Left time derivatives in mathematics, mechanics and control of motion

By the traditional representation accepted in mathematics, mechanics and theoretical physics, the time-derivatives are defined as the right derivatives and are used in this way in differential equations describing processes in nature and technology. The fact that even for infinitely smooth x(t), the right time-derivatives do not physically exist, due to the positive orientation of time, somehow escaped the attention of scientists. This led to misconceptions and omissions in mechanics, physics and engineering, with unexpected consequences in some cases. All measurements and experiments contain and use only left time-derivatives, thereby with time delays. All processes require some kind of transmittal of information (forces, actions) which takes time, so the expressions that define their evolution from a current state actually contain the left and delayed time derivatives, even if they are written with the exact right time-derivatives, according to the classical tradition. In this paper, the causal representations of physical processes by differential equations with the left time-derivatives on the right-hand side are considered for some basic problems in classical mechanics, physics and technology. The use of the left time-derivatives explicitly takes into account the causality of processes depending on the transmission of information and defines the motions subject to external forces that may depend on accelerations and higher order derivatives of velocities. Such forces are exhibited in Weber’s electro-dynamic law of attraction; they are produced by the Kirchhoff-Thomson adjoint fluid acceleration resistance acting on a body moving in a fluid, and they are also involved in the manual control of aircraft or spacecraft that depends on accelerations of the craft itself. The consistency condition is presented, and the existence of solutions for equations of motion driven by forces with higher order derivatives of velocity is proved. The inclusion of such forces in the autopilot design is proposed to assure the safety of the aircraft in case of a failure of its outboard velocity sensors. It is demonstrated that the classical form of the 2nd law of Newton is preserved with respect to the effective forces for which the parallelogram law of addition is valid. Then the Lagrange and Hamilton equations are extended to include the generalized forces with the left higher order derivatives, and a method for the solution of such equations with the left and delayed higher order derivatives is presented with the example of a physical pendulum. The results open new avenues in science and technology providing the basis for correct design in the projects sensitive to information transmittal.     

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.     

On canonical transformations between equivalent hamiltonian formulations of general relativity

Two Hamiltonian formulations of general relativity, due to Pirani, Schild and Skinner (Phys. Rev. 87, 452, 1952) and Dirac (Proc. Roy. Soc. A 246, 333, 1958), are considered. Both formulations, despite having different expressions for the constraints, allow one to derive four-dimensional diffeomorphism invariance. The relation between these two formulations at all stages of the Dirac approach to constrained Hamiltonian systems is analyzed. It is shown that the complete sets of their phase-space variables are related by a transformation which satisfies the ordinary condition of canonicity known for unconstrained Hamiltonians and, in addition, converts one total Hamiltonian into another, thus preserving form-invariance of generalized Hamiltonian equations for the constrained systems.     

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.     

Information transmission between two communicators with relative motion

In information transmission between two communicators with relative motion, the information transmission rate and efficiency are influenced by the Doppler effect and the relativistic effect. The received signal power P′ = αP is established for a sound or radio channel; then, taking the Doppler effect into account, the capacity of the AWGN Shannon channel and the capacity region for the AWGN multiple-access channel with and without feedback are determined. Influences of the Doppler effect and the relativistic effect on the asymmetry in information transmission efficiency are also discussed; it is found that the asymmetry favors the information transmission from a stationary transmitter to a moving receiver for the AWGN channel with the relativistic Doppler effect, while it favors that from the moving to the stationary for the AWGN channel with only the classical Doppler effect.     

Motion of spin-half particles in the axially symmetric field of naked singularities of the static <Emphasis Type="Italic">q</Emphasis>-metric

Quantum-mechanical motion of a spin-half particle is examined in the axially symmetric fields of static naked singularities formed by a mass distribution with a quadrupole moment (q-metric). The analysis is performed by means of the method of effective potentials of the Dirac equation generalized to the case where radial and angular variables are not separated. If ?1 < q < qlim, |qlim| ? 1, where q is the quadrupolemoment in proper units, the naked singularities do not exclude the existence of stationary bound states of Dirac particles for a prolate mass distribution in the q-metric along the axial axis. For an oblate mass distribution, the naked singularities of the q-metric are separated from a Dirac particle by infinitely large repulsive barriers followed by a potential well which deepens while moving apart from the equator (from θ = θ _min or θ = π ? θ _min) toward the poles. The poles make an exception, and at 0 < q < q*, there are some points θ _i for particle states with j ≥ 3/2.     

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.     

Unfinished history and paradoxes of quantum potential. I. Non-relativistic origin, history and paradoxes

This is the first of two related papers analyzing and explaining the origin, manifestations and parodoxical features of the quantum potential (QP) from the non-relativistic and relativistic points of view. The QP arises in the quantum Hamiltonian under various procedures of quantization of natural systems, i.e., those whose Hamilton functions are positive-definite quadratic forms in momenta with coefficients depending on the coordinates in (n-dimensional) configurational space V _n thus endowed with a Riemannian structure. The result of quantization may be considered as quantum mechanics (QM) of a particle in V _n in the normal Gaussian coordinate system in the globally static space-time V _1,n . Contradiction of the QP to the General Covariance and Equivalence principles is discussed. It is found that actually the historically first Hilbert space-based quantization by E. Schrödinger (1926), after revision in the modern framework of QM, also leads to a QP in the form that B. DeWitt found 26 years later. Efforts to avoid the QP or to reduce its drawbacks are discussed. The general conclusion is that some form of QP and a violation of the principles of general relativity which it induces are inevitable in the non-relativistic quantum Hamiltonian. It is also shown that Feynman (path-integral) quantization of natural systems singles out two versions of the QP, which both determine two bi-scalar (independent of a choice of coordinates) propagators fixing two different algorithms of path integral calculation. The accompanying paper under the same general title and the subtitle “The Relativistic Point of View” (published in the same issue of the journal and referred to as Paper II) considers a relation of the nonrelativistic QP to the quantum theory of a scalar field non-minimally coupled to the curved space-time metric.     

Motion planning with inertial constraints

A bodyB must move from a placementZ ₀ to a placementZ ₁, while avoiding collision with a setS of moving obstacles. The motion must satisfy an inertial constraint: the acceleration cannot exceed a given boundM. The problem is analyzed, and polynomial-time motion-planning algorithms are given for the case of a particle moving in one dimension.     

Relative Finite Element Coordinates in Multibody System Simulation

In the paper a numerical approach for deriving the nonlinear explicitform dynamic equations of rigid and flexible multibody systems ispresented. The dynamic equations are obtained as Ordinary DifferentialEquations for generalized coordinates and without algebraic constraints.The Finite Element Theory is applied for discretization of flexiblebodies. The minimal set of the generalized coordinates includesindependent joint motions, as well as independent small flexibledeflections of finite element nodes. The node deflections and stiffnessmatrices are calculated with respect to the moving relative coordinatesystems of the flexible bodies. The positions and orientations ofelement and substructure coordinate systems are updated according to thenode deflections. A major step of the numerical process is the kinematicanalysis and calculation of matrices of partial derivatives of thequasi-coordinates (dependent joint motions and coordinates of points andnodes) with respect to the generalized coordinates. The inertia terms inthe dynamic equations are obtained multiplying the matrices of thepartial derivatives by the mass matrices of the rigid and flexiblebodies. Stiffness properties of flexible bodies are presented in thedynamic equations by stiff forces that depend on the generalizedrelative flexible deflections only. Several examples of large motion ofbeam structures show the effectiveness of the algorithm.     

Formation of inverse Chladni patterns in liquids at microscale: roles of acoustic radiation and streaming-induced drag forces

While Chladni patterns in air over vibrating plates at macroscale have been well studied, inverse Chladni patterns in water at microscale have recently been reported. The underlying physics for the focusing of microparticles on the vibrating interface, however, is still unclear. In this paper, we present a quantitative three-dimensional study on the acoustophoretic motion of microparticles on a clamped vibrating circular plate in contact with water with emphasis on the roles of acoustic radiation and streaming-induced drag forces. The numerical simulations show good comparisons with experimental observations and basic theory. While we provide clear demonstrations of three-dimensional particle size-dependent microparticle trajectories in vibrating plate systems, we show that acoustic radiation forces are crucial for the formation of inverse Chladni patterns in liquids on both out-of-plane and in-plane microparticle movements. For out-of-plane microparticle acoustophoresis, out-of-plane acoustic radiation forces are the main driving force in the near-field, which prevent out-of-plane acoustic streaming vortices from dragging particles away from the vibrating interface. For in-plane acoustophoresis on the vibrating interface, acoustic streaming is not the only mechanism that carries microparticles to the vibrating antinodes forming inverse Chladni patterns: In-plane acoustic radiation forces could have a greater contribution. To facilitate the design of lab-on-a-chip devices for a wide range of applications, the effects of many key parameters, including the plate radius R and thickness h and the fluid viscosity μ, on the microparticle acoustophoresis are discussed, which show that the threshold in-plane and out-of-plane particle sizes balanced from the acoustic radiation and streaming-induced drag forces scale linearly with R and \(\sqrt \mu\), but inversely with \(\sqrt h\).     

Feedback control of nonlinear stochastic systems for targeting a specified stationary probability density

In the present paper, an innovative procedure for designing the feedback control of multi-degree-of-freedom (MDOF) nonlinear stochastic systems to target a specified stationary probability density function (SPDF) is proposed based on the technique for obtaining the exact stationary solutions of the dissipated Hamiltonian systems. First, the control problem is formulated as a controlled, dissipated Hamiltonian system together with a target SPDF. Then the controlled forces are split into a conservative part and a dissipative part. The conservative control forces are designed to make the controlled system and the target SPDF have the same Hamiltonian structure (mainly the integrability and resonance). The dissipative control forces are determined so that the target SPDF is the exact stationary solution of the controlled system. Five cases, i.e., non-integrable Hamiltonian systems, integrable and non-resonant Hamiltonian systems, integrable and resonant Hamiltonian systems, partially integrable and non-resonant Hamiltonian systems, and partially integrable and resonant Hamiltonian systems, are treated respectively. A method for proving that the transient solution of the controlled system approaches the target SPDF as t→∞ is introduced. Finally, an example is given to illustrate the efficacy of the proposed design procedure.     

Symbolic computation of the phoretic acceleration of convex particles suspended in a non-uniform gas

A package has been developed for calculating analytic expressions for forces and torques onto an arbitrarily shaped convex tracer (aerosol) particle small compared to the mean free path of the surrounding nonequilibrium gas. The package Phoretic allows to compute analytical (and also numerical) expressions for forces and torques stemming from elastic and diffusive scattering processes parameterized by an accommodation coefficient. The method is based on calculating half-sphere integral tensors of arbitrary rank and on integrating forces and torques acting on surface elements. The surrounding gas is completely specified by an arbitrarily shaped velocity distribution function. Accordingly, Phoretic requires two inputs: A particle (surface) geometry and a velocity distribution function. For example, the particle may be a cylinder with flat end caps, and the distribution function the one of Maxwell (isotropic) or Grad (13th moment approximation). The package reproduces analytic results for spheres which were available in the literature, and the ones for other geometries (cylinders, cuboids, ellipsoids) which were, however, only partially available (some works considered only elastic collisions, others temperature, or pressure, or only velocity gradients, etc.). In addition, Phoretic takes into account angular velocities which have been usually neglected and become relevant for non-spherical particles. The package is geared towards the implementation of dynamical equations for aerosol particles suspended in dilute or semidilute gases and as such helps to obtain concentration profiles and mobilities of aerosol particles depending on their shape (distribution) and environmental conditions.

Program summary

Title of program:PhoreticCatalogue identifier:ADYI_v1_0Program summary URL:http://cpc.cs.qub.ac.uk/summaries/ADYI_v1_0Program obtainable from: CPC Program Library, Queen's University of Belfast, N. IrelandLicensing provisions: Persons requesting the program must sign the standard CPC-non-profit use license (see license agreement printed in every issue)Computer for which the program is designed and others on which it has been tested: All platforms with a monitorOperating systems or monitors under which the program has been tested: Linux, Windows XP, Unix, Mac-OSProgram language used: Mathematica^®, version 5.2 or later. Phoretic makes use of the DiscreteMath‘Combinatorica’ Mathematica^® packageMemory required to execute with typical data: 10 MByteNo. of lines in distributed program, including test data, etc.: 22 410No. of bytes in distributed program, including test data, etc.: 114 657Distribution format:tar.gzNature of physical problem: Starting from a non-uniform velocity distribution function of a gas in terms of its moments, i.e. field variables, and field gradients such as temperature, pressure, or velocity field, the problem is to analytically calculate forces and torques acting onto arbitrarily shaped convex tracer (aerosol) particles small in size compared to the mean free path of the gas. The collision process is modeled as a superposition of elastic and diffusive scattering processes (parameterized by 0?α?1).Method of solution: We implemented the solution to this problem in the symbolic programming language Mathematica^®. The program allows to specify an arbitrary shape of the tracer particle and an arbitrary distribution function of the gas and returns symbolic or numerical expressions for forces and torques. The solution requires the calculation of half-sphere and base surface integrals and subsequent symbolic algebraic and tensorial manipulations.Restrictions on the complexity of the problem: Not known. In case the software cannot calculate surface integrals analytically it offers the possibility to proceed with a numerical evaluation of the corresponding terms.Typical running time: Typical running times mostly depend on the shape of the tracer particle. For all examples coming together with the software distribution run times are below 5 minutes on a modern single-processor platform.     

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.     

Switched Modeling and Task–Priority Motion Planning of Wheeled Mobile Robots Subject to Slipping

This study is devoted to the modelling and control of Wheeled Mobile Robots moving with longitudinal and lateral slips of all wheels. Due to wheel slippage we have to deal with systems with changing dynamics. Wheeled Mobile Robots can be thus modeled as switched systems with both autonomous switches (due to wheel slippage) and smooth controls (due to control algorithm). It is assumed that the slipping is counteracted by the slip reaction forces acting at contact points of the wheels with the ground. A model of these reaction forces, borrowed from the theory of automotive systems, has been adopted and included into the Lagrangian dynamic equations of the robot. A framework for designing motion planning schemes devoid of chattering effects for systems with changing dynamics is presented. A task–priority motion planning problem for wheeled mobile robots subject to slipping is addressed and solved by means of Jacobian motion planning algorithm based on the Endogenous Configuration Space Approach. Performance of the algorithm is presented in simulations of the Pioneer 2DX mobile platform. The robot dynamics equations are derived and 4 variants of motion are distinguished. The motion planning problem is composed of two sub-tasks: robot has to reach a desired point in the task space (proper motion planning) and the motion should minimize either the control energy expendinture or the wheel slippage. Performance of the motion planning algorithm is illustrated by a sort of the parking maneuver problem.     

悬臂梁大变形的向量式有限元分析

为分析悬臂梁的几何非线性行为,用向量式有限元法将结构离散成质点系以及质点间的连接单元.根据牛顿第二定律得到每个质点在内力和外载荷作用下的运动方程以及悬臂梁在每个时刻的变形用该时刻质点系的运动表示.结合刚架元的节点内力和等效质量得出质点位移的迭代计算公式,采用FORTRAN编制计算程序,对悬臂梁分别承受集中载荷和弯矩下的大变形进行算例分析.计算结果与理论解吻合较好,表明该方法能很好地模拟分析悬臂梁的大变形.