Coordinates Of Configuration Space Research Articles

Energy contour exploration with potentiostatic kinematics

We introduce a method of exploring potential energy contours (PECs) in complex dynamical systems based on potentiostatic kinematics wherein the systems are evolved with minimal changes to their potential energy. We construct a simple iterative algorithm for performing potentiostatic kinematics, which uses an estimate curvature to predict new configuration-space coordinates on the PEC and a potentiostat term component to correct for errors in prediction. Our methods are then applied to atomic structure models using an interatomic potential for energy and force evaluations as would commonly be invoked in a molecular dynamics simulation. Using several model systems, we assess the stability and accuracy of the method on different hyperparameters in the implementation of the potentiostatic kinematics. Our implementation is open source and available within the atomic simulation environment package.

The Poincaré Group in a Demisemidirect Product with a Non-associative Algebra with Representations that Include Particles and Quarks—II

The quarks and particles’ mass and mass/spin relations are provided with coordinates in configuration space and/or momentum space by means of the marriage of ordinary Poincare group representations with a non-associative algebra made through a demisemidirect product, in the notation of Leibniz algebras. Thus, we circumvent the restriction that the Poincare group cannot be extended to a larger group by any means (including the (semi)direct product) to get even the mass relations. Finally, we will discuss a connection between the phase space representations of the Poincare group and the phase space representations of the associated Leibniz algebra.

A spherical harmonic expansion of the Boltzmann equation for nonhydrodynamic weakly ionized plasma in the presence of both electric and magnetic fields

The Boltzmann kinetic equation for nonhydrodynamic weakly ionized plasma in the presence of both electric and magnetic fields is considered. The charged particles distribution function is decomposed in terms of spherical harmonics in momentum space. After substituting the expansion into the Boltzmann equation an infinite hierarchy of differential equations for the distribution function expansion coefficients is derived. The cases of Cartesian, cylindrical and spherical coordinates in configuration space are studied. We applied obtained equations to the description of electron transport in nitrogen at high values of rf electric field intensity to number density ratio E/N.

Poisson structure on a space with linear SU(2) fuzziness

The Poisson structure is constructed for a model in which spatial coordinates of configuration space are noncommutative and satisfy the commutation relations of a Lie algebra. The case is specialized to that of the group SU(2), for which the counterpart of the angular momentum, as well as the Euler parameterization of the phase space are introduced. SU(2)-invariant classical systems are discussed, and it is observed that the path of particle can be obtained by the solution of a first-order equation, as the case with such models on commutative spaces. The examples of free particle, rotationally-invariant potentials, and specially the isotropic harmonic oscillator are investigated in more detail.

The use of quadratic forms in the calculation of ground state electronic structures

There are many examples in theoretical physics where a fundamental quantity can be considered a quadratic form ρ=∑iρi=∣Ψ∣2 and the corresponding linear form Ψ=∑iψi is highly relevant for the physical problem under study. This, in particular, is the case of the density and the wave function in quantum mechanics. In the study of N-identical-fermion systems we have the additional feature that Ψ is a function of the 3N configuration space coordinates and ρ is defined in three-dimensional real space. For many-electron systems in the ground state the wave function and the Hamiltonian are to be expressed in terms of the configuration space (CS), a replica of real space for each electron. Here we present a geometric formulation of the CS, of the wave function, of the density, and of the Hamiltonian to compute the electronic structure of the system. Then, using the new geometric notation and the indistinguishability and equivalence of the electrons, we obtain an alternative computational method for the ground state of the system. We present the method and discuss its usefulness and relation to other approaches.

Finding the Hamiltonian for Cosmological Models in Fourth-Order Gravity Theories Without Resorting to the Ostrogradski or Dirac Formalism

The Hamilton formalism of cosmological models in fourth-order theories of gravity is considered. An approach to constructing the Hamilton function is presented which starts by replacing the second order derivatives of configuration space coordinates by functions depending on these coordinates, its first order derivatives, and additional variables playing the role of configuration space coordinates. This formalism, which does not resort to the Ostrogradski or Dirac formalism, is elucidated and applied to examples. For a special class of Lagrange functions, it is demonstrated that the canonical coordinates of the considered formalism and of the Ostrogradski formalism are related via a canonical transformation. The canonical transformation is a transformation of the configuration space coordinates and a transformation of momentum components induced by the transformation of the configuration space coordinates for a special element of the class of Lagrange functions mentioned. The Wheeler-DeWitt equations belonging to this Lagrange function are related via minisuperspace coordinate transformations.

Alternative Sturmian bases and momentum space orbitals: an application to the hydrogen molecular ion

Abstract The relationship between alternative separable solutions of the Coulomb problem both in configuration and in momentum space is exploited in order to obtain Sturmian orbitals of use as expansion basis sets in atomic and molecular problems. The usual spherical basis is obtained by separation in polar coordinates. The mathematical properties are explored for a type of basis set for quantum mechanical problems with axial symmetry, examples being diatomic molecules, or atoms under the influence of a uniform electric field. Because of its appropriateness for treatment of the Stark effect in atomic physics, this alternative basis set is called Stark basis . The Stark basis corresponds to separation in parabolic coordinates in configuration space and in cylindrical coordinates in momentum space. Fock's projection onto the surface of a sphere in the four dimensional hyperspace allows us to establish the connections of the momentum space wave functions with hyperspherical harmonics. As an application, the Shobuya and Wulfman analysis of momentum space molecular orbitals is reformulated and monoelectronic multicenter integrals are calculated by angular momentum algebra. Properties of the Stark basis, which exhibits less coupling at long range, are illustrated with reference to the case of the hydrogen molecular ion in the fixed nuclei approximation.

Gauge mechanical model inN dimensions

The Christ-Lee mechanical model is generalized toN spatial dimensions. Its quantization as a gauge system is carried out, emphasizing the relationship between gauge-fixing and curvilinear coordinates in configuration space.

General classical variational treatment of the rates of bimolecular exchange and unimolecular reactions involving three bodies

The classical variational theory of chemical reaction rates gives the rate as the equilibrium flux of systems through a trial surface in the phase space of the reaction system. The surface divides the phase space into reactant and product regions and is varied to obtain a least upper bound for the rate of product formation. For bimolecular reactions of the type A+BC→AB+C and the high pressure limit of unimolecular reactions of the type ABC→products, we have derived expressions which give the canonical and microcanonical rate coefficients for the most general dividing surface defined by internal configuration space coordinates. For bimolecular reactions, we have also derived an expression for the energy dependent mean reaction cross section for the most general dividing surface. Expressions for the rate coefficients and mean reaction cross section for any of the more restricted formulations of the dividing surface used in earlier work, and more flexible ones, can be obtained by the substitution of two terms in the appropriate general equation; namely, the partial derivatives of the function that defines the surface with respect to the internal coordinates that define the surface. For example, the flexibility of the surface can be improved systematically by introducing terms in a power series expansion of the surface. The application of a simplex algorithm to determine the coefficients (variational parameters) in an expansion of the surface that includes terms up to the second and third order may give a surface which is close to the best one possible. The minimization procedure corresponds to a search of the potential-energy function for a reaction coordinate that satisfies the variational condition. The variationally determined dividing surface identifies regions of configuration space in which the potential energy must be accurate in order to obtain accurate classical rate coefficients. Thecalculus of variations was applied to the general equations for the rate coefficients to obtain differential equations which give the best dividing surface defined by internal coordinates for the canonical and microcanonical formulations of transition state theory. The corresponding rate coefficients or mean reaction cross section are as close as possible to convergence with the results of classical trajectory calculations. The flux can thus be minimized completely by obtaining a numerical solution for the differential equations which define the best surface or nearly minimized for a series approximation to the best surface. Computational studies are required to determine the more tractable method. By using an appropriate generating function, the procedure described in this work can be extended to reactions involving more than three bodies. The variationally determined dividing surface, reaction coordinate, and reaction surface Hamiltonian could provide a basis for a semiclassical theory of reaction rates.

A Generalized Solution to the Inverse Kinematics of Robotic Manipulators

In the context of kinematic control of a robotic manipulator if a certain set of task space coordinates (end effector position and orientation) are commanded then the corresponding configuration space coordinates (joint variables) must be provided. The joint variables are obtained by solving the “inverse kinematics problem.” Typically a solution to the problem can be obtained in closed-form; however, such a solution is inherently manipulator-dependent. The paper presents an approach for providing a generalized inverse kinematics solution which is manipulator-independent. The solution is based on an iterative procedure. An algorithm was developed and demonstrated using the PUMA-560 and Stanford manipulator models.

Renormalization of configuration space into phase space (Can nonrealizable Hamiltonians be realized?)

We show that gauge-type coupling can be the basis for new renormalization phenomena in which ordinary configuration-space coordinates become effectively canonically conjugate. This is demonstrated first for the case of a particle in a very strong magnetic field, where some of the experimental consequences can be surprising. Another example is presented which suggests a possible new approach to the quark-confinement problem.