Showing papers on "Configuration space published in 2001"

Partial observables

Abstract: We discuss the distinction between the notion of partial observable and the notion of complete observable Mixing up the two is frequently a source of confusion The distinction bears on several issues related to observability, such as (i) whether time is an observable in quantum mechanics, (ii) what are the observables in general relativity, (iii) whether physical observables should or should not commute with the Wheeler-DeWitt operator in quantum gravity We argue that the extended configuration space has a direct physical interpretation, as the space of the partial observables This space plays a central role in the structure of classical and quantum mechanics and the clarification of its physical meaning sheds light on this structure, particularly in context of general covariant physics

Gauge field theory coherent states (GCS): II. Peakedness properties

Abstract: In this article we apply the methods outlined in the previous paper of this series to the particular set of states obtained by choosing the complexifier to be a Laplace operator for each edge of a graph. The corresponding coherent state transform was introduced by Hall for one edge and generalized by Ashtekar, Lewandowski, Marolf, Mourao and Thiemann to arbitrary, finite, piecewise analytic graphs. However, both of these works were incomplete with respect to the following two issues : (a) The focus was on the unitarity of the transform and left the properties of the corresponding coherent states themselves untouched. (b) While these states depend in some sense on complexified connections, it remained unclear what the complexification was in terms of the coordinates of the underlying real phase space. In this paper we complement these results : First, we explicitly derive the com- plexification of the configuration space underlying these heat kernel coherent states and, secondly, prove that this family of states satisfies all the usual properties : i) Peakedness in the configuration, momentum and phase space (or Bargmann-Segal) representation. ii) Saturation of the unquenched Heisenberg uncertainty bound. iii) (Over)completeness. These states therefore comprise a candidate family for the semi-classical anal- ysis of canonical quantum gravity and quantum gauge theory coupled to quantum gravity. They also enable error-controlled approximations to difficult analytical cal- culations and therefore set a new starting point for numerical canonical quantum general relativity and gauge theory. The text is supplemented by an appendix which contains extensive graphics in order to give a feeling for the so far unknown peakedness properties of the states constructed.

A Geometrical Interpretation and Uniform Matrix Formulation of Multibody System Dynamics

Abstract: The purpose of this paper is to associate the multibody dynamics procedures with a geometrical picture involving the concepts of configuration manifolds, tangent vector spaces, and orthogonality of constraint reactions to the constraint surfaces. An unconstrained mechanical system is assigned a free configuration manifold and is treated as a generalized particle on the manifold. The system dynamics is then formulated in the local tangent space to the manifold at the system representation point. Imposed constraints on the system, the tangent space splits into the velocity restricted and velocity admissible subspaces, while the system configuration manifold confines to the holonomic constraint manifold. Based on these geometrical concepts, a uniform vector-matrix formulation is developed. Both holonomic and nonholonomic systems are treated in a unified way, and the dynamic equations are expressible either in generalized velocities or in quasi-velocities. Using a geometrically grounded projection method, compact schemes for obtaining different types of equations of motion and for determination of constraint reactions are provided. Some fresh contributions to the theory of constrained systems are reported. A relationship between the present formulation and the other classical methods of analytical dynamics is shown.

System-level exploration for pareto-optimal configurations in parameterized systems-on-a-chip

Abstract: In this work, we provide a technique for efficiently exploring the configuration space of a parameterized system-on-a-chip (SOC) architecture to find all Pareto-optimal configurations. These configurations represent the range of meaningful power and performance tradeoffs that are obtainable by adjusting parameter values for a fixed application mapped onto the SOC architecture. Our approach extensively prunes the potentially large configuration space by taking advantage of parameter dependencies. We have successfully incorporated our technique into the parameterized SOC tuning environment (Platune) and applied it to a number of applications.

Configuration spaces with summable labels

Abstract: An n-monoid is the appropriate extension of an A ∞-space for the theory of n-fold loop spaces We define spaces of configurations on n-manifolds with summable labels in partial n-monoids In particular we obtain an n-fold delooping machinery, that extends the construction of the classifying space by Stasheff Our configuration spaces cover also symmetric products, spaces of rational curves and spaces of labelled subsets A configuration space with connected space of labels has the homotopy type of the space of sections of a certain bundle This extends and unifies results by Bodigheimer, Guest, Kallel and May

On the dynamics of parallel manipulators

Abstract: Studies the dynamics of parallel manipulators. We first have a brief review and discussion on different dynamics formulations in the literature (Newton-Euler, direct Lagrangian, and Lagrange-D'Alembert formulation on the reduced system). Then we show the equivalence of these methods. Based on the concepts from differential manifolds, we prove that away from configuration singularity, there exists a projection from the joint space to parameterize the configuration space. The fact that the dynamics is well defined even at actuators singularity, end-effector singularity and other kinds of parameterization singularity is highlighted. For the method of reduced systems, there are two main drawbacks. Firstly the joints being cut for forming the tree system are presumed to have no external torque. Secondly the force and torque applied to other links of the manipulator is not considered. We propose two methods to remedy the situation. Firstly by cutting a link instead of a joint, all the joints torque can be incorporated into our equations of motion. This is useful not only for the case of actuating all the joints, but also if we consider compensating the joints friction. Secondly we propose a concept of transforming force to the generalized force space so that all the other forces and torque can be considered.

Realizing Holonomic Constraints in Classical and Quantum Mechanics

Abstract: We consider the problem of constraining a particle to a smooth compact submanifold Σ of configuration space using a sequence of increasing potentials. We compare the classical and quantum versions of this procedure. This leads to new results in both cases: an unbounded energy theorem in the classical case, and a quantum averaging theorem. Our two step approach, consisting of an expansion in a dilation parameter, followed by averaging in normal directions, emphasizes the role of the normal bundle of Σ, and shows when the limiting phase space will be larger (or different) than expected.

Quantum spin dynamics (QSD): VII. Symplectic structures and continuum lattice formulations of gauge field theories

Abstract: Interesting nonlinear functions on the phase spaces of classical field theories can never be quantized immediately because the basic fields of the theory become operator-valued distributions. Therefore, one is usually forced to find a smeared substitute for such a function which corresponds to a regularization. The smeared functions define a new symplectic manifold of their own which is easy to quantize. Finally, one must remove the regulator and establish that the final operator, if it exists, has the correct classical limit. In this paper we begin the investigation of these steps for diffeomorphism-invariant quantum field theories of connections. We introduce a (generalized) projective family of symplectic manifolds, coordinatized by the smeared fields, which is labelled by a pair consisting of a graph and another graph dual to it. We show that a subset of the corresponding projective limit can be identified with the symplectic manifold that one started from. Then we illustrate the programme outlined above by applying it to the Gauss constraint. This paper also complements, as a side result, earlier work by Ashtekar, Corichi and Zapata who observed that certain operators are non-commuting on certain states, although the Poisson brackets between the corresponding classical functions vanish. These authors showed that this is not a contradiction provided that one refrains from a phase space quantization but rather applies a quantization based on the Lie algebra of vector fields on the configuration space of the theory. Here we show that one can provide a phase space quantization, that is, one can find other functions on the classical phase space which give rise to the same operators but whose Poisson algebra precisely mirrors the quantum commutator algebra. The framework developed here is the classical cornerstone on which the semiclassical analysis in a new series of papers called `gauge theory coherent states' is based.

Quantum dynamics for a system coupled to slow baths: On-the-fly filtered propagator method

Abstract: An on-the-fly filtered propagator functional path integral scheme is introduced as an efficient way of calculating the iterative dynamics of complex condensed systems. Time evolution of the reduced density matrix of a dissipative quantum system is evaluated iteratively by filtering the negligible propagator elements at each propagation step. This on-the-fly filtering along with the finiteness of the bath memory dramatically reduces the configuration space to be integrated without losing numerical accuracy of the results. The required computational storage space for the configuration and the weight of the survived path segments increases linearly with the bath memory length. Up to the bath memory time, it is found that a strikingly small fraction of the configurations survives the on-the-fly filtering process and the number of surviving configurations increases algebraically with the propagation time. At times longer than the bath memory time, the number of surviving configurations required for numerically accurate results is essentially saturated and is less than 0.1% of the total number of the configurations. This new scheme is extremely useful for the problems of low-frequency solvents for which the bath memory spans many time steps of the propagation.

Gauge fields and extrapotentials in constrained quantum systems

Abstract: We derive an effective Hamiltonian for a quantum system constrained to a submanifold (the constraint manifold) of configuration space (the ambient space) by an infinite restoring force. We pay special attention to how this Hamiltonian depends on quantities which are external to the constraint manifold, such as the extrinsic curvature of the constraint manifold, the curvature of the ambient space, and the constraining potential. In particular, we find the remarkable fact that the twisting of the constraining potential appears as a gauge potential in the constrained Hamiltonian. This gauge potential is closely related to the geometric phase originally discussed by Berry. The constrained Hamiltonian also contains an effective potential depending on the extrinsic curvature of the constraint manifold, the curvature of the ambient space, and the twisting of the constraining potential. The general nature of our analysis allows applications to a wide variety of problems, such as rigid molecules, the evolution of molecular systems along reaction paths, and quantum strip waveguides.

Disassembly sequencing using a motion planning approach

Abstract: Our motion planning based approach treats the parts in the assembly as robots and operates in the composite configuration space of the parts' individual configuration spaces. Randomized techniques inspired by recent motion planning methods are used to sample configurations in this space. Since typical assemblies consist of many parts, the corresponding composite C-spaces have high dimensionality. Also, since many important configurations for the disassembly sequence will involve closely packed parts, the disassembly problem suffers from the so-called narrow passage problem. We bias the sampling by computing potential movement directions based on the geometric characteristics of configurations known to be reachable from the assembled configuration. We construct a disassembly tree which is rooted at the starting assembled configuration. Our experimental results with several non-trivial puzzle-like assemblies show the potential of this approach.

Families of Periodic Solutions of Resonant PDEs

Abstract: We construct some families of small amplitude periodic solutions close to a completely resonant equilibrium point of a semilinear reversible partial differential equation. To this end, we construct, using averaging methods, a suitable map from the configuration space to itself. We prove that to each nondegenerate zero of such a map there corresponds a family of small amplitude periodic solutions of the system. The proof is based on Lyapunov-Schmidt decomposition. This establishes a relation between Lyapunov-Schmidt decomposition and averaging theory that could be interesting in itself. As an application, we construct countable many families of periodic solutions of the nonlinear string equation utt-uxx± u3=0 (and of its perturbations) with Dirichlet boundary conditions. We also prove that the fundamental periods of solutions belonging to the nth family converge to 2π/n when the amplitude tends to zero.

Planning motion patterns of human figures using a multi-layered grid and the dynamics filter

Abstract: Presents a practical motion planner for humanoids and animated human figures. Modeling human motions as a sum of rigid body and cyclic motions, we identify body postures that represent the rigid-body part of typical motion patterns. This leads to a model of the configuration space that consists of a multi-layered grid, each layer corresponding to a single posture. A global search through this reduced configuration space yields a feasible path and the corresponding postures along the path. A velocity profile is calculated along the optimal path, subject to the speed and acceleration limits assumed for each posture. Cyclic motions, generated from "primitive" cyclic motion patterns for each posture, are then added to the trajectory produced by the path planner. This "kinematic" motion is then modified by a dynamics filter to result in dynamically consistent behavior. Examples are presented which demonstrate the use of this planner in an office environment.

The Phase Space Structure Around L4 in the Restricted Three-Body Problem

Abstract: The phase space structure around L 4 in the restricted three-body problem is investigated. The connection between the long period family emanating from L 4 and the very complex structure of the stability region is shown by using the method of Poincare’s surface of section. The zero initial velocity stability region around L 4 is determined by using a method based on the calculation of finite-time Lyapunov characteristic numbers. It is shown that the boundary of the stability region in the configuration space is formed by orbits suffering slow chaotic diffusion.

Hyperspherical harmonics as Sturmian orbitals in momentum space: A systematic approach to the few-body Coulomb problem

Abstract: To exploit hyperspherical harmonics (including orthogonal transformations) as basis sets to obtain atomic and molecular orbitals, Fock projection into momentum space for the hydrogen atom is extended to the mathematical d dimensional case, higher than the physical case d = 3 For a system of N particles interacting through Coulomb forces, this method allows us to work both in a d = 3( N - 1) dimensional configuration space (on eigenfunctions expanded on a Sturmian basis) and in momentum space (using a ( d + 1)-dimensional hyperspherical harmonics basis set) Numerical examples for three-body problems are presented Performances of alternative basis sets corresponding to different coupling schemes for hyperspherical harmonics have also been explicitly obtained for bielectronic atoms and H 2 + (in the latter case, also in the Born-Oppenheimer approximation extending the multicentre technique of Shibuya and Wulfman) Among the various generalizations and applications particularly relevant is the introduction

A new method for measuring the splitting of invariant manifolds

Abstract: We study the so-called Generalized Arnol'd Model (a weakly hyperbolic near-integrable Hamiltonian system), with d+1 degrees of freedom (d⩾2), in the case where the perturbative term does not affect a fixed invariant d-dimensional torus. This torus is thus independent of the two perturbation parameters which are denoted e (e>0) and μ. We describe its stable and unstable manifolds by solutions of the Hamilton–Jacobi equation for which we obtain a large enough domain of analyticity. The splitting of the manifolds is measured by the partial derivatives of the difference ΔS of the solutions, for which we obtain upper bounds which are exponentially small with respect to e. A crucial tool of the method is a characteristic vector field, which is defined on a part of the configuration space, which acts by zero on the function ΔS and which has constant coefficients in well-chosen coordinates. It is in the case where |μ| is bounded by some positive power of e that the most precise results are obtained. In a particular case with three degrees of freedom, the method leads also to lower bounds for the splitting.

Controllability of kinematic control systems on stratified configuration spaces

Abstract: This paper considers nonlinear kinematic controllability of a class of systems called stratified. Roughly speaking, such stratified systems have a configuration space which can be decomposed into sub-manifolds upon which the system has different sets of equations of motion. For such systems, considering the controllability is difficult because of the discontinuous form of the equations of motion. The main result in this paper is a controllability test, analogous to Chow's theorem, is based upon a construction involving distributions, and the extension thereof to robotic gaits.

Trajectories for the wave function of the universe from a simple detector model

Abstract: Inspired by Mott's (1929) analysis of particle tracks in a cloud chamber, we consider a simple model for quantum cosmology which includes, in the total Hamil- tonian, model detectors registering whether or not the system, at any stage in its en- tire history, passes through a series of regions in configuration space. We thus derive a variety of well-defined formulas for the probabilities for trajectories associated with the solutions to the Wheeler-DeWitt equation. The probability distribution is peaked about classical trajectories in configuration space. The "measured" wave functions still satisfy the Wheeler-DeWitt equation, except for small corrections due to the disturbance of the measuring device. With modified boundary conditions, the measurement amplitudes es- sentially agree with an earlier result of Hartle derived on rather different grounds. In the special case where the system is a collection of harmonic oscillators, the interpretation of the results is aided by the introduction of "timeless" coherent states - eigenstates of the Hamiltonian which are concentrated about entire classical trajectories.

Complexity of Nonholonomic motion planning

Abstract: The complexity of motion planning amidst obstacles is a well modeled and understood notion. What is the increase of the complexity when the problem is to plan the trajectories of a nonholonomic robot? We show that this quantity can be seen as a function of paths and of the distance between the paths and the obstacles. We propose various definitions of it, from both topological and metric points of view, and compare their values. For two of them we give estimates which involve some E-norm on the tangent space to the configuration space. Finally we apply these results to compute the complexity needed to park a car-like robot with trailers.

Configurations of points

Abstract: Berry & Robbins, in their discussion of the spin–statistics theorem in quantum mechanics, were led to ask the following question. Can one construct a continuous map from the configuration space of n distinct particles in 3–space to the flag manifold of the unitary group U ( n )? I shall discuss this problem and various generalizations of it. In particular, there is a version in which U ( n ) is replaced by an arbitrary compact Lie group. It turns out that this can be treated using Nahm9s equations, which are an integrable system of ordinary differential equations arising from the self–dual Yang-Mills equations. Our topological problem is therefore connected with physics in two quite different ways, once at its origin and once at its solution.

Electronically nonadiabatic transitions in a collinear H2 + H+ system : Quantum mechanical understanding and comparison with a trajectory surface hopping method

Abstract: Electronically nonadiabatic transitions in a collinear H2 + H+ system have been studied both quantum mechanically and with classical mechanics using a diatomics-in-molecules type potential energy surface fitted to recent ab initio data. The quantum dynamical calculations are carried out by employing a standard close-coupling method in hyperspherical coordinates and the quasiclassical trajectory calculations were carried out with a basic surface hopping method. Special emphasis is placed on qualitative analysis of the physical mechanisms of the electronically nonadiabatic transitions. To make such an analysis, the basic idea proposed by Nobusada et al. (K. Nobusada, O. I. Tolstikhin and H. Nakamura, J. Chem. Phys., 1998, 108, 8922) in the study of vibrationally nonadiabatic transitions is applied, and also the reaction dynamics is visualized in detail by using classical trajectories. It is found that the electronically nonadiabatic transition occurs in a rather narrow configuration space and its reactivity depends strongly on the initial vibrational state of H2.

Estimates for homological dimension of configuration spaces of graphs

Abstract: We show that the homological dimension of a configuration space of a graph Γ is estimated from above by the number b of vertices in Γ whose valence is greater than 2. We show that this estimate is optimal for the n-point configuration space of Γ if n ≥ 2b. 0. Introduction. Let Γ be a finite graph and n a natural number. The marked n-point configuration space of Γ is a subspace CnΓ in the nth cartesian power of Γ defined by CnΓ := {(x1, . . . , xn) ∈ Γ n : xi 6= xj for i 6= j}. Consider the natural free action of the symmetric group Sn on the space CnΓ defined by σ(x1, . . . , xn) = (xσ(1), . . . , xσ(n)) and put CnΓ := CnΓ/Sn. The space CnΓ is called the (unmarked) n-point configuration space of Γ . This paper reports on partial progress towards understanding the homology of configuration spaces of graphs, or even more generally of compact polyhedra. For another recent result in that direction, see [G]. We call a vertex v of Γ branched if it is adjacent to at least three edges. We denote by b = b(Γ ) the number of branched vertices in Γ . The main result of this paper is the following. 0.1. Theorem. Let Γ be a finite graph and n a natural number. (1) There exists a cube complex KnΓ of dimension min(b(Γ ), n) which embeds as a deformation retract into the configuration space CnΓ . (2) The fundamental group π1(CnΓ ) contains a subgroup isomorphic to the free abelian group Z with k = min(b(Γ ), [n/2]), where [x] denotes the integer part of x. 2000 Mathematics Subject Classification: Primary 55M10; Secondary 20J05, 51F99. The author was supported by the Polish State Committee for Scientific Research (KBN) grant 2 P03A 023 14.

Dynamics of supercooled water in configuration space.

Abstract: We study the potential energy surface ~PES! sampled by a liquid modeled via the widely studied extended simple point charge ~SPC/E! model for water. We characterize the curvature of the PES by calculating the instantaneous normal mode ~INM! spectrum for a wide range of densities and temperatures. We discuss the information contained in the INM density of states, which requires additional processing to be unambiguously associated with the long-time dynamics. For the SPC/E model, we find that the slowing down of the dynamics in the supercooled region—where the ideal mode coupling theory has been used to describe the dynamics—is controlled by the reduction in the number of directions in configuration space that allow a structural change. We find that the fraction f dw of the double-well directions in configuration space determines the value of the diffusion constant D, thereby relating a property of the PES to a macroscopic dynamic quantity; specifically, it appears that AD is approximately linear in f dw . Our findings are consistent with the hypothesis that, at the mode coupling crossover temperature, dynamical processes based on the free exploration of configuration space vanish, and processes requiring activation dominate. Hence, the reduction of the number of directions allowing free exploration of configuration space is the mechanism of diffusion implicitly implemented in the ideal mode coupling theory. Additionally, we find a direct relationship between the number of basins sampled by the system and the number of free directions. In this picture, diffusion appears to be related to geometrical properties of the PES, and to be entropic in origin.

A strategy for analysis of (molecular) equilibrium simulations: Configuration space density estimation, clustering, and visualization

Abstract: We propose an approach for summarizing the output of long simulations of complex systems, affording a rapid overview and interpretation. First, multidimensional scaling techniques are used in conjunction with dimension reduction methods to obtain a low-dimensional representation of the configuration space explored by the system. A nonparametric estimate of the density of states in this subspace is then obtained using kernel methods. The free energy surface is calculated from that density, and the configurations produced in the simulation are then clustered according to the topography of that surface, such that all configurations belonging to one local free energy minimum form one class. This topographical cluster analysis is performed using basin spanning trees which we introduce as subgraphs of Delaunay triangulations. Free energy surfaces obtained in dimensions lower than four can be visualized directly using iso-contours and -surfaces. Basin spanning trees also afford a glimpse of higher-dimensional to...

The three-dimensional beam theory: finite element formulation based on curvature

Abstract: A new finite element formulation of the 'geometrically exact finite-strain beam theory' is presented. The formulation employs the generalized virtual work principle in which the main role is played by the pseudo-curvature vector. The solution of the governing equations is obtained by using a combined Galerkin-collocation algorithm. The collocation assures that the equilibrium and the constitutive internal force and moment vectors are equal at a set of the chosen discrete points. A special update procedure for the pseudo-curvature and rotation vectors is employed in Newton's iteration because of the non-linearity of the configuration space. The accuracy and the efficiency of the derived numerical algorithm are demonstrated by several examples.

Null Space Integration Method for Constrained Multibody Systems with No Constraint Violation

Abstract: A method for integrating equations of motion of constrained multibodysystems with no constraint violation is presented. A mathematical model,shaped as a differential-algebraic system of index 1, is transformedinto a system of ordinary differential equations using the null-spaceprojection method. Equations of motion are set in a non-minimal form.During integration, violations of constraints are corrected by solvingconstraint equations at the position and velocity level, utilising themetric of the system's configuration space, and projective criterion to thecoordinate partitioning method. The method is applied to dynamicsimulation of 3D constrained biomechanical system. The simulation resultsare evaluated by comparing them to the values of characteristicparameters obtained by kinematic analysis of analyzed motion based onmeasured kinematic data.

Sensor-based construction of a retract-like structure for a planar rod robot

Abstract: Sensor-based planning for rod-shaped robots is necessary for the realistic deployment of noncircularly symmetric robots into unknown environments. Whereas circularly symmetric robots have two-dimensional Euclidean configuration spaces, planar rod robots posses three degrees-of-freedom, two for position and one for orientation, and hence have a three-dimensional configuration space, SE(2). In this work, we define the rod hierarchical generalized Voronoi graph (rod-HGVG) which is a roadmap of the rod's configuration space. Prior work in Voronoi-based roadmaps use a retraction of the robot's free space to define the roadmap; here, we break apart the robot's free space into regions where fragments of the roadmap are defined and then connect the fragments. The primary advantage of the rod-HGVG is that it is defined in terms of workspace distance measurements, which makes it amenable to sensor-based planning. This paper also includes a numerical procedure that generates the rod-HGVG edge fragments using only information that is within line of sight of the rod robot. It is worth noting that this procedure does not require an explicit definition of configuration space, i.e., this procedure constructs a roadmap of rod configuration space without ever constructing the configuration space itself.

Localization properties of groups of eigenstates in chaotic systems

Abstract: In this paper we study in detail the localized wave functions defined in Phys. Rev. Lett. 76, 1613 (1994), in connection with the scarring effect of unstable periodic orbits in highly chaotic Hamiltonian system. These functions appear highly localized not only along periodic orbits but also on the associated manifolds. Moreover, they show in phase space the hyperbolic structure in the vicinity of the orbit, something that translates in configuration space into the structure induced by the corresponding self-focal points. On the other hand, the- quantum dynamics of these functions are also studied. Our results indicate that the probability density first evolves along the unstable manifold emanating from the periodic orbit, and localizes temporarily afterwards on only a few, short related periodic orbits. We believe that this type of study can provide some keys to disentangle the complexity associated with the quantum mechanics of these kind of systems, which permits the construction of a simple explanation in terms of the dynamics of a few classical structures.