Showing papers on "Configuration space published in 2012"

Symmetries, Topology and Resonances in Hamiltonian Mechanics

Abstract: I Hamiltonian Mechanics.- 1 The Hamilton Equations.- 2 Euler-Poincare Equations on Lie Algebras.- 3 The Motion of a Rigid Body.- 4 Pendulum Oscillations.- 5 Some Problems of Celestial Mechanics.- 6 Systems of Interacting Particles.- 7 Non-holonomic Systems.- 8 Some Problems of Mathematical Physics.- 9 The Problem of Identification of Hamiltonian Systems.- II Integration of Hamiltonian Systems.- 1 Integrals. Classes of Integrals of Hamiltonian Systems.- 2 Invariant Relations.- 3 Symmetry Groups.- 4 Complete Integrability.- 5 Examples of Completely Integrable Systems.- 6 Isomorphisms of Some Integrable Hamiltonian Systems.- 7 Separation of Variables.- 8 The Heisenberg Representation.- 9 Algebraically Integrable Systems.- 10 Perturbation Theory.- 11 Normal Forms.- III Topological and Geometrical Obstructions to Complete Integrability.- 1 Topology of the Configuration Space of an Integrable System.- 2 Proof of Nonintegrability Theorems.- 3 Geometrical Obstructions to Integrability.- 4 Systems with Gyroscopic Forces.- 5 Generic Integrals.- 6 Topological Obstructions to the Existence of Linear Integrals.- 7 Topology of the Configuration Space of Reversible Systems with Nontrivial Symmetry Groups.- IV Nonintegrability of Hamiltonian Systems Close to Integrable Ones.- 1 The Poincare Method.- 2 Applications of the Poincare Method.- 3 Symmetry Groups.- 4 Reversible Systems With a Torus as the Configuration Space.- 5 A Criterion for Integrability in the Case When the Potential is a Trigonometric Polynomial.- 6 Some Generalizations.- 7 Systems of Interacting Particles.- 8 Birth of Isolated Periodic Solutions as an Obstacle to Integrability.- 9 Non-degenerate Invariant Tori.- 10 Birth of Hyperbolic Invariant Tori.- 11 Non-Autonomous Systems.- V Splitting of Asymptotic Surfaces.- 1 Asymptotic Surfaces and Splitting Conditions.- 2 Theorems on Nonintegrability.- 3 Some Applications.- 4 Conditions for Nonintegrability of Kirchhoff's Equations.- 5 Bifurcation of Separatrices.- 6 Splitting of Separatrices and Birth of Isolated Periodic Solutions.- 7 Asymptotic Surfaces of Unstable Equilibria.- 8 Symbolic Dynamics.- VI Nonintegrability in the Vicinity of an Equilibrium Position.- 1 Siegel's Method.- 2 Nonintegrability of Reversible Systems.- 3 Nonintegrability of Systems Depending on Parameters.- 4 Symmetry Fields in the Vicinity of an Equilibrium Position.- VII Branching of Solutions and Nonexistence of Single-Valued Integrals.- 1 The Poincare Small Parameter Method.- 2 Branching of Solutions and Polynomial Integrals of Reversible Systems on a Torus.- 3 Integrals and Symmetry Groups of Quasi-Homogeneous Systems of Differential Equations.- 4 Kovalevskaya Numbers for Generalized Toda Lattices.- 5 Monodromy Groups of Hamiltonian Systems with Single-Valued Integrals.- VIII Polynomial Integrals of Hamiltonian Systems.- 1 The Birkhoff Method.- 2 Influence of Gyroscopic Forces on the Existence of Polynomial Integrals.- 3 Polynomial Integrals of Systems with One and a Half Degrees of Freedom.- 4 Polynomial Integrals of Hamiltonian Systems with Exponential Interaction.- 5 Perturbations of Hamiltonian Systems with Non-Compact Invariant Surfaces.- References.

The Status of our Ordinary Three Dimensions in a Quantum Universe 1

Abstract: There are now several, realist versions of quantum mechanics on offer. On their most straightforward, ontological interpretation, these theories require the existence of an object, the wavefunction, which inhabits an extremely high-dimensional space known as configuration space. This raises the question of how the ordinary three-dimensional space of our acquaintance fits into the ontology of quantum mechanics. Recently, two strategies to address this question have emerged. First, Tim Maudlin, Valia Allori, and her collaborators argue that what I have just called the ‘most straightforward’ interpretation of quantum mechanics is not the correct one. Rather, the correct interpretation of realist quantum mechanics has it describing the world as containing objects that inhabit the ordinary three-dimensional space of our manifest image. By contrast, David Albert and Barry Loewer maintain the straightforward, wavefunction ontology of quantum mechanics, but attempt to show how ordinary, three-dimensional space may in a sense be contained within the high-dimensional configuration space the wavefunction inhabits. This paper critically examines these attempts to locate the ordinary, three-dimensional space of our manifest image “within” the ontology of quantum mechanics. I argue that we can recover most of our manifest image, even if we cannot recover our familiar three-dimensional space.

Density of States for a Specified Correlation Function and the Energy Landscape

Abstract: The degeneracy of two-phase disordered microstructures consistent with a specified correlation function is analyzed by mapping it to a ground-state degeneracy. We determine for the first time the associated density of states via a Monte Carlo algorithm. Our results are explained in terms of the roughness of an energy landscape, defined on a hypercubic configuration space. The use of a Hamming distance in this space enables us to define a roughness metric, which is calculated from the correlation function alone and related quantitatively to the structural degeneracy. This relation is validated for a wide variety of disordered structures.

Quantum heat engine in the relativistic limit: the case of a Dirac particle.

Abstract: We studied the efficiency of two different schemes for a quantum heat engine, by considering a single Dirac particle trapped in an infinite one-dimensional potential well as the ``working substance.'' The first scheme is a cycle, composed of two adiabatic and two isoenergetic reversible trajectories in configuration space. The trajectories are driven by a quasistatic deformation of the potential well due to an external applied force. The second scheme is a variant of the former, where isoenergetic trajectories are replaced by isothermal ones, along which the system is in contact with macroscopic thermostats. This second scheme constitutes a quantum analog of the classical Carnot cycle. Our expressions, as obtained from the Dirac single-particle spectrum, converge in the nonrelativistic limit to some of the existing results in the literature for the Schr\"odinger spectrum.

Statistical consistency of quantum-classical hybrids

Abstract: After formulating a no-go theorem for perfect quantum-classical hybrid systems, a consistency requirement based on standard statistical considerations is noted. It is shown that such requirement is not fulfilled by the mean-field approach or by the statistical ensemble in configuration space approach. Further unusual features of the latter scheme are pointed out.

Graviton vertices and the mapping of anomalous correlators to momentum space for a general conformal field theory

Abstract: We investigate the mapping of conformal correlators and of their anomalies from configuration to momentum space for general dimensions, focusing on the anomalous correlators TOO, TVV — involving the energy-momentum tensor (T) with a vector (V) or a scalar operator (O) — and the 3-graviton vertex TTT. We compute the TOO, TVV and TTT one-loop vertex functions in dimensional regularization for free field theories involving conformal scalar, fermion and vector fields. Since there are only one or two independent tensor structures solving all the conformal Ward identities for the TOO or TVV vertex functions respectively, and three independent tensor structures for the TTT vertex, and the coefficients of these tensors are known for free fields, it is possible to identify the corresponding tensors in momentum space from the computation of the correlators for free fields. This works in general d dimensions for TOO and TVV correlators, but only in 4 dimensions for TTT, since vector fields are conformal only in d = 4. In this way the general solution of the Ward identities including anomalous ones for these correlators in (Euclidean) position space, found by Osborn and Petkou is mapped to the ordinary diagrammatic one in momentum space. We give simplified expressions of all these correlators in configuration space which are explicitly Fourier integrable and provide a diagrammatic interpretation of all the contact terms arising when two or more of the points coincide. We discuss how the anomalies arise in each approach. We then outline a general algorithm for mapping correlators from position to momentum space, and illustrate its application in the case of the VVV and TOO vertices. The method implements an intermediate regularization — similar to differential regularization — for the identification of the integrands in momentum space, and one extra regulator. The relation between the ordinary Feynman expansion and the logarithmic one generated by this approach are briefly discussed.

Statistics of Wave Functions for a Point Scatterer on the Torus

Abstract: Quantum systems whose classical counterpart have ergodic dynamics are quantum ergodic in the sense that almost all eigenstates are uniformly distributed in phase space. In contrast, when the classical dynamics is integrable, there is concentration of eigenfunctions on invariant structures in phase space. In this paper we study eigenfunction statistics for the Laplacian perturbed by a delta-potential (also known as a point scatterer) on a flat torus, a popular model used to study the transition between integrability and chaos in quantum mechanics. The eigenfunctions of this operator consist of eigenfunctions of the Laplacian which vanish at the scatterer, and new, or perturbed, eigenfunctions. We show that almost all of the perturbed eigenfunctions are uniformly distributed in configuration space.

Proving path non-existence using sampling and alpha shapes

Abstract: In this paper, we address the problem determining the connectivity of a robot's free configuration space. Our method iteratively builds a constructive proof that two configurations lie in disjoint components of the free configuration space. Our algorithm first generates samples that correspond to configurations for which the robot is in collision with an obstacle. These samples are then weighted by their generalized penetration distance, and used to construct alpha shapes. The alpha shape defines a collection of simplices that are fully contained within the configuration space obstacle region. These simplices can be used to quickly solve connectivity queries, which in turn can be used to define termination conditions for sampling-based planners. Such planners, while typically either resolution complete or probabilistically complete, are not able to determine when a path does not exist, and therefore would otherwise rely on heuristics to determine when the search for a free path should be abandoned. An implementation of the algorithm is provided for the case of a 3D Euclidean configuration space, and a proof of correctness is provided.

Avoided crossings, conical intersections, and low-lying excited states with a single reference method: the restricted active space spin-flip configuration interaction approach.

Abstract: The restricted active space spin-flip CI (RASCI-SF) performance is tested in the electronic structure computation of the ground and the lowest electronically excited states in the presence of near-degeneracies. The feasibility of the method is demonstrated by analyzing the avoided crossing between the ionic and neutral singlet states of LiF along the molecular dissociation. The two potential energy surfaces (PESs) are explored by means of the energies of computed adiabatic and approximated diabatic states, dipole moments, and natural orbital electronic occupancies of both states. The RASCI-SF methodology is also used to study the ground and first excited singlet surface crossing involved in the double bond isomerization of ethylene, as a model case. The two-dimensional PESs of the ground (S0) and excited (S1) states are calculated for the complete configuration space of torsion and pyramidalization molecular distortions. The parameters that define the state energetics in the vicinity of the S0/S1 conical ...

A numerical test of differential equations for one- and two-loop sunrise diagrams using configuration space techniques

Abstract: We use configuration space methods to write down one-dimensional integral representations for one- and two-loop sunrise diagrams (also called Bessel moments) which we use to numerically check on the correctness of the second order differential equations for one- and two-loop sunrise diagrams that have recently been discussed in the literature.

PolyDepth: Real-time penetration depth computation using iterative contact-space projection

Abstract: We present a real-time algorithm that finds the Penetration Depth (PD) between general polygonal models based on iterative and local optimization techniques. Given an in-collision configuration of an object in configuration space, we find an initial collision-free configuration using several methods such as centroid difference, maximally clear configuration, motion coherence, random configuration, and sampling-based search. We project this configuration on to a local contact space using a variant of continuous collision detection algorithm and construct a linear convex cone around the projected configuration. We then formulate a new projection of the in-collision configuration onto the convex cone as a Linear Complementarity Problem (LCP), which we solve using a type of Gauss-Seidel iterative algorithm. We repeat this procedure until a locally optimal PD is obtained. Our algorithm can process complicated models consisting of tens of thousands triangles at interactive rates.

Classical and quantum modes of coupled Mathieu equations

Abstract: We expand the solutions of linearly coupled Mathieu equations in terms of infinite-continued matrix inversions, and use it to find the modes which diagonalize the dynamical problem. This allows obtaining explicitly the (Floquet–Lyapunov) transformation to coordinates in which the motion is that of decoupled linear oscillators. We use this transformation to solve the Heisenberg equations of the corresponding quantum-mechanical problem, and find the quantum wavefunctions for stable oscillations, expressed in configuration space. The obtained transformation and quantum solutions can be applied to more general linear systems with periodic coefficients (coupled Hill equations, periodically driven parametric oscillators), and to nonlinear systems as a starting point for convenient perturbative treatment of the nonlinearity.

Randomized path planning on manifolds based on higher-dimensional continuation

Abstract: Despite the significant advances in path planning methods, highly constrained problems are still challenging. In some situations, the presence of constraints defines a configuration space that is a non-parametrizable manifold embedded in a high-dimensional ambient space. In these cases, the use of sampling-based path planners is cumbersome since samples in the ambient space have low probability to lay on the configuration space manifold. In this paper, we present a new path planning algorithm specially tailored for highly constrained systems. The proposed planner builds on recently developed tools for higher-dimensional continuation, which provide numerical procedures to describe an implicitly defined manifold using a set of local charts. We propose to extend these methods focusing the generation of charts on the path between the two configurations to connect and randomizing the process to find alternative paths in the presence of obstacles. The advantage of this planner comes from the fact that it directly operates into the configuration space and not into the higher-dimensional ambient space, as most of the existing methods do.

Analysis of a generalized kinematic impact law for multibody-multicontact systems, with application to the planar rocking block and chains of balls

Abstract: In this paper, we analyze the capabilities of a generalized kinematic (Newton’s like) restitution law for the modeling of a planar rigid block that impacts a rigid ground. This kinematic restitution law is based on a specific state transformation of the Lagrangian dynamics, using the kinetic metric on the configuration space. It allows one to easily derive a restitution rule for multiple impacts. The relationships with the classical angular velocity restitution coefficient r for rocking motion are examined in detail. In particular, it is shown that r has the interpretation of a tangential restitution coefficient. The case when Coulomb’s friction is introduced at the contact impulse level together with an angular velocity restitution is analyzed. A simple chain of aligned balls is also examined, illustrating that the impact law applies to various types of multibody systems.

Presentations of graph braid groups

Abstract: Let be a graph. The (unlabeled) configuration space UCn of n points on is the space of n-element subsets of . The n-strand braid group of , denoted Bn, is the fundamental group of UCn. This paper extends the methods and results of (11). Here we compute presentations for Bn, where n is an arbitrary natural number and is an arbitrary finite connected graph. Particular attention is paid to the case n = 2, and many examples are given.

Rigid Body Energy Minimization on Manifolds for Molecular Docking.

Abstract: Virtually all docking methods include some local continuous minimization of an energy/scoring function in order to remove steric clashes and obtain more reliable energy values. In this paper, we describe an efficient rigid-body optimization algorithm that, compared to the most widely used algorithms, converges approximately an order of magnitude faster to conformations with equal or slightly lower energy. The space of rigid body transformations is a nonlinear manifold, namely, a space which locally resembles a Euclidean space. We use a canonical parametrization of the manifold, called the exponential parametrization, to map the Euclidean tangent space of the manifold onto the manifold itself. Thus, we locally transform the rigid body optimization to an optimization over a Euclidean space where basic optimization algorithms are applicable. Compared to commonly used methods, this formulation substantially reduces the dimension of the search space. As a result, it requires far fewer costly function and gradi...

Implications of the two nodal domains conjecture for ground state fermionic wave functions

Abstract: The nodes of many-body wave functions are mathematical objects important in many different fields of physics. They are at the heart of the quantum Monte Carlo methods but outside this field their properties are neither widely known nor studied. In recent years a conjecture, already proven to be true in several important cases, has been put forward related to the nodes of the fermionic ground state of a many-body system, namely that there is a single nodal hypersurface that divides configuration space into only two connected domains. While this is obviously relevant to the fixed node diffusion Monte Carlo method, its repercussions have ramifications in various fields of physics as diverse as density functional theory or Feynman and Cohen's backflow wave function formulation. To illustrate this we explicitly show that, even if we knew the exact Kohn-Sham exchange correlation functional, there are systems for which we would obtain the exact ground state energy and density but a wave function quite different from the exact one. This paradox is only apparent since the Hohenberg-Kohn theorem relates the energy directly to the density and the wave function is not guaranteed to be close to the exact one. The aim of this paper is to stimulate the investigation of the properties of the nodes of many-body wave functions in different fields of physics. Furthermore, we explicitly show that this conjecture is related to the phenomenon of avoided nodal crossing but it is not necessarily caused by electron correlation, as sometimes has been suggested in the literature. We explicitly build a many-body uncorrelated example whose nodal structure shows the same phenomenon.

Critical thresholds in multi-dimensional euler-poisson equations with radial symmetry ∗

Abstract: We study the global regularity of multi-dimensional repulsive Euler-Poisson equa- tions in the radial setup. We show that the question of global regularity vs. finite breakdown of smooth solutions depends on whether the initial configuration crosses an initial critical threshold in configuration space. Specifically, there exists a global-in-time smooth solution if and only if the initial configuration of density �0, radial velocity R0, and electrical charge e0 satisfies R ' ≥F(�0,e0,R0) for a certain threshold F. Similarly, we characterize the critical threshold for global smooth solutions subject to two-dimensional radially symmetric data with swirl. We also discuss a possible framework for global regularity analysis beyond the radial case, which indicates that the main difficulty lies with bounding the spectral gap, �2(∇u) −�1(∇u).

Hierarchical Motion Planning in Topological Representations

Abstract: Motion can be described in alternative represen- tations, including joint configuration or end-effector spaces, but also more complex topological representations that imply a change of Voronoi bias, metric or topology of the motion space. Certain types of robot interaction problems, e.g. wrapping around an object, can suitably be described by so-called writhe and interaction mesh representations. However, considering mo- tion synthesis solely in topological spaces is insufficient since it does not cater for additional tasks and constraints in other representations. In this paper we propose methods to combine and exploit different representations for motion synthesis, with specific emphasis on generalization of motion to novel situations. Our approach is formulated in the framework of optimal con- trol as an approximate inference problem, which allows for a direct extension of the graphical model to incorporate multiple representations. Motion generalization is similarly performed by projecting motion from topological to joint configuration space. We demonstrate the benefits of our methods on problems where direct path finding in joint configuration space is extremely hard whereas local optimal control exploiting a representation with different topology can efficiently find optimal trajectories. Further, we illustrate the successful online motion generalization to dynamic environments on challenging, real world problems.

An intrinsic formulation of the problem on rolling manifolds

Abstract: We present an intrinsic formulation of the kinematic problem of two n-dimensional manifolds rolling one on another without twisting or slipping. We determine the configuration space of the system, which is an n(n?+?3)/2-dimensional manifold. The conditions of no-twisting and no-slipping are encoded by means of a distribution of rank n. We compare the intrinsic point of view versus the extrinsic one. We also show that the kinematic system of rolling the n-dimensional sphere over $ {\mathbb{R}^n} $ is controllable. In contrast with this, we show that in the case of SE(3) rolling over $ \mathfrak{s}\mathfrak{e}(3) $ the system is not controllable, since the configuration space of dimension 27 is foliated by submanifolds of dimension 12.

Massive Gravity on Curved Background

Abstract: We investigate generally covariant theories which admit a Fierz–Pauli mass term for metric perturbations around an arbitrary curved background. For this we restore the general covariance of the Fierz–Pauli mass term by introducing four scalar fields which preserve a certain internal symmetry in their configuration space. It is then apparent that for each given spacetime metric this construction corresponds to a completely different generally covariant massive gravity theory with different symmetries. The proposed approach is verified by explicit analysis of the physical degrees of freedom of massive graviton on de Sitter space.

Numerical computation of manipulator singularities

Abstract: This paper provides a method to compute all types of singularities of non-redundant manipulators with non-helical lower pairs and designated instantaneous input and output speeds. A system of equations describing each singularity type is given. Using a numerical method based on linear relaxations, the configurations in each type are computed independently. The method is general and complete: it can be applied to manipulators with arbitrary geometry; and will isolate singularities with the desired accuracy. As an example, the entire singularity set and its complete classification are computed for a two-degree-of-freedom mechanism. The complex partition of the configuration space by various singularities is illustrated by three-dimensional projections.

Nonlinear behavior of baryon acoustic oscillations from the zel'dovich approximation using a non-fourier perturbation approach

Abstract: Baryon acoustic oscillations are an excellent technique to constrain the properties of dark energy in the universe. In order to accurately characterize the dark energy equation of state, we must understand the effects of both the nonlinearities and redshift space distortions on the location and shape of the acoustic peak. In this paper, we consider these effects using the Zel'dovich approximation and a novel approach to second-order perturbation theory. The second-order term of the Zel'dovich power spectrum is built from convolutions of the linear power spectrum with polynomial kernels in Fourier space, suggesting that the corresponding term of the Zel'dovich correlation function can be written as a sum of quadratic products of a broader class of correlation functions, expressed through simple spherical Bessel transforms of the linear power spectrum. We show how to systematically perform such a computation. We explicitly prove that our result is the Fourier transform of the Zel'dovich power spectrum and compare our expressions to numerical simulations. Finally, we highlight the advantages of writing the nonlinear expansion in configuration space, as this calculation is easily extended to redshift space, and the higher-order terms are mathematically simpler than their Fourier counterparts.

Learning local linear Jacobians for flexible and adaptive robot arm control

Abstract: Successful planning and control of robots strongly depends on the quality of kinematic models, which define mappings between configuration space (e.g. joint angles) and task space (e.g. Cartesian coordinates of the end effector). Often these models are predefined, in which case, for example, unforeseen bodily changes may result in unpredictable behavior. We are interested in a learning approach that can adapt to such changes--be they due to motor or sensory failures, or also due to the flexible extension of the robot body by, for example, the usage of tools. We focus on learning locally linear forward velocity kinematics models by means of the neuro-evolution approach XCSF. The algorithm learns self-supervised, executing movements autonomously by means of goal-babbling. It preserves actuator redundancies, which can be exploited during movement execution to fulfill current task constraints. For detailed evaluation purposes, we study the performance of XCSF when learning to control an anthropomorphic seven degrees of freedom arm in simulation. We show that XCSF can learn large forward velocity kinematic mappings autonomously and rather independently of the task space representation provided. The resulting mapping is highly suitable to resolve redundancies on the fly during inverse, goal-directed control.

Hamiltonian dynamics and Noether symmetries in Extended Gravity Cosmology

Abstract: We discuss the Hamiltonian dynamics for cosmologies coming from Extended Theories of Gravity. In particular, minisuperspace models are taken into account searching for Noether symmetries. The existence of conserved quantities gives selection rule to recover classical behaviors in cosmic evolution according to the so called Hartle criterion, that allows to select correlated regions in the configuration space of dynamical variables. We show that such a statement works for general classes of Extended Theories of Gravity and is conformally preserved. Furthermore, the presence of Noether symmetries allows a straightforward classification of singularities that represent the points where the symmetry is broken. Examples of nonminimally coupled and higher-order models are discussed.

A Tulczyjew triple for classical fields

Abstract: The geometrical structure known as the Tulczyjew triple has proved to be very useful in describing mechanical systems, even those with singular Lagrangians or subject to constraints. Starting from basic concepts of the variational calculus, we construct the Tulczyjew triple for first-order field theory. The important feature of our approach is that we do not postulate ad hoc the ingredients of the theory, but obtain them as unavoidable consequences of the variational calculus. This picture of field theory is covariant and complete, containing not only the Lagrangian formalism and Euler–Lagrange equations but also the phase space, the phase dynamics and the Hamiltonian formalism. Since the configuration space turns out to be an affine bundle, we have to use affine geometry, in particular the notion of the affine duality. In our formulation, the two maps α and β which constitute the Tulczyjew triple are morphisms of double structures of affine-vector bundles. We also discuss the Legendre transformation, i.e. the transition between the Lagrangian and the Hamiltonian formulation of the first-order field theory.