Showing papers on "Canonical transformation published in 1980"
TL;DR: In this paper, the canonical formalism for constrained systems with a finite number of degrees of freedom was developed by making use of the Poincare-Cartan integral invariant method.
Abstract: In this work we develop the canonical formalism for constrained systems with a finite number of degrees of freedom by making use of the Poincare–Cartan integral invariant method. A set of variables suitable for the reduction to the physical ones can be obtained by means of a canonical transformation. From the invariance of the Poincare–Cartan integral under canonical transformations we get the form of the equations of motion for the physical variables of the system.
30 citations
01 Jan 1980
TL;DR: In this paper, the authors discuss another aspect of the theory of nonlinear evolution equations which are integrable by the inverse scattering method, and they show that these equations are infinite-dimensional Hamiltonian systems, and their explicit solvability has the following interpretation in the language of Hamiltonian system: a transform from the initial Cauchy data to the scattering data which underlies the inverse scatter method represents a nonlinear canonical transformation to variables of the action-angle type.
Abstract: In this chapter we shall discuss another aspect of the theory of nonlinear evolution equations which are integrable by the inverse scattering method. Those having applications of physical interest prove to be infinite-dimensional Hamiltonian systems. Their explicit solvability has the following interpretation in the language of Hamiltonian systems: a transform from the initial Cauchy data to the scattering data which underlies the inverse scattering method represents a nonlinear canonical transformation to variables of the action-angle type. This interpretation was originally suggested by ZAKHAROV and this author [11.1] for the case of the Korteweg-de Vries equation. Its most interesting applications are in the quantization problem for nonlinear equations. It also played an important heuristic role in clarifying the slow stochastization of an oscillator lattice [11.2] and in deriving the integrability of finite-dimensional systems of stationary points for the higher conservation laws [11.3]. In this connection we note that the N-dimensional rigid body equations have recently been shown to be integrable by MANAKOV [11.4].
21 citations
TL;DR: In this article, a model which describes a relativistic interaction between two point particles via an action at a distance is derived from a set of hypotheses on the relativism dynamics.
Abstract: In this paper a model which describes a relativistic interaction between two point particles via an action at a distance is derived from a set of hypotheses on the relativistic dynamics. From this set of hypotheses a singular Lagrangian is obtained. The aim of this paper is to find a link between the singular-Lagrangian approach and other approaches to the relativistic dynamics of two particles. The connection of this Lagrangian model with the predictive approach of the relativistic mechanics is studied, by showing that it is possible to calculate the instantaneous forces, at least in principle. An explicit canonical transformation is given, such that a subset of the new canonical variables becomes free of constraints. In this way the instant form of the relativistic dynamics found by Bakamjan and Thomas and by Foldy is recovered.
15 citations
TL;DR: In this paper, the authors consider the SL(2,R) Lie algebra of second-order differential operators given by the Schrodinger Hamiltonians of the harmonic, repulsive, and free particle, all with a strong centripedal core placing them in the Ceq continuous series of representations.
Abstract: We consider the sl(2,R) Lie algebra of second‐order differential operators given by the Schrodinger Hamiltonians of the harmonic, repulsive, and free particle, all with a strong centripedal core placing them in the Ceq continuous series of representations. The corresponding SL(2,R) Lie group is shown to be a group of integral transforms acting on a (two‐component) space of square‐integrable functions, with an integral (matrix) kernel involving Hankel and Macdonald functions. The subgroup bases for irreducible representations consist of Whittaker, power, Hankel, and Macdonald functions. We construct the operator which intertwines this realization of SL(2,R) with the more familiar Bargmann realization on functions on the unit circle. This operator implements the canonical transformation of the above Schrodinger systems to action and angle variables.
15 citations
TL;DR: In this paper, a canonical transformation of an infinite system of differential equations describing the motion of the three-body system in the adiabatic basis is suggested, and the physical meaning of the transformation and the solutions obtained are discussed.
Abstract: For pt.I see ibid., vol.12, no.4, p.567 (1979). A canonical transformation of an infinite system of differential equations describing the motion of the three-body system in the adiabatic basis is suggested. This transformation allows one to reduce the original problem to the solution of a finite set of differential equations. As an example, a system of two differential equations is constructed which represents the infinite (or finite) system of equations within the accuracy (2M)-2, where M-1=mc/M0 is the ratio of the mass, mc, of the negative charged particle, c, to the reduced mass, M0, of two positively charged particles a and b. The physical meaning of the transformation and the solutions obtained is discussed.
15 citations
TL;DR: In this article, a simple canonical transformation is shown that makes the transition from the classical Delaunay elements to the Scheifele variables, and the meaning of the new variables, especially the use of the true anomaly as one of the variables, is given.
Abstract: It is noted that a satellite theory, based on extended phase space and on the true anomaly, was introduced by Scheifele (1970). In the present paper a simple canonical transformation is shown that makes the transition from the classical Delaunay elements to the Scheifele variables. It is stressed that neither spherical coordinates nor Hamilton-Jacobi theory is used. Finally, attention is given to the meaning of the new variables, especially the use of the true anomaly as one of the variables.
12 citations
12 citations
10 citations
TL;DR: This paper showed that the generalized backlund transformations of Calogero and Degasperis form the group of canonical transformations that keep integrable nonlinear evolution equations in 1 + 1 dimensions invariant.
7 citations
6 citations
TL;DR: In this article, a positive Hamiltonian expressed in terms of unconstrained canonical variables is obtained for the gauge field with an external source, where the Gauss-law constraints are second class.
Abstract: When the gauge field interacts with an external source, the Gauss-law constraints are second class. The difficulties associated with this are resolved by a canonical transformation. A positive Hamiltonian expressed in terms of unconstrained canonical variables is obtained.
01 Jan 1980
TL;DR: In this paper, the classical theory of canonical transformations is briefly reviewed and applied to the generation of angle-action variables for one vibrational degree of freedom, and the origins of the primitive semiclassical and various uniform approximations for the S matrix are outlined.
Abstract: The classical theory of canonical transformations is briefly reviewed and applied to the generation of angle-action variables for one vibrational degree of freedom. Analytical aspects of classical inelastic scattering are discussed in a modified angle-action representation. The origins of the primitive semiclassical and various uniform approximations for the S matrix are outlined. Numerical comparisons between exact and semiclassical transition probabilities are given for collinear vibrationally inelastic and reactive models.
TL;DR: In this article, a new phase transition to a state with macroscopic transverse polarization for a system of N two-levels atoms described by the Dicke hamiltonian is discussed.
01 Jan 1980
TL;DR: In this article, the authors presented five lectures on solitons, including a canonical structure of all AKNS-ZS systems of the type considered in this paper, as well as a canonical transformation for the sine-Gordon equation.
Abstract: Five lectures on solitons are presented. The first summarises the discovery of the inverse scattering method for solving the initial value problem of the Korteweg de Vries equation by Kruskal and colleagues: the polynomial conserved densities of the KdV are introduced. The second lecture treats Backlund transformations and conserved densities, especially for the sine-Gordon equation, and extends the inverse scattering method to the 2x2 scheme of Zakharov and Shabat and of Ablowitz, Kaup, Newell and Segur. It is shown that all AKNS-ZS systems represent surfaces of constant negative Gaussian curvature. The geometrical analysis is used to derive non-local conserved densities for the sine-Gordon equation. In the third and fourth lectures the canonical structure of these systems is exhibited: all AKNS-ZS systems of the types considered are infinite dimensional completely integrable Hamiltonian systems. The AKNS-ZS scattering transformation is shown to be a canonical transformation. New canonical co-ordinates expressed in terms of scattering data are found. These are used for the semi-classical quantisation of these systems. The sine-Gordon equation is quantised explicitly and its eigen spectrum found. The relation of this to the eigen spectrum of the spin-00BD;’ x-y-z model of statistical mechanics and the massive Thirring model of massive fermions is sketched. In the final lecture the double sine-Gordon equations uxx-u tt=± (sin u + 00BD; sin λ sin 00BD; u) which arise in resonant non-linear optics and in the theory of spin waves in 3He below 2.6 mK are mentioned. The optical problem (+ ve sign) is used as a vehicle for singular perturbation theory about the sine-Gordon equation.
TL;DR: In this paper, the properties of the generator coordinate subspaces of the many-body Hilbert space, which are associated with the use of generator coordinate method in connection with one-parameter and two-conjugateparameter families of generator states, are examined in detail.
Abstract: The properties of the subspaces of the many-body Hilbert space, which are associated with the use of the generator coordinate method in connection with one-parameter and with two-conjugate-parameter families of generator states, are examined in detail. These families are obtained by letting unitary displacement operators, having as generators canonical operators $\stackrel{^}{P}$ and $\stackrel{^}{Q}$, defined in the many-body Hilbert space, act on a reference state. We show that natural orthonormal base vectors in each case are immediately related to Peierls-Yoccoz and Peierls-Thouless projections, respectively. Through the formal consideration of a canonical transformation to collective, $\stackrel{^}{P}$ and $\stackrel{^}{Q}$, and intrinsic degrees of freedom, we discuss in detail the properties of the generator-coordinate-method subspaces with respect to the kinematical separation of these degrees of freedom. An application is made, using the ideas developed in this paper, (a) to translations, (b) to illustration of the qualitative understanding of the content of existing generator-coordinate-method calculations of giant resonances in light nuclei, and (c) to the definition of appropriate asymptotic states in current generator-coordinate-method descriptions of scattering.
TL;DR: In this paper, the quantum-mechanical current density operator for general Hamiltonian operators was defined and the electric and magnetic multipole-moment operators in close correspondence with the classical case were defined.
Abstract: The authors define the quantum-mechanical current density operator for general Hamiltonian operators and hence define electric and magnetic multipole-moment operators in close correspondence with the classical case. General Hamiltonian operators are transformed to multipole form by canonical transformation. As examples we treat the charge exchange interaction and relativistic corrections to the magnetic moment of an electron moving in a central potential.
01 Jan 1980
TL;DR: In this article, the authors discuss the formal Hamiltonian development with special emphasis on the freedom allowed by canonical transformations and make explicit canonical transformations analogous to those used in quantum electrodynamics.
Abstract: There have been recenCt attempts(1–4) to study radiation theory from a classical point of view. One of particular interest that has been investigated extensively is the formulation due to Jaynes and co-workers(5) called the neoclassical theory. This theory can be developed from the Hamiltonian viewpoint from which equations of motion can be derived. For example, Maxwell’s equations form one set of these equations of motion where the driving currents are c-numbers formed by taking expectation values of the quantum operator currents. Much of the previous work has been devoted to examining the solutions of these equations. Here we discuss the formal Hamiltonian development with special emphasis on the freedom allowed by canonical transformations. In particular, we make explicit canonical transformations analogous to those used in quantum electrodynamics.(6–9) For example, one such transformation gives a flexibility in the choice of the field E or D for the canonical momentum. This corresponds to the transformation from minimal coupling to a Hamiltonian in multipolar form. Despite the classical nature of the Hamiltonian, a detailed analysis of the transformation shows that the Schrodinger character of the underlying dynamics requires closure relations and other sum rules that reflect basic quantum behavior.
Abstract: It is shown that the excitation spectrum of a realistic superfluid Bose system (He II) can be derived from a purely microscopic RPA calculation involving a simple pseudopotential which contains the essential features of the exact interparticle potential, i.e., a repulsive core and an attractive well. The theory is founded on a temperature dependent single-particle picture introduced by a canonical transformation of the exact Hamiltonian and Bogoliubov's variational principle, which was presented in a foregoing work. The “phonon-roton” spectrum results from a certain term of the retarded density-density correlation function in the dispersion relation; the analogous term yields the plasma oscillations in a charged Fermi system.
TL;DR: In this paper, a canonical transformation was found that enables one in the first quasirelativistic approximation (which takes into account orders c−2) to separate the internal motion in a classical system of two particles from its motion as a whole in the case of arbitrary interaction between the particles.
Abstract: A canonical transformation is found that enables one in the first quasirelativistic approximation (which takes into account orders c−2) to separate the internal motion in a classical system of two particles from its motion as a whole in the case of arbitrary interaction between the particles.