Showing papers on "Canonical coordinates published in 1978"

Quantization and independent coordinates

Abstract: This paper reviews canonical and other quantization methods, gives a quantization rule that gives results that agree with those obtained by direct transformation of the Laplacian, and thus expresses the correct Hamiltonian operator in terms of proper conjugate momentum operators. It gives applications to a number of curvilinear coordinates, introduces the class of independent coordinates, presents generalized Jacobi coordinates, and discusses the molecular vibration‐rotation Hamiltonian.

A Lagrangian for two interacting relativistic particles: Canonical formulation

Abstract: The canonical formalism of a singular-Lagrangian model describing the interaction between two relativistic particles is studied. Instead of following the Dirac method, we make use of a canonical transformation that enables us to work in the complete phase space. The covariance and the quantization of the model are discussed.

Nonbijective canonical transformations and their representation in quantum mechanics

Abstract: In the present paper we analyze the representations in quantum mechanics of classical canonical transformations that are nonbijective, i.e., not one‐to‐one onto. We take as the central example the canonical transformation that changes the Hamiltonian of a one‐dimensional oscillator of frequency κ−1 into one of frequency k−1 where κ, k are relatively prime integers. For the particular case k=1, the mapping of the original phase space (x,p) onto the new one (x, p) is κ to 1 and the equivalent points in (x,p) are related by a cyclic group Cκ of linear canonical transformations. When formulating this problem in Bargmann Hilbert space, the canonical transformation can be related to the conformal transformation w=zκ, which again is κ to 1 and where a group Cκ also appears. This cyclic group proves fundamental for the determination of representations of the conformal transformation in Bargmann Hilbert space. To begin with, it suggests that while we can take in the original Bargmann Hilbert space a single compo...

The canonical variables, the symplectic structure and the initial value formulation of the generalized Einstein-Cartan theory of gravity

Abstract: Canonical variables for the generalized (non-metric) Einstein-Cartan theory of gravity are defined. The space of solutions is equipped with a closed differential 2-form Ω. The symplectic 2-form Ω has a diagonal representation in terms of canonical variables. A geometric interpretation of the canonical variables is presented and the 3+1 formulation of the field equations is given.

Hamiltonian Theory of The Libration of The Moon

Abstract: The feasibility of applying the Lie transform method to the problem of the physical libration of the Moon is investigated. By a succession of canonical transformations, the Hamiltonian of the problem is brought under a form suitable for perturbation technique. The mean value of the inclination of the angular momentum upon the ecliptic and the frequencies of the free libration are computed.

On canonical formalism in field theory with derivatives of higher order-canonical transformations

Abstract: For a canonical formalism with derivatives of higher order, the corresponding theory of canonical transformations is given in the most general case for classical and covariant field theory. Relations with the generating functional, infinitesimal transformations, Hamilton-Jacobi method, Lagrange and Poisson brackets, as well as integral invariants of the first and higher orders and the corresponding Liouville theorem are considered.

Canonical Transformations in the Optimal Control of Mechanical Systems

Abstract: In this paper the canonical perturbation method, which is widely used in analytical mechanics, is applied to optimal control problems. It is shown that the state and adjoint equations present additional interesting symmetries if the state equations are themselves of the Hamiltonian type, which is frequently the case if a mechanical system is to be controlled. The application of the canonical perturbation method to optimal control problems turns out to be particularly simple, if the optimal control is piecewise constant. Several examples are considered.

On a geometric interpretation of Heisenberg’s commutation rule and the algebraic structure of the Pauli equation

Abstract: It is shown that Heisenberg's commutation rule between the position co-ordinate and the corresponding canonically conjugate momentum may be interpreted by noncommuting geometrical structures. As in the absence of a magnetic field the Euclidean norm of the momentum space directly enters the kinetic energy, the momentum space can be mapped onto the quaternion fieldU2. Such a mapping preserves the norm of the momentum space. By that, the geometric and algebraic structure of the Pauli equation can be obtained and the relationship between the Pauli and the Dirac equation can be made apparent by noncommuting algebraic structures. In an appendix it will also be shown that the extension of the procedure to vector spaces equipped with Riemannian geometry makes no difficulties and a covariant quantization procedure can be formulated.

An extended particle with interacting constituents

Abstract: It is shown that a manifestly covariant description of a system of particles can accommodate pairwise scalar interactions without sacrificing the canonical integrity of each particle co-ordinate and its conjugate momentum. The proof requires the introduction of a center-of-mass four-vector co-ordinate. The deduced functional relationship between this center-of-mass proper time and the proper time of each constituent particle yields the expected binding-energy defect for the system rest mass.

Mathematical Results of the General Planetary Theory in Rectangular Coordinates

Abstract: Mathematical construction of the general planetary theory has led to the series of two forms for the coordinates of eight major planets (excluding Pluto). The series of the first form are Poisson series where all orbital elements except the semi-major axes occur in literal, shape. The series of the second form are polynomial-exponential series with respect to the time and serve to calculate the ephemerides. The arbitrary constants of the theory are related to the Keplerian elements. The terms of the zero and first degree in eccentricities and inclinations have been found in the second approximation with, respect to the disturbing masses. Among those of particular interest are the resonant terms caused by the commensurabilities of the mean notations of triplets of planets.

On certain families of rational functions arising in dynamics

Abstract: In the last decade, it has become apparent that linear systems, depending on parameters, can occur in surprisingly diverse situations, including families of rational solutions to the Korteweg-de Vries equation or to the finite Toda lattice. Now, it turns out that the "inverse scattering" method employed by Moser [12] to obtain canonical coordinates for the finite, homogeneous Toda lattice is precisely the network synthesis question for RC networks, with parameters. The corresponding question has a negative answer in the multivariable RC or for RLC network synthesis, due to global topological obstructions. The multivariable RC setting is ideal for the analysis of the periodic Toda lattice and the topological obstructions are, in fact, generated by tori.

On the theory of canonical forms for control systems with delay

Abstract: Some canonical forms for pointwise controllable systems with retarded argument are considered. Bibliography: 4 titles.

Classical mechanics in phase space revisited

Abstract: A Bose type of classical Hamilton algebra, i.e., the algebra of the canonical formalism of classical mechanics, is represented on a linear space of functions of phase space variables. The symplectic metric of the phase space and possible algorithms of classical mechanics (which include the standard one) are derived. It is shown that to each of the classical algorithms there is a corresponding one in the phase space formulation of quantum mechanics.

New Formulation of de Sitter’s Theory of Motion for Jupiter I–IV. I. Equations of Motion and the Disturbing Function

Abstract: A brief discussion is given of the basic features of de Sitter’s theory. The main advantage of his theory is that it contains no small divisors, thanks to the use of elliptic rather than circular intermediate orbits in the first approximation. A 50-year extension of the satellite observations available to de Sitter makes it desirable to rederive the elements of his intermediate orbits, whose perijoves have a common retrograde motion. Furthermore, the theory suffers from a convergence problem, which can be avoided by reformulating the theory in terms of canonical variables, a task that is begun here. We adopt a formulation in Poincare’s canonical relative coordinates rather than, as customary, in ordinary relative coordinates or in the Jacobian canonical coordinates. By means of the generalized Newcomb operators devised by Izsak, the disturbing function is expanded in a form that is very convenient for use with the modified Delaunay variables, G, L - G, H - G, l + ω + Ω, l, and Ω and their associated Poincare variables.

Canonical transformation of polynomial hamiltonians

Abstract: This thesis is an investigation of time-dependent canonical transformations as applied to polynomial hamiltonians. The motivation came from two sources. The first was the work of Lewis [74, 7 5, 16, 7 7] on an exact invariant for the time-dependent harmonic oscillator. The second was the construction by Fradkin ([?]; see also [4]) of an invariant tensor for the time-independent three dimensional isotropic oscillator. This tensor, together with the angular momentum, was used as a basis for the generators of the dynamical symmetry group of that system, namely SU(3) .