Showing papers on "Canonical coordinates published in 1970"

On Canonical Forms and Simplification

Abstract: This paper deals with the simplification problem of symbolic mathematics. The notion of canonical form is defined and presented as a well-defined alternative to the concept of simplified form. Following Richardson it is shown that canonical forms do not exist for sufficiently rich classes of mathematical expressions. However, with the aid of a nmnber- theoretic conjecture, a large subclass of the negative classes is shown to possess a canonical form.

Derivations of Lie brackets and canonical quantisation

Abstract: An extensive analysis of the Dirac problem of canonical quantisation is reported. In this a known solution [1] has been found to be unique to within a canonical transformation under a certain prescribed condition. This proves a conjecture due to Streater [2]. A further canonically inequivalent solution is obtained by relaxing this condition. The results obtained are discussed in terms of the derivation algebras pertaining to the Classical and Quantum Lie brackets. Applications to the study of higher symmetries and to realisations of Lie algebras as polynomial functions of canonical operators are pointed out.

On nonclassical canonical systems

Abstract: Generalizations in the canonical theory of dynamics are made; at first transformations which augment the number of canonical variables, and secondly differential transformations of the independent variable are outlined. This is applied to the perturbed two-body problem. The results are canonical systems using independent variables other than time. This leads to ‘Delaunay-similar’ sets of 8 canonical elements when the Jacobian equation is separable. The application of the theory to the KS-transformation yields a completely regular canonical system in a 10-dimensional phase-space, using the eccentric anomaly as independent variable. Subsequently sets of 10 regular canonical elements are introduced.

Direct Canonical Transformations

Abstract: Some of the perturbation methods in classical Hamiltonian mechanics lead to near‐identity transformations of the variables, with the new variables explicitly given as functions of the old ones. Two methods are used for identifying and characterizing the subclass of all such transformations which are also canonical: one approach is related to the conventional method of generating canonical transformations, while the other one uses the properties of Poisson brackets and is related to an operator method of Lie. Either of the methods may be used to derive certain steps in a perturbation method devised by Lacina, inadvertently omitted by that author.

Symmetric dirac bracket in classical mechanics.

Abstract: It is known that, in classical systems that have second‐class constraints which relate the canonical coordinates and momentum, the ordinary skew‐symmetric Poisson bracket must be replaced by the skew‐symmetric Dirac bracket. It is also known that in the process of quantization of such systems, the Dirac bracket replaces the Poisson bracket in its correspondence with the quantum commutators. In this paper we obtain the symmetric partner of the Dirac bracket, which is of interest not only to classical mechanics but also in regard to the quantization procedure; i.e., the quantization rules for systems which are restricted by second‐class constraints such that the commutation rules involve anticommutators (instead of commutators, as in certain fields with Fermi‐Dirac statistics) can be given in terms of this new symmetric bracket. This symmetric bracket is related to the Poisson‐Droz‐Vincent symmetric bracket.

Natural coordinates for electrons in solids

Abstract: THE DESCRIPTION OF A PROBLEM in physics can often be significantly simplified by a suitable choice of coordinates. Usually the choice of the best coordinates is dictated by the symmetry of the problem. Thus for a spherically symmetric potential it is most convenient to use spherical coordinates. In some cases it is very easy to choose the right coordinates; in other cases, the problem is not trivial at all. Here we will discuss the choice of suitable coordinates for the motion of electrons in solids.

A unique canonical form for muItivariable linear systems

Abstract: A new canonical form for linear time-invariant multivariable systems is developed. This canonical form is distinguished by the fact that it is uniquely related to the system's input-output behaviour (transfer function matrix). Furthermore, the number of parameters in the canonical form which are not pre-specified (as either zero or one) is minimal.

An integral invariant formulation of a canonical equation set for non-linear electrical networks

Abstract: A canonical form of non-linear network equations is derived from two dual integral invariants for a. non-linear network. The equations hold for time-invariant non-linear networks driven by constant-voltage and constant-current Sources.

Continuity properties of the representations of the canonical commutation relations

Abstract: We prove that a given representation of the canonical commutation relations can be extended uniquely by continuity to larger test function spaces which are maximal in the sense that no further extension is possible. For irreducible tensor product representations of the canonical commutation relations we give a necessary and a sufficient condition for the admissible test functions. We consider the problem of finding topologies on the test function spaces such that this extension can be obtained by a topological completion. Various examples are discussed.

Hamiltonians and Wave Equations for Particles of Spin 0 and Spin 12 with Nonzero Mass

Abstract: The formal mathematical structure of the truly Hamiltonian formulations of nonrelativistic and relativistic quantum mechanics is critically examined for both spin 0 and spin 12 particles with nonzero mass. It is shown that the relativistic quantum mechanics can, in principle, at least be fitted into a truly covariant Hamiltonian procedure where the Hamiltonian is a Lorentz-invariant world scalar (and is the negative of 12 the rest energy of the particle, regardless of its spin). In particular, a truly covariant Hamiltonian theory of the electron is presented. It will be seen that the Lorentz-invariant Dirac theory of the electron does not fall within the framework of a truly covariant Hamiltonian procedure. Further, the dynamical variables associated with the translational degree of freedom of the particle (such as the position, the canonical momentum, and the velocity four-vectors of the relativistic quantum mechanics) correspond precisely to the same variables of the truly covariant formulation of relat...

Charge and Pole: Canonical Coordinates without Potentials

Abstract: For particles having both magnetic and electric charge it is shown that (a) in the nonrelativistic many‐particle problem where only Coulomb and Biot‐Savart fields need be considered and (b) in the one‐particle relativistic problem (orbital pole‐charge moving around a fixed pole‐charge), the well‐set classical dynamics can be reduced directly from the equations of motion to Hamiltonian form without the introduction of potentials and Dirac strings. The Lie‐Koenigs theorem, which can give Hamiltonian format to any dynamics, is invoked for this. The essential feature is that canonical coordinates cannot be physical particle coordinates. For (a) and (b), suitable canonical variables are explicitly constructed. Using only Bohr‐Sommerfeld quantization, the Schwinger charge‐pole quantum condition is obtained for pure‐charge‐pure‐pole interactions; but when Coulomb forces are additionally considered, no quantum restriction on charge and pole strength is required.

Further Developments in Generalized Classical Mechanics

Abstract: Transformation theory for mechanics with higher derivatives is investigated. Poisson brackets previously introduced are proved not to be invariant under canonical transformations. A new definition of Poisson brackets is proposed and it is found to be invariant. Hamilton-Jacobi theory is established.

A brief derivation of the Heisenberg commutation relations

Abstract: The commutation relations for the canonical coordinates of a mechanical system are here derived from a suitably defined form of translational invariance. Let us assume that we are dealing with a physical system which can be adequately described in terms of a finite set of canonical coordinates, (xi, * * *, x'), whose expected values in every admissible state of the system are given by a positive linear functional on the moment algebra A (Xi, , * * Xn) generated by the xi over the complex field C [1i]. We may take for A the algebra T of all noncommutative polynomials with complex coefficients in the xi modulo the ideal J of relations on which all expectations vanish. Let us introduce, for each real a E C, the translation (or virtual displacement) rj(a) of the jth coordinate by the formula rj(a)xi=xj + a if i =j, = X otherwise. Each such translation assigns to every polynomial p in the xi a new polynomial rj(a)p obtained by replacing x; by xj+a throughout. Let us now assume that the ideal J defining our moment algebra A satisfies two natural requirements: (a) if p--0 mod J, then r (a)p_O mod J, for allj and all a. This says that J is stable under translations, and hence that A admits these translations as automorphisms. (b) If rj(a)p-p mod J for all a, then p-q mod J, where q is independent of xj. This says that any polynomial invariant, mod J, under rj(a) for all a can be expressed, mod J, as a polynomial in the xi for isj. Both of these requirements appear to be essential to any variational formulation of the laws of motion of the system. Let us call any moment algebra whose defining ideal of relations satisfies these two requirements acceptable. It is easy to verify that the moment algebras associated with the systems of both classical and Received by the editors May 23, 1969 and, in revised form, March 10, 1970. AMS 1969 subject classifications. Primary 8122.