Showing papers on "Canonical transformation published in 1984"

A nine-fold canonical decomposition for linear systems

Abstract: The zero structure for non-minimal proper systems in state-space form is investigated. The approach is ‘ geometric ’, and a complete characterization in geometric terms is given of the invariant, decoupling, system and transmission zeros, as defined by Rosenbrock. The first main result is a formula for the transmission zeros. Second, a ‘ canonical ’ lattice diagram is presented of a decomposition of the state space which can be viewed as the ‘ product ’ of the Kalman canonical decomposition and the Morse canonical decomposition. This decomposition gives a straightforward characterization of all zeros just mentioned in terms of spectral properties of subspaces under a certain class of feedback and injection mappings. Via this diagram a number of equivalent formulae for the transmission zeros are derived. The freedom in pole assignment leads to new characterizations for the invariant and system zeros in terms of greatest common divisors of characteristic polynomials. Finally, the relation is demonstrated be...

Motion of a particle in a ring-shaped potential: an approach via a nonbijective canonical transformation

Abstract: The problem of a particle in the three-dimensional ring-shaped potential ησ2(2a0/r − ηa/r2 sin2 θ)e0 introduced by Hartmann is transformed into the problem of a coupled pair two-dimensional harmonic oscillators with inverse quadratic potentials by using a nonbijective canonical transformation, viz., the Kustaanheimo–Stiefel transformation. The energy E of the levels for the ring-shaped potential is obtained in a straightforward way from the one for the two-dimensional potential — (4Eρ2 + η2σ2a e0/ρ2).

Functions analytic on the half‐plane as quantum mechanical states

Abstract: A transform between the state space of one‐dimensional quantum mechanical systems and a direct sum of two spaces of square integrable functions analytic on the open upper half‐plane is constructed. It gives a representation of usual quantum mechanics on which the free evolution is trivial and the representation of canonical transformation very simple. Generalizations to higher dimensions are also discussed.

Closing and abelizing operatorial gauge algebra generated by first class constraints

Abstract: A canonical transformation of operators of relativistic phase space is constructed, which abelizes the basis of the operator‐valued gauge algebra of first class constraints. The new operators of contraints commute among themselves. The new Hamiltonian commutes with the constraints. Dynamics of the new operators is physically equivalent to the initial one.

Gaussian beam synthetic seismograms in a vertically varying medium

Abstract: Summary We develop the Gaussian beam summation method for a medium in which velocity is only a function of depth. We show that in such a medium the kinematic and dynamic ray tracing equations, i.e. trajectories and amplitudes, may be solved in closed form for any initial wavefield specified at the source. The solution for an individual Gaussian beam is written in terms of the usual functions of ray theory: distance, travel time, intercept time and geometrical spreading. An important result of this analysis, confirmed by numerical experiments, is that one of the base functions selected by Cervený, Popov & Psencik to solve the dynamic ray tracing equations should be modified to avoid causality problems. Finally, by means of a simple canonical transformation we rewrite all the equations in a geographical coordinate system independent of the particular ray trajectories. We then show that Gaussian beam summation is an analytical continuation to complex values of position and slowness of the WKB method proposed by Chapman. A simple computational method is developed in which it is not necessary to determine the coordinates of the observer in ray centred coordinates. This simplifies the computational effort so that Gaussian beam calculation becomes only slightly more expensive than WKB. With respect to the latter method Gaussian beam summation has the advantage that it is possible to control the amplitudes of the cut-off phases due to a finite range of slowness integration.

Vector fields, line integrals, and the Hamilton-Jacobi equation: Semiclassical quantization of bound states

Abstract: In 1972, P. Pechukas [J. Chem. Phys. 57, 5577 (1972)] proposed "classical states" associated with Miller-Good transformations, semiclassical quantization, and the Hamilton-Jacobi equation. In this paper we use some of the concepts established by Pechukas and extend them to define generalized curvilinear coordinates $\stackrel{\ensuremath{\rightarrow}}{\ensuremath{\gamma}}$ associated with nonlinear dynamical systems with Hamiltonians of the form $H=T(\stackrel{\ensuremath{\rightarrow}}{\mathrm{p}})+V(\stackrel{\ensuremath{\rightarrow}}{\mathrm{q}})$. For reasons discussed in the paper we call these coordinates "nodal" coordinates. We show that a transformation to nodal coordinates is to a very good approximation dependent only on Cartesian variables. Using this approximation we demonstrate that, within the regular regime of phase space, canonical transformation to $\stackrel{\ensuremath{\rightarrow}}{\ensuremath{\gamma}}$ coordinates simplifies the analysis of the classical and quantum mechanics of dynamical systems. Our fundamental conclusions are as follows: (1) The solution of the Hamilton-Jacobi equation is separable in $\stackrel{\ensuremath{\rightarrow}}{\ensuremath{\gamma}}$ coordinates; (2) the WKB wave function ${\ensuremath{\Psi}}^{\mathrm{WKB}}$ is separable in $\stackrel{\ensuremath{\rightarrow}}{\ensuremath{\gamma}}$ coordinates; and (3) if the transformation to $\stackrel{\ensuremath{\rightarrow}}{\ensuremath{\gamma}}$ coordinates is made conformal (which we show we are allowed to do within the approximation stated above), then the Schr\"odinger equation and wave functions become separable. Finally, the concepts in this paper are discussed for systems of two degrees of freedom but can be generalized to more degrees of freedom.

Coulombic and ring-shaped potentials treated in a unified way via a nonbijective canonical transformation

Abstract: This paper is concerned with the three-dimensional potentialV q =ησ2 (2a 0/r−q ηa 0 2/r 2 sin2 θ) e0 which comprises as particular cases the ring-shaped potential (q = 1) and the Coulomb potential (q = 0). The Schrodinger equation for the potentialV q is transformed via a nonbijective canonical transformation, viz., the Kustaanheimo-Stiefel transformation, into a coupled pair of Schrodinger equations for two-dimensional harmonic oscillators with inverse-square potentials. As a consequence, the discrete spectrum for the potentialV q is obtained in a straightforward way. A special attention is paid to the caseq = 0. In particular, the coupled pair of Schrodinger equations for two-dimensional harmonic oscillators is tackled in the situations where the spectrum for the potentialV 0 is discrete, continuous, or reduced to the zero point. Finally, some group-theoretical questions about the potentialV q are mentioned as well as a connection, via the Kustaanheimo-Stiefel and the Levi-Civita transformations, between the quantum-mechanical problems for the potentialV q and the Sommerfeld and Kratzer potentials.

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.

Lee Hwa Chung theorem for presymplectic manifolds. Canonical transformations for constrained systems

Abstract: We generalize the analogous of Lee Hwa Chung’s theorem to the case of presymplectic manifolds. As an application, we study the canonical transformations of a canonical system (M, S, Ω). The role of Dirac brackets as a test of canonicity is clarified.

Dynamics of Orbiting Dust under Radiation Pressure

Abstract: For a three-dimensional Keplerian system in the presence of a homogeneous field possibly in uniform rotation, action and angle variables are introduced by canonical transformation in the averaged Hamiltonian truncated at the first order. After substitution, the first order averaged system proves to be integrable. More precisely, it is shown how the orbit space decomposes into a pair of spheres in a three-dimensional space, on which the representative curves are the small circles induced by a finite rotation about a fixed axis. From this intuitive geometric picture follow simple formulas for solving the initial value problem.

On the path integral quantization of the damped harmonic oscillator

Abstract: A time‐dependent canonical transformation is performed in the path integral expression of the propagator for a damped harmonic oscillator to reduce it to a harmonic oscillator with modified frequency. The transformation is carried out in a properly time‐symmetrized expression of the lattice space path integral by making expansions about the midpoint of each time interval.

A singular lagrangian model for N-body relativistic interactions

Abstract: We reformulate a multitemporal model for N-particles by means of a constrained system described by a singular lagrangian. We deduce the canonical transformation adapted to second-class constraints and discuss the quantization of the model. RESUME. On reformule un modele a plusieurs temps pour N particules sous forme d’un systeme avec contraintes decrit par un Lagrangien singulier. On obtient la transformation canonique adaptee aux contraintes de deuxieme espèce et on discute la quantification du modele.

A Microscopic Theory of Superfluids. I Quasi-Particles and the Extended Bogoliubov Transformation

Abstract: Creation and annihilation operators of quasi·particles are introduced into the description of a finite temperature system of degenerate bosons with the aid of a canonical transformation which generalizes the one due to Bogoliubov. A variational calculation taking into account the interaction between non· zero momentum particles shows that the quasi·particles are phonons in the long wave· length limit. The state of the system, which is approximated by the grand canonical ensemble of the quasi· particles, approaches the Hartree·Fock state when the condensate is depleted completely at the critical temperature. This is the first part of a series of papers presenting a microscopic theory of the superfiuid system of degenerate bosonsweakly interacting with each other. The series proposes a formalism whereby the system can be regarded as an assembly of real energy quasi-particles which are phonons in the long wave-length limit and which remain phonons when the interaction between them is taken into consideration. We shall discuss here an approximate determination of the equilibrium state and elementary excitations (quasi­ particles) of the system at finite temperature. The phonon spectrum derived here incor­ porates the interaction between thermally excited particles. The remaining part of the series will deal with the residual interaction between quasi-particles as well as the correlation functions pertaining to superfiuidity. It will be shown that these functions have the forms which are expected on grounds of the two-fluid model and the linear response theory in the long wave-length limit. The phonon spectrum which has been laid by Landau l ) to the basis of the phenomena of superfluidity, was related to the Bose-Einstein condensation for the first time by Bogoliubov. 2 ) Introducing approximate quasi-particles with the aid of the Bogoliubov transformation, he has obtained the phonon spectrum by assuming that (a) the spectrum may be calculated neglecting the interaction between non-zero momentum particles and (b) the depletion of the condensate is small. For the study of the system at elevated temperatures in which we are interested, these assumptions are not reasonable however weak the interaction may be. But when assumption (a) is removed, direct application of the Bogoliubov transformation does not yield a phonon spectrum. 3 ) In order to construct a theory unrestricted by assumption (a), the present author and his collaborators 4 ) have proposed to employ improved quasi-particles which are defined by a canonical transfor­ mation generalizing the one due to Bogoliubov. In the present work we shall employ quasi-particles further improved so as to take into account appreciable depletion of the condensate.

On the phonon mediated electron-electron interactions

Abstract: We have applied a canonical transformation to get phonon mediated electron-electron interactions As examples, the Hubbard model, periodic Anderson model and spinless Anderson model were considered as being relevant for a wide class of systems Corrections to the displaced oscillator transformation and to the results of other authors have been derived The significance of these differences for the effective interactions has been discussed

A group-theoretic treatment of Gaussian optics and third-order aberrations

Abstract: Optical systems produce canonical transformations on the phase space of position and direction of light rays. Contractions of this pseudogrup cut the approximation order automatically to the desired terms throughout. The group Is Sp(2,R) accounts for Gaussian optics and for up-to-third order aberrations. Calculations on concatenation of optical components involve rather simple 2x2 -matrix plus 5-vector algebra.

Stoichiometry as canonical transformation of chemical kinetics

Abstract: Applying the apparatus of differential manifolds and that of classical conservative point mechanics, it is shown that stoichiometry plays the role of canonical transformations in chemical reaction kinetics.

First integrals via polynomial canonical transformations

Abstract: Maharatna, Dutt, and Chattarji [J. Math. Phys. 20, 2221 (1979)] discussed the use of time‐dependent canonical transformations for the determination of first integrals for time‐dependent Hamiltonian systems. One particular proposal that successive time‐dependent polynomial canonical transformations will enable first integrals to be found for a wider variety of time‐dependent polynomial Hamiltonians than can be obtained using time‐dependent linear canonical transformations is shown to be not true for the paradigm which they selected. It is suggested that their ansatz is ill‐founded in general.

A Factor Analysis for Time Series.

Abstract: : This paper studies how to identify hidden factors in multivariate time series process. It is shown that the number of factors must be equal to the rank of both the covariance matrices and the parameter matrices of the infinite moving average representation of the process. A canonical transformation is derived which can recover such factors. The method is illustrated with several examples. Originator-supplied keywords include: Multivariate ARMA process, Canonical analysis, Eigenvalues and Eigenvectors.

Generalization of Equation of Collective Submanifold -- A Theory of Large Amplitude Collective Motion and Its Coupling with Intrinsic Degrees of Freedom--

Abstract: A theory is proposed with the aim of describing large amplitude collective motion and its coupling with intrinsic degrees of freedom. A canonical transformation is investigated in the full time·dependent Hartree·Fock theory, i.e., in the classical image of boson expansion theory. With the aid of the transfor· mation, the whole system is separated into collective and intrinsic degrees of freedom. Under the condition of the unique separation, two types of equations of collective submanifold, which are canonically invariant, are obtained. One is of the same form as that of the conventional equation of collective submanifold. A principle of the specification of coordinate system is discussed. Recently, the present authors, together with !ida, showed that the discussion concerning the non-uniqueness is based on a certain misunderstanding of the equation of collective suhmanifold and the solution exists uniquely.3) In this sense, the theory of collective submanifold is self-contained in the present foim and a powerful method for the descrip­ tion of collective motion. However, if we include intrinsic degrees of freedom in the description, the situation becomes different. In this case, the simplest approach may be the random phase approx­ imation (RPA). The RPA theory gives us the equation of collective submanifold in small amplitude limit. Further, the degrees of freedom orthogonal to the collective one, which we will call generally intrinsic degrees of freedom, are introduced in a natural way. In RPA, fluctuations around the equilibrium point given by the static Hartree-Fock (- Bogoliubov ) method are taken into account in terms of linear effects. Therefore, RP A is not applicable to large amplitude collective motion, in which non-linear effects of the fluctuations are expected to be large. With the aim of making the basic idea of RP A applicable to the collective motion with non-linear effects, boson expansion (BE) theory has been developed. The typical example is the Holstein-Primakoff type representation. The BE theory is a kind of quantum canonical theory constructed by boson operators or their equivalent coordinate and momentum operators. As one of theoretical interests, we can see that the c-number replacement of these operators leads us to full time-dependent

The Influence of Molecular Motions on Multipulse NMR Spectra in Solids

Abstract: A canonical transformation technique generalized for the case of a non-periodical time dependence of the Hamiltonian is used for the analysis of molecular motion influence on multipulse NMR high resolution spectra in solids. The linewidth is shown to be described in some cases by a temperature-dependent effective Hamiltonian. [Russian Text Ignored].

Electromagnetic effects in magnetic superconductors

Abstract: A canonical transformation method in all orders of the spin-photon coupling is used for the calculation of the effective exchange integral in a system consisting of localized spins, superconducting electrons, and an equilibrium radiation field. The condition for the appearance of an incommensurate magnetic order is discussed for a cubic magnetic symmetry, with emphasis on the lattice effects. The necessity of calculating the higher-order effects is proved in the case of perfect spin localization.