Theoretical methods for the study of chemical dynamics

Fortran programs for the time-dependent Gross–Pitaevskii equation in a fully anisotropic trap

Recently, the robot with many links and degrees of freedoms is actively produced. Generally, the robot that has many degrees of freedoms owns many redundant joints. So we make a suggestion for using redundant joints effectively. The paper proposes using redundant joint as a counter weight. We make a trajectory in momentum space, and we use Fourier Basis Algorism as the method for an optimization. This trajectory generating method has an effect on reducing a driving energy for throwing motion.

In this thesis, we present a dynamical systems approach to adaptive controllers for locomotion control. The approach is based on a rigorous mathematical framework using nonlinear dynamical systems and is inspired by theories of self-organization. Nonlinear dynamical systems such as coupled oscillators are an interesting approach for the on-line generation of trajectories for robots with many degrees of freedom (e.g. legged locomotion). However, designing a nonlinear dynamical system to satisfy a given specification and goal is not an easy task, and, hitherto no methodology exists to approach this problem in a unified way. Nature presents us with satisfactory solutions for the coordination of many degrees of freedom. One central feature observed in biological subjects is the ability of the neural systems to exploit natural dynamics of the body to achieve efficient locomotion. In order to be able to exploit the body properties, adaptive mechanisms must be at work. Recent work has pointed out the importance of the mechanical system for efficient locomotion. Even more interestingly, such well suited mechanical systems do not need complicated control. Yet, in robotics, in most approaches, adaptive mechanisms are either missing or they are not based on a rigorous framework, i.e. they are based on heuristics and ad-hoc approaches. Over the last three decades there has been enormous progress in describing movement coordination with the help of Synergetic approaches. This has led to the formulation of a theoretical framework: the theory of dynamic patterns. This framework is mathematically rigorous and at the same time fully operational. However, it does not provide any guidelines for synthetic approaches as needed for the engineering of robots with many degrees of freedom, nor does it directly help to explain adaptive systems. We will show how we can extend the theoretical framework to build adaptive systems. For this purpose, we propose the use of multi-scale dynamical systems. The basic idea behind multi-scale dynamical systems is that a given dynamical system gets extended by additional slow dynamics of its parameters, i.e. some of the parameters become state variables. The advantages of the framework of multi-scale dynamical systems for adaptive controllers are 1) fully dynamic description, 2) no separation of learning algorithm and learning substrate, 3) no separation of learning trials or time windows, 4) mathematically rigorous, 5) low dimensional systems. However, in order to fully exploit the framework important questions have to be solved. Most importantly, methodologies for designing the feedback loops have to be found and important theoretical questions about stability and convergence properties of the devised systems have to be answered. In order to tackle this challenge, we first introduce an engineering view on designing nonlinear dynamical systems and especially oscillators. We will highlight the important differences and freedom that this engineering view introduces as opposed to a modeling one. We then apply this approach by first proposing a very simple adaptive toy-system, consisting of a dynamical system coupled to a spring-mass system. Due to its spring-mass dynamics, this system contains clear natural dynamics in the form of resonant frequencies. We propose a prototype adaptive multi-scale system, the adaptive frequency oscillator, which is able to adapt its intrinsic frequency to the resonant frequency of the body dynamics. After a small sidetrack to show that we can use adaptive frequency oscillators also for other applications than for adaptive controllers, namely for frequency analysis, we then come back to further investigation of the adaptive controller. We apply the same controller concept to a simple spring-mass hopper system. The spring-mass system consists of a body with two legs attached by rotational joints. The legs contain spring-damper elements. Finally, we present results of the implementation of the controller on a real robot, the experimental robot PUPPY II. This robot is a under-actuated robot with spring dynamics in the knee joints. It will be shown, that due to the appropriate simplification and concentration on relevant features in the toy-system the controller concepts works without a fundamental change on all systems from the toy system up to the real robot.

A Path Integral (PI) formulation of linear elastostatics is first presented. For this, Navier equations are modified by adding a fictitious 'time' derivative of displacements so that equilibrium corresponds to the steady state of the resulting diffusion-like equations. The evolution of displacement is then represented as the propagation, through the fictitious time co-ordinate, of an initial displacement field corresponding to the unloaded state. The resulting procedure somehow mimics the well-known Feynman path integral of quantum mechanics, which is equivalent to the differential formulation embodied in Schrödinger equation. However, the path integral for elastostatics is formulated in terms of infinitesimal propagators of local support. In its simplest form, the formulation can be used as a relaxation method of solution, by updating displacements until convergence. This may be advantageous for problems involving a very large number of unknowns. On the other hand, by equating the updated displacement field to the actual one a direct method of solution is obtained, which leads to non-symmetric (but sparse and banded) discrete equations. Unlike variational principles this formulation does not require integration over the whole domain, effectively eliminating the need of a background mesh for integration. Also, it only requires continuity of the displacement field on the propagator's support. As a consequence, the formulation lends itself to very flexible meshless implementations. To demonstrate this we describe a simple numerical method in which displacements around each node are approximated by quadratic bivariate polynomials, which is the simplest approximation technique. The feasibility of the method is assessed through a number of numerical examples and comparisons with analytical solutions and other meshless methods. Copyright © 2000 John Wiley & Sons, Ltd.

We consider the problem of potential interaction between a finite-dimensional linear Lagrangian system and an infinite-dimensional one (a system of linear oscillators and a thermostat). We study the final dynamics of the system. Under natural assumptions, this dynamics turns out to be very simple and admits an explicit description because the thermostat produces an effective dissipation despite the energy conservation and the Lagrangian nature of the system. We use the methods of [1], where the final dynamics of the finite-dimensional subsystem is studied in the case when it has one degree of freedom and a linear potential or (under additional assumptions) polynomial potential. We consider the case of finite-dimensional subsystems with arbitrarily many degrees of freedom and a linear potential and study the final dynamics of the system of oscillators and the thermostat. The necessary assertions from [1] are given with proofs adapted to the present situation.

We address the impossibility of achieving exact time reversal in a system with many degrees of freedom. This is a particular example of the difficult task of “aiming” an initial classical state so as to become a specific final state. We also comment on the classical-to-quantum transition in any non-separable closed system of \(n \ge 2\) degrees of freedom. Even if the system is initially in a well-defined WKB, semi-classical state, quantum evolution and, in particular, multiple reflections at classical turning points make it completely quantum mechanical with each particle smeared almost uniformly over all the configuration space. The argument, which is presented in the context of \(n\) hard discs, is quite general. Finally, we briefly address more complex quantum systems with many degrees of freedom and ask when can they provide an appropriate environment to the above simpler systems so that quantum spreading is avoided by continuously leaving “imprints” in the environment. We also discuss the possible connections with the pointer systems that are needed in the quantum-to-classical “collapse” transitions.

The geometric phase (GP) is a fundamental quantum effect arising from conical intersections (CIs), with profound consequences for vibronic energy levels. Standard imaginary-time path integral molecular dynamics (PIMD) based on the Born-Oppenheimer approximation does not account for the GP, potentially leading to significant errors in low-temperature thermodynamic properties. In this Perspective, we demonstrate that the multi-electronic-state path integral (MES-PI) formulation in imaginary time─developed in J. Chem. Phys. 2018, 148, 102319─naturally captures the GP effect through the electronic trace of the product of statistically weighted overlap matrices between successive imaginary-time slices. This crucial capability was already implicit in the benchmark MES-PIMD simulations in that foundational work. To isolate this topological effect from other nonadiabatic effects, we introduce a geometric signature matrix (for the CI) and a winding-number-induced phase factor, constructing an ad hoc GP-excluded MES-PI method. Comparing this ad hoc baseline against the rigorous MES-PI approach allows us to unambiguously quantify the impact of the GP on thermodynamic properties. While simpler approximations exist when only the ground electronic state is considered, MES-PIMD is the most general and accurate approach applicable to real complex systems, where the location and topology of CI seams are often not known a priori.

Ab initio molecular orbital (MO) methods can provide accurate descriptions of potential energy surfaces for chemical reactions. For triatomic systems such as H3 and H2F extensive calculations of very accurate potential surfaces have been carried out. For systems having four atoms or more, however, the accomplishment so far has been rather limited. One of the reasons for this is that the full potential energy surface is a function of 3N-6 coordinates where N is the number of atoms, and this is too many degrees of freedom for the full surface to be mapped. Even to optimize the geometrical parameters for all the degrees of freedom to determine the reactant and saddle point geometries is very difficult. However, a revolution is taking place in the way quantum chemists probe a potential surface. It is based on the development of new theoretical methods and computer programs for analytically calculating the gradient of the energy with respect to the nuclear coordinates.1 Quantum chemists are now learning how to use this energy gradient, a concept more familiar in the context of collision theory, to study the characteristics of potential energy surfaces. In this review we will first summarize briefly how to calculate the energy gradient analytically. Then we will discuss methods of determining geometries, force constants, and normal vibrations for both equilibrium configurations and saddle points, and we will show several examples of such calculations we have recently carried out.

A methodology to capture and post-process bat flight 3-D Stereo Triangulation data to formulate an approximated rigid body kinematic model was investigated. Bat flight is unique in nature due to the bats inherent agility and many degrees of freedom when compared to other flying animals. This complexity makes capturing accurate aerodynamic data very difficult. Unlike insects, which utilize few degrees of freedom and a high flap frequency for sustained flight and maneuverability, the agility of bats comes in part from the many degrees of freedom present in the bat wing. In order to better understand the aerodynamics present in bat flight, bats Hipposideridae (Old World leaf-nosed bats) were examined. The trajectories of critical points along the bat wings were recorded using 3D stereo triangulation techniques to capture the complexities of the bat flight. Markers were placed at all the joint locations along the bat wing. The resulting trajectories were then translated into a periodic kinematic model for future computational use.

Evaluation of the absolute configurational entropy of tri-O-methyl-β-cyclodextrin, a molecule with many degrees of freedom

The standard C*-formalism of representations of commutations relations for systems of infinitely many degrees of freedom over a real numbers field is extended to the case of a p-adic numbers field. It is proved that any positive function on the Weyl algebra over a p-adic symplectic space is regular and restriction of the generating function of a state of the Weyl algebra on any nongenerate finite-dimensional subspace of p-adic symplectic space is continuous. Some nontrivial class of nonequivalent representations of commutations relations is described and analogies of Fock representations and representations with number operators are constructed. The criteria for the existence of free-field unitary quantization of symplectic transformations is given.

A fast algorithm is presented that can handle the motion planning problem for articulated figures with branches and many degrees of freedom. The algorithm breaks down the degrees of freedom of the figure into Cspace groups and compute the free motion for each of these groups in a sequential fashion. It traverses the tree in a depth first order to compute the motion for all the branches. A special playback routine is then used to traverse the tree again in a reverse order to playback the final motion. The planner runs in linear time with respect to the total number of Cspace groups without backtracking. We believe that the planner would find a path in most cases and is fast enough for practical use in a wide range of applications.

Many real-world systems are often regarded as weakly coupled limit-cycle oscillators, in which each oscillator corresponds to a dynamical system with many degrees of freedom that have collective oscillations. One of the most practical methods for investigating the synchronization properties of such a rhythmic system is to statistically extract phase coupling functions between limit-cycle oscillators directly from observed time-series data. In Particular, using a method that combines phase reduction theory and Bayesian inference, the phase coupling functions can be extracted from the time-series data of even just one variable in each oscillatory dynamical system with many degrees of freedom. However, it remains unclear how the choice of the observed variables affects the statistical inference for the phase coupling functions. In this study, we examine the influence of observed variable types on the extraction of phase coupling functions using some typical dynamical elements under various conditions. We demonstrate that our method can consistently extract the macroscopic phase coupling functions between two phases representing collective oscillations in a fully locked state, regardless of the observed variable types; for example, even using one variable of any element in one system and the mean-field value over all the elements in another system. We also study the case of globally coupled phase oscillators in a partially locked state. Our results reveal directional asymmetry in the robustness of extracting the macroscopic phase coupling function between two networks. For instance, when an asynchronous oscillator in network $A$ and the macroscopic collective oscillation of network $B$ is observed, the macroscopic phase coupling function from network $A$ to network $B$ can be extracted more robustly than in the opposite direction.

The method of multiple scales is used to analyze parametrically excited, linear systems having many degrees of freedom and distinct natural frequencies. Explicit second-order expressions are obtained for the characteristic exponents which yield the transition curves. Various combination resonances are treated. The results are applied to the buckling of free-clamped columns under the influence of sinusoidally varying axial loads.

From metastable to stable states in a magnetic system with many degrees of freedom