internal motions. The commuting operators are: (i) the corresponding Hamiltonian in the infinite-nuclear-mass approximation, (ii) a small additional Zeeman term, and (iii) a centre-of-mass operator. In the hydrogenic-ion case, the coupling term is treated as a perturbation and does not contribute to first order. Schematically, the perturbation parameter is the product of the reduced field B(2.35*105 T) and the electron-to-nuclear mass ratio. The commuting operators provide an exact first-order centre-of-mass correction to the energy, valid for any field strength. Higher order corrections are calculated approximately for strong fields. Accurate quantum excesses are also determined with a simple variational basis.
01 Jan 1986
TL;DR: In this article, the authors present a brief survey of considerations and results of a study under-taken by the author and his group, on the links between the Hamiltonian formalism and the numerical methods for solving dynamical problems expressed in the form of the canonical system of diierential equations.
Abstract: It is an honor and a pleasure for me to present the inaugural talk at the Tenth International Conference on Numerical Methods in Fluid Dynamics in Beijing. We present a brief survey of considerations and results of a study 1, 2, 3, 4, 6], under-taken by the author and his group, on the links between the Hamiltonian formalism and the numerical methods for solving dynamical problems expressed in the form of the canonical system of diierential equations dp i The canonical system (1.1) with remarkable elegance and symmetry was introduced by Hamilton as a general mathematical scheme, rst for problems of geometrical optics in 1824, then for conservative dynamical problems in 1834. The approach was followed and developed further by Jacobi into a well-established mathematical formalism for analytical dynamics, which is an alternative of, and equivalent to, the Newtonian and Lagrangian formalisms. The geometrization of the Hamiltonian formalism was undertaken by Poincare in 1890's and by Cartan, Birkhoo, Weyl, Siegel, etc., in the 20th century; this gave rise a new discipline, called symplectic geometry, which serves as the mathematical foundation of the Hamiltonian formalism. It is known that, Hamiltonian formalism, apart from its classical links with analytical mechanics, geometrical optics, calculus of variations and non-linear PDE of rst order, has inherent connections also with unitary representations of Lie groups, geometric quantization, pseudo-diierential and Fourier integral operators, classiication of singularities, integrability of non-linear evolution equations, optimal control theory, etc.. It is also under extension to innnite dimensions for various eld theories, including uid dynamics, elasticity, elec-trodynamics, plasma physics, relativity, etc.. Now it is almost certain that all real physical
01 Jan 1986
TL;DR: In this paper, the authors introduce the reader to single particle dynamics in circular accelerators with an emphasis on nonlinear resonances, and discuss the effects of the nonlinear terms.
Abstract: The purpose of this paper is to introduce the reader to single particle dynamics in circular accelerators with an emphasis on nonlinear resonances. We begin with the Hamiltonian and the equations of motion in the neighborhood of the design orbit. In the linear theory this yields linear betatron oscillations about a closed orbit. It is useful then to introduce the action-angle variables of the linear problem. Next we discuss the nonlinear terms which are present in an actual accelerator, and in particular, we motivate the inclusion of sextupoles to cure chromatic effects. To study the effects of the nonlinear terms, we next discuss canonical perturbation theory which leads us to nonlinear resonances. After showing a few examples of perturbation theory, we abandon it when very close to a resonance. This leads to the study of an isolated resonance in one degree of freedom with a 'time'-dependent Hamiltonian. We see the familiar resonance structure in phase space which is simply closed islands when the nonlinear amplitude dependence of the frequency or 'tune' is included. To show the limits of the validity of the isolated resonance approximation, we discuss two criteria for the onset of chaotic motion. Finally, we study an isolatedmore » coupling resonance in two degrees of freedom with a 'time'-dependent Hamiltonian and calculate the two invariants in this case. This leads to a surface of section which is a 2-torus in 4-dimensional phase space. However, we show that it remains a 2-torus when projected into particular 3-dimensional subspaces, and thus can be viewed in perspective.« less
TL;DR: In this paper, the zonal Earth satellite theory is studied in the polar nodal canonical set of variables (τ, ϑ, υ,R, i,N) taking advantage of the radial intermediaries and the regularization and linearization methods.
Abstract: Taking advantage of the radial intermediaries and the regularization and linearization methods, the zonal Earth satellite theory is studied in the polar nodal canonical set of variables (τ, ϑ, υ,R, i,N).
TL;DR: In this paper, a characterization of the infinitesimal generator of one-parameter Lie Groups of non-univalent canonical transformations is given, which is used to derive a general form of the virial theorem.
TL;DR: In this article, it was shown that if the resonances are sparsely distributed through the system most of the sites in the transformed Hamiltonian behave essentially like an integrable system, at least for as long a time as the trajectory of the system lies within the region where the canonical transformation is defined.
Abstract: Given a weakly coupled Hamiltonian system with short range, one dimensional interactions, andany initial conditions a canonical change of variables is constructed which yields a new Hamiltonian consisting of three parts—an integrable term, a resonant term whose effects are localized in those regions of the system which give small denominators in the Kolmogorov-Arnol'd-Moser iteration scheme and a non-resonant interaction term which is very small. (In particular, much, much smaller than our original interactions.) The conditions which allow such a transformation to be constructed are independent of the number of degrees of freedom in the system, as are the estimates on the size of the various terms. Thus, if the resonances are “sparsely” distributed through the system most of the sites in the transformed Hamiltonian behave essentially like an integrable system, at least for as long a time as the trajectory of the system lies within the region where the canonical transformation is defined. In subsequent work it is shown that this time is long, and once again independent of the number of degrees of freedom in the system.
TL;DR: In this paper, a new normal form, called versal, for the linearized Hamiltonian vector field of the planar restricted three-body problem at the Lagrange equilibrium point L4 depending smoothly on the mass ratio for all values close to the critical Routh's ratio is described.
TL;DR: In this paper, a canonical transformation was carried out on the system consisting of the bulk modes in interaction with a charged particle, which effectively incorporated the charge and the bulk field into redefined states of the system in a manner analogous to the Power-Zienau transformation of quantum optics.
Abstract: It is shown how explicit field quantisation of bulk plasma modes can be carried out from first principles when the theory is based on the continuum jellium model with spatial dispersion. A canonical transformation, motivated by classical correspondence, is first carried out on the system consisting of the bulk modes in interaction with a charged particle. This transformation effectively incorporates the charge and the bulk field into redefined states of the system in a manner which is analogous to the Power-Zienau transformation of quantum optics. When the transformed theory is applied to the scattering of electrons by the bulk plasmons it more readily enables the screening effects to be taken into account. The shape of the energy-loss-spectrum and its dependence on the momentum transfer are shown to be in satisfactory agreement with available experimental data on metals.
TL;DR: In this article, a theory based on the time-dependent Hartree-Fock theory is proposed with the aim of describing large amplitude collective motion and its coupling with intrinsic degrees of freedom.
Abstract: A theory based on the time·dependent Hartree·Fock theory is proposed with the aim of describing large amplitude collective motion and its coupling with intrinsic degrees of freedom. With the use of time·dependent canonical transformation, the whole system is separated into two types of degrees of freedom. Under the condition of unique separation, two kinds of equations of collective submanifold are obtained. One js for the collective motion and of the same form as that previously given by the present authors. The other is for the intrinsic degrees of freedom and consists of a set of linear partial differential equations. It is reduced to the old one, the form of which is non-linear.
TL;DR: The quantum canonical transformation between the spherical polar and action-angle coordinates and momenta is constructed, and the Hamiltonian and energies in terms of the action variables are determined.
Abstract: Angular-momentum theory is formulated in terms of quantum action-angle variables. The quantum canonical transformation between the spherical polar and action-angle coordinates and momenta is constructed, and the Hamiltonian and energies in terms of the action variables are determined. Various sets of action-angle variables are considered.
TL;DR: In this paper, the authors extended the results to Hamiltonians with periodic potentials which have a band spectra and showed that the canonical transformations are non-linear and non-bijective.
TL;DR: In this article, a perturbation theory for acoustic ray propagation is presented which is based on the analogy between Hamilton's Principle of Least Action (independent variable: time) and Fermat's principle of least time (independent variables: space coordinate).
Abstract: A perturbation theory for acoustic ray propagation is presented which is based on the analogy between Hamilton’s Principle of Least Action (independent variable: time) and Fermat’s Principle of Least Time (independent variable: space coordinate). In a vertical ocean section with x positive to the right and z positive upward, the Lagrangian variables z and z (a dot denotes differentiation with respect to x; z=dz/dx) are first replaced by the canonical variables (z, p) of the corresponding Hamiltonian formulation. For the zeroth‐order case, where the sound c(x,z) reduces to the form c0(z), a canonical transformation, z, p → ω0, J0, can be made. The new variables are the analogs of angles‐action variables in dynamics, and are constants. The generator for this transformation is just the Eikonal function for zeroth order and, hence, is recognized as the travel time between a source and receiver of sound. Finally, when the x‐dependent perturbations are turned on, ω0 and J0 cease to be constants and a second, ...
CERN1
TL;DR: In this article, it was shown that all integrable systems on R 2n are canonically equivalent, and that the canonical transformation that does the job can be used to transform the moment and cross-sequences associated to each Hamiltonian onto each other.
TL;DR: In this article, a Lagrangian formulation for a general Hamiltonian theory with constraints is constructed and a modification of the standard procedure of the Hamiltonianization is proposed for the case of primary constraints.
Abstract: A Lagrangian formulation is constructed for a general Hamiltonian theory with constraints. A modification is proposed of the standard procedure of the “Hamiltonianization” of a Lagrangian theory in the case when the Lagrangian theory has primary constraints. The obtained results are used to establish the Lagrangian form of infinitesimally small canonical transformations in the Hamiltonian formalism.
TL;DR: In this article, the elementary-excitation spectrum of a system of bosons interacting with a spherical shell potential function from a theory combining the advantages of the correlated-basis-function and canonical transformation was studied.
Abstract: We study the elementary-excitation spectrum of a system of bosons interacting with a spherical-shell potential function from a theory combining the advantages of the method of correlated-basis-function and canonical transformation. By fitting physical parameters with those of liquid helium four into our derived equations, the computed results agree reasonably well with the experimental ones obtained from neutron scattering.
01 Jan 1986
TL;DR: In this paper, the Lie-a-gebraic formulation of geometrical rays for imaging systems is transcribed to a form suitable for wave fields and the merits of a coherent-state wave field formulation are stressed along with a related path-integral representation suitable for a study of general aberrations.
Abstract: The Lie-aIgebraic formulation of geometrical rays for imaging systems is transcribed to a form suitable for wave fields. The merits of a coherent-state wave field formulation are stressed along with a related path-integral representation suitable for a study of general aberrations.
Posted Content•