Euclidean Configuration Space Research Articles

Engineered swift equilibration for arbitrary geometries.

Engineered swift equilibration (ESE) is a class of driving protocols that enforce an equilibrium distribution with respect to external control parameters at the beginning and end of rapid state transformations of open, classical nonequilibrium systems. ESE protocols have previously been derived and experimentally realized for Brownian particles in simple, one-dimensional, time-varying trapping potentials; one recent study considered ESE in two-dimensional Euclidean configuration space. Here we extend the ESE framework to generic, overdamped Brownian systems in arbitrary curved configuration space and illustrate our results with specific examples not amenable to previous techniques. Our approach may be used to impose the necessary dynamics to control the full temporal configurational distribution in a wide variety of experimentally realizable settings.

Rational maps from Euclidean configuration spaces to spheres

In this note we give an algorithm to determine the rational homotopy type of the free and pointed mapping spaces $\map(F(\br^m,k),\bs^n)$ and $\map^*(F(\br^m,k),\bs^n)$. An explicit description of these spaces is given for $k=3$. The general case for $n$ odd is also presented as an immediate consequence of the rational version of a classical result of Thom.

Non-Formality of Planar Configuration Spaces in Characteristic 2

AbstractWe prove that the ordered configuration space of four points or more in the plane has a nonformal singular cochain algebra in characteristic 2. This is proved by constructing an explicit nontrivial obstruction class in the Hochschild cohomology of the cohomology ring of the configuration space, by means of the Barratt–Eccles–Smith simplicial model. We also show that if the number of points does not exceed its dimension then a Euclidean configuration space is intrinsically formal over any ring.

Quasi-momentum theorem in Riemann-Cartan space

The geometric formulation of motion of the first-order linear homogenous scleronomous nonholonomic system subjected to active forces is studied with the nonholonomic mapping theory. The quasi-Newton law, the quasi-momentum theorem, and the second kind Lagrange equation of dynamical systems are obtained in the Riemann-Cartan configuration spaces. By the nonholonomic mapping, a Euclidean configuration space or a Riemann configuration space of a dynamical system can be mapped into a Riemann-Cartan configuration space with torsion. The differential equations of motion of the dynamical system can be obtained in its Riemann-Cartan configuration space by the quasi-Newton law or the quasi-momentum theorem. For a constrained system, the differential equations of motion in its Riemann-Cartan configuration space may be simpler than the equations in its Euclidean configuration space or its Riemann configuration space. Therefore, the nonholonomic mapping theory can solve some constrained problems, which are difficult to be solved by the traditional analytical mechanics method. Three examples are given to illustrate the effectiveness of the method.

Quantization of systems with temporally varying discretization. II. Local evolution moves

Several quantum gravity approaches and field theory on an evolving lattice involve a discretization changing dynamics generated by evolution moves. Local evolution moves in variational discrete systems (1) are a generalization of the Pachner evolution moves of simplicial gravity models, (2) update only a small subset of the dynamical data, (3) change the number of kinematical and physical degrees of freedom, and (4) generate a dynamical (or canonical) coarse graining or refining of the underlying discretization. To systematically explore such local moves and their implications in the quantum theory, this article suitably expands the quantum formalism for global evolution moves, constructed in Paper I [P. A. Höhn, “Quantization of systems with temporally varying discretization. I. Evolving Hilbert spaces,” J. Math. Phys. 55, 083508 (2014); e-print arXiv:1401.6062 [gr-qc]], by employing that global moves can be decomposed into sequences of local moves. This formalism is spelled out for systems with Euclidean configuration spaces. Various types of local moves, the different kinds of constraints generated by them, the constraint preservation, and possible divergences in resulting state sums are discussed. It is shown that non-trivial local coarse graining moves entail a non-unitary projection of (physical) Hilbert spaces and “fine grained” Dirac observables defined on them. Identities for undoing a local evolution move with its (time reversed) inverse are derived. Finally, the implications of these results for a Pachner move generated dynamics in simplicial quantum gravity models are commented on.

Motion planning algorithms for configuration spaces

We provide explicit motion planners for Euclidean configuration spaces. This allows us to recover some known values of the topological complexity and the Lusternik–Schinirelman category of these spaces.

Flat minimal quantizations of Stäckel systems and quantum separability

In this paper, we consider the problem of quantization of classical Stäckel systems and the problem of separability of related quantum Hamiltonians. First, using the concept of Stäckel transform, natural Hamiltonian systems from a given Riemann space are expressed by some flat coordinates of related Euclidean configuration space. Then, the so-called flat minimal quantization procedure is applied in order to construct an appropriate Hermitian operator in the respective Hilbert space. Finally, we distinguish a class of Stäckel systems which remains separable after any of admissible flat minimal quantizations.

Natural star-products on symplectic manifolds and related quantum mechanical operators

In this paper is considered a problem of defining natural star-products on symplectic manifolds, admissible for quantization of classical Hamiltonian systems. First, a construction of a star-product on a cotangent bundle to an Euclidean configuration space is given with the use of a sequence of pair-wise commuting vector fields. The connection with a covariant representation of such a star-product is also presented. Then, an extension of the construction to symplectic manifolds over flat and non-flat pseudo-Riemannian configuration spaces is discussed. Finally, a coordinate free construction of related quantum mechanical operators from Hilbert space over respective configuration space is presented.

Corrections to “Singularity-Free Dynamic Equations of Open-Chain Mechanisms With General Holonomic and Nonholonomic Joints”

This is a correction to [Duindam and Stramigioli, “Singularity-free dynamic equations of open-chain mechanisms with general holonomic and nonholonomic joints,” IEEE Trans. Robot., vol. 24, no. 3, pp. 527-526, Jun. 2008] where the singularity-free dynamic equations of mechanical systems with Euclidean or non-Euclidean configuration spaces are presented. We present the correct explicit expressions of the equations presented in the above referenced paper.

Topological constraints in search-based robot path planning

There are many applications in motion planning where it is important to consider and distinguish between different topological classes of trajectories. The two important, but related, topological concepts for classifying manifolds that are of importance to us are those of homotopy and homology. In this paper we consider the problem of robot exploration and planning in Euclidean configuration spaces with obstaclees to (a) identify and represent different homology classes of trajectories; (b) plan trajectories constrained to certain homology classes or avoiding specified homology classes; and (c) explore different homotopy classes of trajectories in an environment and determine the least cost trajectories in each class. We exploit theorems from complex analysis and the theory of electromagnetism to solve the problem 2-dimensional and 3-dimensional configuration spaces respectively. Finally, we describe the extension of these ideas to arbitrary D-dimensional configuration spaces. We incorporate these basic concepts to develop a practical graph-search based planning tool with theoretical guarantees by combining integration theory with search techniques, and illustrate it with several examples.

Noncommutative Integrability from Classical to Quantum Mechanics

We propose in this work a definition of integrable quantum system, which is based upon the correspondence with the concept of noncommutative integrability for classical mechanical systems. We then determine sufficient conditions under which, given an integrable classical system, it is possible to construct an integrable quantum system by means of a quantization procedure based on the symmetrized product of operators. As a first example of application of such an approach, we will consider the possible cases of noncommutative integrability for systems with rotational symmetry in an n-dimensional Euclidean configuration space.

Superintegrability on curved spaces, orbits and momentum hodographs: revisiting a classical result by Hamilton

The equation of the orbits (in the configuration space) and of the hodographs (in the ‘momentum’ plane) for the ‘curved’ Kepler and harmonic oscillator systems, living in a configuration space of any constant curvature and either signature type, are derived by purely algebraic means. This result extends to the ‘curved’ Kepler or harmonic oscillator for the classical Hamilton derivation of the orbits of the Euclidean Kepler problem through its hodographs. In both cases, the fundamental property allowing these derivations to work is the superintegrability of the ‘curved’ Kepler and harmonic oscillator, no matter whether the constant curvature of the configuration space is zero or not, or whether the configuration space metric is Riemannian or Lorentzian. In the ‘curved’ case the basic result does not refer to the ‘velocity hodograph’ but to the ‘momentum hodograph’; both coincide in a Euclidean configuration space, but only the latter is unambiguously defined in all curved spaces.

A Basic Power Decomposition in Lagrangian Mechanics

A Basic Power Decomposition in Lagrangian Mechanics

Convolution of Lorentz Invariant Ultradistributions and Field Theory

In this work, a general definition of convolution between two arbitrary four dimensional Lorentz invariant (fdLi) Tempered Ultradistributions is given, in both: Minkowskian and Euclidean Space (Spherically symmetric tempered ultradistributions). The product of two arbitrary fdLi distributions of exponential type is defined via the convolution of its corresponding Fourier Transforms. Several examples of convolution of two fdLi Tempered Ultradistributions are given. In particular we calculate exactly the convolution of two Feynman's massless propagators. An expression for the Fourier Transform of a Lorentz invariant Tempered Ultradistribution in terms of modified Bessel distributions is obtained in this work (Generalization of Bochner's formula to Minkowskian space). At the same time, and in a previous step used for the deduction of the convolution formula, we obtain the generalization to the Minkowskian space, of the dimensional regularization of the perturbation theory of Green Functions in the Euclidean configuration space given in ref.[12]. As an example we evaluate the convolution of two n-dimensional complex-mass Wheeler's propagators.

Safe Cooperative Robot Dynamics on Graphs

This paper introduces the use of vector fields to design, optimize, and implement reactive schedules for safe cooperative robot patterns on planar graphs. We consider automated guided vehicles (AGVs) operating upon a predefined network of pathways. In contrast to the case of locally Euclidean configuration spaces, regularization of collisions is no longer a local procedure, and issues concerning the global topology of configuration spaces must be addressed. The focus of the present inquiry is the definition, design, and algorithmic construction of controllers for achievement of safe, efficient, cooperative patterns in the simplest nontrivial example (a pair of robots on a Y-network) by means of a hierarchical event-driven state feedback law.

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

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.

On generalized Runge-Lenz vector and conserved symmetric tensor for central-potential systems with a monopole field on spaces of constant curvature

In classical mechanics, it was shown by Fradkin that there exist O(4) andSU(3) dynamical symmetries in a local sense for all central-potential problems in the three-dimensional Euclidean space. He constructed the generalized Runge-Lenz vector and the generalized conserved symmetric tensor for central-potential systems concretely. This article deals with central-potential systems with a monopole field on a space of constant curvature. An extension of Frandkin’s method together with the Boulware-Brown-Cahn-Ellis-Lee transformation is applied to obtain the generalized Runge-Lenz vector and the generalized conserved symmetric tensor for the present dynamical system. Global consideration is given to those conserved quantities for the Kepler motion and the harmonic oscillator. Further, the symmetry problem in the case of the Euclidean configuration space will be touched upon.

Robot motion planning with uncertainty in control and sensing

Most of the research in robot motion planning has focused on finding a collision-free path connecting two configurations of the robot among obstacles, under the assumptions that geometric models are complete and accurate and that robot control is perfect. In this paper we consider the problem of planning motion strategies in the presence of uncertainty in both control and sensing, using the preimage backchaining approach. Though attractive, this approach raises difficult computational issues. One of them, preimage computation, is the main topic of this paper. We describe two practical methods for computing preimages for a robot having a two-dimensional Euclidean configuration space. We also propose a combination of these two methods which yields the most powerful practical method devised so far for computing preimages. A motion planner based on these methods has been implemented and experimental results are given in this paper. We discuss non-implemented improvements of this planner, including a new way to embed knowledge in a motion command in order to produce larger preimages.

Weyl-invariant random Hamiltonians and their relation to translational-invariant random potentials on Landau levels

The authors construct and discuss classes of random Hamiltonians on the infinite-dimensional Hilbert space of a quantum system with a Euclidean configuration space. By construction, the corresponding probability distributions are invariant under the action of the Weyl group. For particular classes of Weyl-invariant random Hamiltonians they establish a relation to translational-invariant random potentials restricted to the Hilbert space of a single Landau level, that is, an eigenspace of the standard Hamiltonian for a charged particle confined to the plane and subjected to a perpendicular constant magnetic field.

On the global nature of the semiclassical wavefunction describing elastic scattering in several coordinate dimensions

The short-wavelength (semiclassical) limit of the quantal wavefunction in a multidimensional euclidean configuration space is discussed. Attentation is drawn to the shortcomings of some previous approaches with regard to the treatment of multivaluedness and caustics. A global representation of the wavefunction based on generalised Kirchoff diffraction integrals is proposed. Application of the saddle-point approximation leads to a local semiclassical representation of the wavefunction as a series of JWKB approximants.