Hamiltonian systems,information transmittal and special relativity

Linear transformations of special relativity considered in [Albert Einstein, Zur Elektrodynamik der bewegte Körper. Annalen der Physik. 17 (1905) 891–921] are applied to Hamiltonian systems and Dirac equations, as presented in [Paul A.M. Dirac, Lectures on Quantum Mechanics, Belfer Graduate School of Science, New York, 1964]. To account for Weber’s electro-dynamic law of particle attraction and for the relativistic increase of the mass in particle accelerators, an extension of the second Newton’s law for motion subject to external forces that may depend on accelerations and higher order derivatives of velocities is considered and the Buquoy–Mestschersky generalization for motion of bodies with variable masses is included. The causality of systems driven by such forces is assured by consideration of left higher order derivatives in the right-hand sides of the equations of motion. The consistency condition is presented, and existence of solutions for the equations of motion driven by forces with left higher order derivatives is proved, leading to the generalized Lagrange and Hamilton equations that incorporate those extensions. Then, non-holonomic Hamiltonian systems with time-invariant constraints introduced by P.G. Bergmann and P.A.M. Dirac are considered, as suggested by Dirac for atomic models. Since one and the same process or particle may be observed as different images, the procedure is developed for identification of processes in moving systems by inverse relativistic transformations applied over small intervals of time to discrete experimental measurements obtained as the images of those processes in the observed (relativistic) coordinates. It is demonstrated that motions and processes evolving in still and/or moving systems can be described in the proper (of a still system) and/or relativistic (of different moving systems) coordinates in one common system of equations under the condition that all components of that system are referred to one and the same time of a still or moving observer. The results open new avenues for further research in relativistic systems theory and provide a basis for development of computing software for process identification in the images transmitted from distant or fast moving systems.     

Information transmittal, Newton’s law of gravitation, and tensor approach to general relativity

The special relativity considered in [A. Einstein, Zur Elektrodynamik der bewegte Körper. Ann. Physik, 17 (1905) 891-921] is based on the concept of finite speed of information transmittal by the available signals (rays of light). It is demonstrated that the same concept applies to Newton’s law of universal gravitation since the magnitude of distances between attracting masses can be physically defined (carried, accounted in acting forces of gravity) only by signals (physical processes) propagating at finite velocities. It follows that the speed of propagation of gravity is finite. The linear transformations of special relativity are applied to Newton’s law of gravitation to take into account the relativistic effects of information transmittal in a field of central forces of attraction. Relativistic representations of Newton’s law are obtained with respect to the center of gravity exposing illusory effects that appear at high velocities. It is verified that in atomic physics the effect of Newtonian gravitation on the motion of elementary particles at high velocities is negligible also in relativistic consideration. Computational methods are developed to measure the intensity of gravitation at a distant space-time location using a body that travels in space, emitting uniform pulses of light that are received by the observer at a different space-time location. It is demonstrated that the tensor approach to the general relativity and the united theory of space, time and gravitation in which the geometrical properties (metric) of the four-dimensional space-time continuum depend on the distribution of gravitating masses in space and their motion represent a transformed Lorentz invariant with a new type of inertia in the field of forces changing in space and time. Real physical processes evolve according to the forces represented in the tensor form by this invariant which is equivalent to the coordinate-free local invariant of relativistic dynamics that defines the field and the motion of a body whose velocities and accelerations can be measured by relativistic identification methods at a point, time and direction of interest. The results open new avenues for research in the general relativity and can be used for software development, field measurements and experimental studies in application to distant or fast moving systems.     

Variational Framework for FIC Formulations in Continuum Mechanics: High Order Tensor-Derivative Transformations and Invariants

This is part of an article series on a variational framework for continuum mechanics based on the Finite Increment Calculus (FIC). The formulation utilizes high order derivatives of the classical fields of continuum mechanics integrated over control regions to construct stabilizing modification terms. Fields may include displacements, body forces, strains, stresses, pressure and volumetric strains. To support observer-invariant FIC formulations, we have catalogued field transformation equations as well as sets of linear and quadratic invariants of fields and of their derivatives up to appropriate order. Attention is focused on the two-dimensional case of a body in plane strain under the drilling-rotation transformation group. Results are presented for displacement and body-force derivatives of orders up to 4, and for stress, strain, pressure and volumetric strain derivatives of order up to 3. The material assembled here is self-contained because this catalog is believed to be useful beyond FIC applications; for example gradient-based, nonlocal constitutive models of multiscale mechanics and physics that involve finite characteristic dimensions analogous to FIC steplengths.     

MPPhys—A many-particle simulation package for computational physics education

In a first course to classical mechanics elementary physical processes like elastic two-body collisions, the mass–spring model, or the gravitational two-body problem are discussed in detail. The continuation to many-body systems, however, is deferred to graduate courses although the underlying equations of motion are essentially the same and although there is a strong motivation for high-school students in particular because of the use of particle systems in computer games. The missing link between the simple and the more complex problem is a basic introduction to solve the equations of motion numerically which could be illustrated, however, by means of the Euler method. The many-particle physics simulation package MPPhys offers a platform to experiment with simple particle simulations. The aim is to give a principle idea how to implement many-particle simulations and how simulation and visualization can be combined for interactive visual explorations.     

Two-dimensional team lifting prediction with floating-base box dynamics and grasping force coupling

The computational procedures of higher-order implicit integrators for mechanical systems with friction are provided in this paper. The dynamic equations are established using the augmented Lagrangian formulation, and set-valued friction forces are described by projection functions. To reduce the accuracy loss caused by event transitions and to eliminate spurious oscillations in the acceleration, a new robust event-driven scheme, which accurately detects event transitions and corrects the friction forces and accelerations at switching points, is proposed. The numerical performance of the proposed scheme is demonstrated by solving several benchmark problems. Numerical results show that the newly developed scheme can approximately achieve second-order accuracy, and they are more accurate than the classical Moreau time-stepping scheme under close computational efforts. Finally, a slider-crank system is simulated to prove the validity of the developed method for nonlinear mechanical systems with friction.

     

Quantum-classical molecular dynamics and its computer implementation

Quantum-classical and quantum-stochastic molecular dynamics (QCMD/QSMD) models are formulated and applied for quantum proton transfer processes. The protein dynamics are described by the time-dependent Schroedinger equation and the motion of classical atoms by the Newtonian or Langevin equations of motion. Instantaneous positions of the classical atoms determine the potential energy surface for the proton dynamics. In turn, the proton wavefunction influences the classical atoms through nonstationary Hellmann-Feynman forces (Bala et al., 1994c). The QCMD/QSMD algorithm is described and numerical results for a proton-bound ammonia-ammonia dimer and an enzyme, phospholipase A₂, are presented. In the case of the enzyme molecule a valence-bond orbital method is used to compute the potential energy function for the proton transfer. The methods are found to be promising tools in studies of molecular and enzymatic reactions in which quantum-dynamical effects cannot be neglected.     

The Newton and Coulomb laws as information transfer by virtual particles

In elementary particle physics the philosophy of virtual particles is widely used. We use this philosophy to obtain the famous inverse square law of classical physics. We define a formal model without fields or forces but with a virtual (auxiliary) particle, information carrier. This formal model admits a very simple (school level) interpretation with two classical particles and one virtual. Then we prove (in a mathematically rigorous way) that the trajectories in our model converge to standard Newtonian trajectories of classical physics.     

SYSTEMS MODEL OF COASTAL CIRCULATION

The numerical model presented simulates two-dimensional hydrodynamic conditions in order to predict both components of the average velocity, vertically integrated, as well as the free surface elevation of the sea in coastal waters. The equations supporting the model are the well-known shallow-water equations, or quasi-static equations. In the classical theory for long waves in shallow water, the vertical accelerations of the fluid particles are neglected because these accelerations are very small with respect to gravity. In the same manner, vertical velocities may be neglected when compared with the horizontal ones. Integration of the system of differential equations in partial derivatives has been performed by using a “leapfrog” finite-difference technique under explicit solving. The computer program was developed in FORTRAN 77. Results and their agreement with coastal configuration are shown for two applications of this model: the Delta del Ebro, on the Mediterranean coast of Spain, and the Ria de Pontevedra, on the Atlantic northwestern coast of Spain. Here attention is focused on the application to the Delta del Ebro area.     

Generalized Variational Problems and Euler–Lagrange equations

This paper introduces three new operators and presents some of their properties. It defines a new class of variational problems (called Generalized Variational Problems, or GVPs) in terms of these operators and derives Euler–Lagrange equations for this class of problems. It is demonstrated that the left and the right fractional Riemann–Liouville integrals, and the left and the right fractional Riemann–Liouville, Caputo, Riesz–Riemann–Liouville and Riesz–Caputo derivatives are special cases of these operators, and they are obtained by considering a special kernel. Further, the Euler–Lagrange equations developed for functional defined in terms of the left and the right fractional Riemann–Liouville, Caputo, Riesz–Riemann–Liouville and Riesz–Caputo derivatives are special cases of the Euler–Lagrange equations developed here. Examples are considered to demonstrate the applications of the new operators and the new Euler–Lagrange equations. The concepts of adjoint differential operators and adjoint differential equations defined in terms of the new operators are introduced. A new class of generalized Lagrangian, Hamiltonian, and action principles are presented. In special cases, these formulations lead to fractional adjoint differential operators and adjoint differential equations, and fractional Lagrangian, Hamiltonian, and action principle. Thus, the new operators introduce a generalized approach to many problems in classical mechanics in general and variational calculus in particular. Possible extensions of the subject and the concepts discussed here are also outlined.     

On the Theories and Numerics of Continuum Models for Adaptation Processes in Biological Tissues

Computational continuum mechanics have been used for a long time to deal with the mechanics of materials. During the last decades researches have been using many of the theoretical models and numerical approaches of classical materials to deal with biological tissue which, in many senses, are a much more sophisticated material. We aim to review the last achievements of continuum models and numerical approaches on adaptation processes in biological tissues. In this review, we are looking, in particular, at growth in terms of changes of density and/or volume as, e.g., in collagen remodeling, wound healing, arterial thickening, etc. Furthermore, we point out some of the most relevant limitations of the current state-of-the-art in terms of these well established computational continuum models. In connection with these limitations, we will finish by discussing the trend lines of future work in the field of modeling biological adaptation, focusing on the computational approaches and mechanics that could overcome the current drawbacks. We would also like to attract the attention of all those researchers in classical materials (metal, alloys, composites, etc), to point out how similar the continuum and computational models between our fields are. We hope we can motivate them for getting their expertize in this challenging field of research.     

Suppression of nonlinear regenerative chatter in milling process via robust optimal control

During the milling process, self-excited vibration or chatter adversely affects tool life, surface quality and productivity rate. In this paper, nonlinear cutting forces of milling process are considered as a function of chip thickness with a complete third order polynomial (instead of the common linear dependency). An optimal control strategy is developed for chatter suppression of the system described through nonlinear delay differential equations. Counterbalance forces exerted by actuators in x and y directions are the control inputs. For optimal control problem, an appropriate performance index is defined such that the regenerative chatter is suppressed while control efforts are minimized. Optimal control law is determined based on variation of extremals algorithm. Results show that under unstable machining conditions, regenerative chatter is suppressed effectively after applying the optimal control strategy. In addition, optimal controller guarantees robust performance of the process in the presence of model parametric uncertainties.     

GeM software package for computation of symmetries and conservation laws of differential equations

We present a recently developed Maple-based “GeM” software package for automated symmetry and conservation law analysis of systems of partial and ordinary differential equations (DE). The package contains a collection of powerful easy-to-use routines for mathematicians and applied researchers. A standard program that employs “GeM” routines for symmetry, adjoint symmetry or conservation law analysis of any given DE system occupies several lines of Maple code, and produces output in the canonical form. Classification of symmetries and conservation laws with respect to constitutive functions and parameters present in the given DE system is implemented. The “GeM” package is being successfully used in ongoing research. Run examples include classical and new results.

Program summary

Title of program: GeMCatalogue identifier: ADYK_v1_0Program summary URL:http://cpc.cs.qub.ac.uk/summaries/ADYK_v1_0Program obtainable from: CPC Program Library, Queen's University of Belfast, N. IrelandLicensing provisions: noneComputers: PC-compatible running Maple on MS Windows or Linux; SUN systems running Maple for Unix on OS SolarisOperating systems under which the program has been tested: Windows 2000, Windows XP, Linux, SolarisProgramming language used: Maple 9.5Memory required to execute with typical data: below 100 MegabytesNo. of lines in distributed program, including test data, etc.: 4939No. of bytes in distributed program, including test data, etc.: 166 906Distribution format: tar.gzNature of physical problem: Any physical model containing linear or nonlinear partial or ordinary differential equations.Method of solution: Symbolic computation of Lie, higher and approximate symmetries by Lie's algorithm. Symbolic computation of conservation laws and adjoint symmetries by using multipliers and Euler operator properties. High performance is achieved by using an efficient representation of the system under consideration and resulting symmetry/conservation law determining equations: all dependent variables and derivatives are represented as symbols rather than functions or expressions.Restrictions on the complexity of the problem: The GeM module routines are normally able to handle ODE/PDE systems of high orders (up to order seven and possibly higher), depending on the nature of the problem. Classification of symmetries/conservation laws with respect to one or more arbitrary constitutive functions of one or two arguments is normally accomplished successfully.Typical running time: 1-20 seconds for problems that do not involve classification; 5-1000 seconds for problems that involve classification, depending on complexity.     

Nonminimal Kane's Equations of Motion for Multibody Dynamical Systems Subject to Nonlinear Nonholonomic Constraints

Nonholonomic constraint equations that are nonlinear in velocities are incorporated with Kane's dynamical equations by utilizing the acceleration form of constraints, resulting in Kane's nonminimal equations of motion, i.e. the equations that involve the full set of generalized accelerations. Together with the kinematical differential equations, these equations form a state-space model that is full-order, separated in the derivatives of the states, and involves no Lagrange multipliers. The method is illustrated by using it to obtain nonminimal equations of motion for the classical Appell–Hamel problem when the constraints are modeled as nonlinear in the velocities. It is shown that this fictitious nonlinearity has a predominant effect on the numerical stability of the dynamical equations, and hence it is possible to use it for improving the accuracy of simulations. Another issue is the dynamics of constraint violations caused by integration errors due to enforcing a differentiated form of the constraint equations. To solve this problem, the acceleration form of the constraint equations is augmented with constraint stabilization terms before using it with the dynamical equations. The procedure is illustrated by stabilizing the constraint equations for a holonomically constrained particle in the gravitational field.     

Quasilinearization approach to quantum mechanics

The quasilinearization method (QLM) of solving nonlinear differential equations is applied to the quantum mechanics by casting the Schrödinger equation in the nonlinear Riccati form. The method, whose mathematical basis in physics was discussed recently by one of the present authors (VBM), approaches the solution of a nonlinear differential equation by approximating the nonlinear terms by a sequence of the linear ones, and is not based on the existence of some kind of a small parameter. It is shown that the quasilinearization method gives excellent results when applied to computation of ground and excited bound state energies and wave functions for a variety of the potentials in quantum mechanics most of which are not treatable with the help of the perturbation theory or the 1/N expansion scheme. The convergence of the QLM expansion of both energies and wave functions for all states is very fast and already the first few iterations yield extremely precise results. The precision of the wave function is typically only one digit inferior to that of the energy. In addition it is verified that the QLM approximations, unlike the asymptotic series in the perturbation theory and the 1/N expansions are not divergent at higher orders.     

Saturation in multivariate simultaneous approximation

In this paper we present a new result on the saturation of sequences of linear operators in a multivariate and simultaneous setting. Specifically, a small o saturation result is obtained for the partial derivatives of the classical Bernstein bivariate operators on the unit simplex. Solutions of boundary value problems for certain partial differential equations of elliptic type play an important role.     

Numerical simulation of the motion of granular material using object-oriented techniques

The objective of this contribution is to present a numerical simulation method to model the motion of a packed bed on a moving grate or in a rotary kiln using object-oriented techniques. The packed bed can be described as granular material consisting of a large number of particles. The method chosen is the Lagrangian time-driven method and it uses the position, the orientation, the velocity and the angular velocity of particles as independent variables. These are obtained by time integration of the three-dimensional dynamics equations which were derived from the classical Newtonian mechanics approach based on the second law of Newton for the translation and rotation of each particle in the granular material. This includes keeping track of all forces and moments acting on each particle at every time-step. Particles are treated as contacting visco-elastic bodies which can overlap each other. Contact forces depend on the overlap geometry, material properties and dynamics of particles and include normal and tangential components of repulsion force with visco-elastic models for energy dissipation through internal and surface friction. The resulting equations of particle motion are solved by the Gear predictor–corrector scheme of fifth-order accuracy.The simulation method is based on object-oriented methodologies and programmed in the programming language C++. This approach supports objects which can be used for three-dimensional particles of various shapes and sizes and for walls as boundaries. The programming modules are implemented in the TOSCA (tools of object-oriented software for continuum mechanic applications) software package which allows for a high degree of flexibility and for shortening the duration of the software development process. As methods for particle motion may deal with particles of different sizes and materials, the approach allows to describe transport processes in technical applications.     

Efficient and accurate modeling of rigid rods

Ten years ago, an original semi-recursive formulation for the dynamic simulation of large-scale multibody systems was presented by García de Jalón et al. (Advances in Computational Multibody Systems, pp. 1–23, 2005). By taking advantage of the cut-joint and rod-removal techniques through a double-step velocity transformation, this formulation proved to be remarkably efficient. The rod-removal technique was employed, primarily, to reduce the number of differential and constraint equations. As a result, inertia and external forces were applied to neighboring bodies. Those inertia forces depended on unknown accelerations, a fact that contributed to the complexity of the system inertia matrix. In search of performance improvement, this paper presents an approximation of rod-related inertia forces by using accelerations from previous time-steps. Additionally, a mass matrix partition is carried out to preserve the accuracy of the original formulation. Three extrapolation methods, namely, point, linear Lagrange and quadratic Lagrange extrapolation methods, are introduced to evaluate the unknown rod-related inertia forces. In order to assess the computational efficiency and solution accuracy of the presented approach, a general-purpose MATLAB/C/C++ simulation code is implemented. A 15-DOF, 12-rod sedan vehicle model with MacPherson strut and multi-link suspension systems is modeled, simulated and analyzed.     

Organic Design of Massively Distributed Systems: A Complex Networks Perspective

The vision of Organic Computing addresses challenges that arise in the design of future information systems that are comprised of numerous, heterogeneous, resource-constrained and error-prone components. The notion organic highlights the idea that, in order to be manageable, such systems should exhibit self-organization, self-adaptation and self-healing characteristics similar to those of biological systems. In recent years, the principles underlying these characteristics are increasingly being investigated from the perspective of complex systems science, particularly using the conceptual framework of statistical physics and statistical mechanics. In this article, we review some of the interesting relations between statistical physics and networked systems and discuss applications in the engineering of organic overlay networks with predictable macroscopic properties.     

Thermodynamics and its prediction and CALPHAD modeling: Review,state of the art,and perspectives

Thermodynamics is a science concerning the state of a system, whether it is stable, metastable, or unstable, when interacting with its surroundings. The combined law of thermodynamics derived by Gibbs about 150 years ago laid the foundation of thermodynamics. In Gibbs combined law, the entropy production due to internal processes was not included, and the 2^nd law was thus practically removed from the Gibbs combined law, so it is only applicable to systems under equilibrium, thus commonly termed as equilibrium or Gibbs thermodynamics. Gibbs further derived the classical statistical thermodynamics in terms of the probability of configurations in a system in the later 1800's and early 1900's. With the quantum mechanics (QM) developed in 1920's, the QM-based statistical thermodynamics was established and connected to classical statistical thermodynamics at the classical limit as shown by Landau in the 1940's. In 1960's the development of density functional theory (DFT) by Kohn and co-workers enabled the QM prediction of properties of the ground state of a system. On the other hand, the entropy production due to internal processes in non-equilibrium systems was studied separately by Onsager in 1930's and Prigogine and co-workers in the 1950's. In 1960's to 1970's the digitization of thermodynamics was developed by Kaufman in the framework of the CALculation of PHAse Diagrams (CALPHAD) modeling of individual phases with internal degrees of freedom. CALPHAD modeling of thermodynamics and atomic transport properties has enabled computational design of complex materials in the last 50 years. Our recently termed zentropy theory integrates DFT and statistical mechanics through the replacement of the internal energy of each individual configuration by its DFT-predicted free energy. The zentropy theory is capable of accurately predicting the free energy of individual phases, transition temperatures and properties of magnetic and ferroelectric materials with free energies of individual configurations solely from DFT-based calculations and without fitting parameters, and is being tested for other phenomena including superconductivity, quantum criticality, and black holes. Those predictions include the singularity at critical points with divergence of physical properties, negative thermal expansion, and the strongly correlated physics. Those individual configurations may thus be considered as the genomic building blocks of individual phases in the spirit of the materials genome®. This has the potential to shift the paradigm of CALPHAD modeling from being heavily dependent on experimental inputs to becoming fully predictive with inputs solely from DFT-based calculations and machine learning models built on those calculations and existing experimental data through newly developed and future open-source tools. Furthermore, through the combined law of thermodynamics including the internal entropy production, it is shown that the kinetic coefficient matrix of independent internal processes is diagonal with respect to the conjugate potentials in the combined law, and the cross phenomena that the phenomenological Onsager flux and reciprocal relationships are due to the dependence of the conjugate potential of a molar quantity on nonconjugate molar quantities and other potentials, which can be predicted by the zentropy theory and CALPHAD modeling.     

Optimization of parameters of nonbonded interactions in a spectroscopically determined force field

A procedure is given by which parameters of nonbonded interactions in a molecular mechanics energy function can be optimized for maximum compatibility with ab initio force fields and structures. The method is based on a previously derived transformation of ab initio valence parameters to the molecular mechanics formalism. Explicit analytical expressions for the derivatives of the molecular mechanics force constants and reference geometry parameters with respect to the parameters of the nonbonded interactions are derived. The form of the goodness-of-fit function is discussed. A first application to a set of alanine dipeptides is described.