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In this chapter we introduce a study of inflationary universe models that are characterized by a single scalar inflation field . The study of these models is based on two dynamical equations: one corresponding to the Klein–Gordon equation for the inflaton field and the other to a generalized Friedmann equation. After describing the kinematics and dynamics of the models under the Hamilton–Jacobi scheme, we determine in some detail scalar density perturbations and relic gravitational waves. We also introduce the study of inflation under the hierarchy of the slow-roll parameters together with the flow equations. We apply this approach to the modified Friedmann equation that we call the Friedmann–Chern–Simons equation, characterized by F ( H ) = H 2 - α H 4 Open image in new window, and the brane-world inflationary models expressed by the modified Friedmann equation.

We review the experimental and theoretical literature on the steady terminal orientation of a body as it settles in a viscous fluid. The terminal orientation of a rigid body is a classic example of a system out of equilibrium. While the dynamical equations are effective in deriving the equilibrium states, they are far too complex and intractable as of yet to resolve questions about the nature of stability of the solutions. The maximum entropy production principle is therefore invoked, as a selection principle, to understand the stable, steady state patterns. Some on-going work and inherent complexities of fluid solid systems are also discussed.

To regulate and optimize the modeling process of lower extremity exoskeleton for rehabilitation, a SolidWorks/SimMechanics-based exoskeleton modeling procedure which includes three stages of modeling, simulation and evaluation, is proposed. Firstly, by importing assembly files of exoskeleton model from SolidWorks to MATLAB Simulink/ SimMechanics, a virtual prototype is generated, which is expediently used for the simulation of exoskeleton mechanism and the calculation of dynamics and kinematics parameters. Secondly the simulation stage is visualized by Simulink/ SimMechanics 3D interpreter and MATLAB’s 2D graphics, which alleviates the derivation of complex dynamical equations. Thirdly the human gait data from C3D database is adopted as the simulation constraint and used to calculate zero moment point (ZMP) to evaluate the stability of virtual prototype. This modeling procedure makes modification of the mechanical structure more efficient and can also be applied to other mechanism simulation.

This chapter provides a fundamental theory of spacecraft dynamics. After a brief survey of gravitational field, the two-body problem is summarized as a simplified model of orbit motion of a spacecraft around the Earth. The main environmental torques acting on spacecraft, the gravitational torque and magnetic torque are introduced. The dynamical equations of attitude motion of a spacecraft are established, where the Euler’s equations and Poisson’s equations are applied for a rigid spacecraft in gravitational field. The stability problems of the relative equilibrium of a rigid spacecraft in circular orbit under gravitational torque are analyzed by using the first approximation method and the Lyapunov’s direct method. The attitude motions of a gyrostat are analyzed as a model of spacecraft with axisymmetric rotors. The permanent rotations and its stability of a spinning spacecraft are discussed under torque-free assumption.

Ropes are widely used in various kinds of fishing gear. In this paper, a numerical method is presented which is competent in predicting the three-dimensional dynamic behavior of cables. The dynamic behavior of cables were simulated using dynamical equations and R language. The method was based on the lump-mass and spring model and the implicit algorithm was applied to solve the stiff differential equations. The mathematics-mechanical analysis was provided on the basic characteristics of the model. The R platform was used in programming, simulation, and visualization of the results. The three-dimensional dynamic behavior of cables in different examples were presented successfully and verify the effectiveness of this method, and the motion process was consistent with the observed data in the ocean.

This chapter treats chaos in spatial attitude motion of spacecraft. The motion of torque-free rigid bodies and gyrostats are discussed in terms of Serret-Andoyer variables, which are introduced to simplify the dynamical models. The influences of the gravitational and magnetic fields on the attitude motion are revisited using the new variables. A rigid body in elliptic orbit and gravitational field, a torque-free rigid body with eccentrically rotating mass, and a magnetic gyrostat in circular orbit acted by gravitational and magnetic torques are discussed with application of Serret-Andoyer variables. The dynamical equations for each model are transformed to an integrable Hamiltonian system with small disturbance. The generalized Melnikov theory is applied to predict transverse heteroclinic points. Poincare maps are numerically calculated to demonstrate the process from regular to chaotic motions.

Here we consider observed and controlled linear dynamical systems in continuous-time. Real systems are rarely linear; they often involve nonlinear behaviors described by nonlinear terms in the dynamical equations of the model. However, the interest of linear dynamical systems is emphasized at the end of this chapter. Indeed, we show that the study of the tangent linear system generally makes it possible to locally stabilize an equilibrium of the nonlinear original system. In contrast to Chaps. 2 and 4, where only two types of variables were considered, namely the state vector and the external input variables, we are now going to take into account output or observed variables which constitute the components of the state directly measured through sensors.

The primitive variable formulation of the moist hydrostatic equations conserves mass and moist total energy due to the property that the divergence and gradient operators are adjoints. Any compatible numerical method, which has a discrete analog of this property will conserve a discrete mass and total energy. We demonstrate this using aqua-planet simulations performed with CAM-HOMME (NCAR’s Community Atmospheric Model with the High-Order Method Modeling Environment dynamical core). CAM-HOMME uses a compatible numerical method on arbitrary unstructured quadrilateral grid. The equations described here are the full set of dynamical equations used by CAM. Aqua-planet simulations use the full suite of physics parametrizations as well. The only simplification is the use of idealized surface conditions. We report on the magnitude of the total energy budget in the dynamical core including estimates for the non-adiabatic processes. The practice of fixing dry total energy as opposed to the conserved total moist energy is shown to generate a forcing of − 0. 56 W/m2.

In this chapter, a novel mathematical analysis for predicting master-slave chaotic synchronization is presented. In most situations when examining this type of synchronization one considers the asymptotic stability of the particular system via Lyapunov’s direct method, or conditional Lyapunov exponents are considered. Initially, in this chapter, Lyapunov’s direct method is used to show the asymptotic stability within the simplest piecewise linear master-slave chaotic flow. However, primarily the master-slave synchronization properties of the simplest quadratic chaotic flow and Ueda chaotic system are examined directly by means of mathematical manipulation of their dynamical equations, where possible, as well as via numerical simulations. In order to achieve this, numerical simulations and theoretical analysis are made use of in conjunction. In this way, it is shown that the synchronization error of the two aforementioned chaotic master-slave systems can indeed be predicted for certain driving signals, without the need for either analytical or numerical evaluation of the conditional Lyapunov exponents or employment of Lyapunov’s direct method.

A wave equation, that governs finite amplitude acoustic disturbances in a thermoviscous Newtonian fluid, and includes nonlinear terms up to second order, is proposed. The equation preserves the Hamiltonian structure of the fundamental fluid dynamical equations in the non-dissipative limit. An exact thermoviscous shock solution is derived. This solution is, in an overall sense, equivalent to the Taylor shock solution of the Burgers equation. However, in contrast to the Burgers equation, the model equation considered here is capable to describe waves propagating in opposite directions. Studies of head-on colliding thermoviscous shocks demonstrate that the propagation speed changes upon collision.

The vehicle systems are modelled mathematically as parameter dependent multi-body systems. The connections between the elements are formulated either as dynamical equations or algebraic, or transcendental or tabulated constraint relations. The connections can rarely be modelled by analytic functions, and the missing analyticity can arise from non-uniqueness or discontinuities in the functions themselves or in their derivatives of any order. In vehicle systems the contact between the vehicle and its support (road or rail) is an important source of missing analyticity. The suspension systems of the vehicles consist of passive and active elements such as springs, dampers and actuators, and their characteristics are only analytic functions within certain intervals of operation. Unilateral contacts in the suspension systems may give rise to changes of the degrees of freedom of the system during operation, and cause impacts or sliding contact during the operation.

Neural networks of the q-state Potts type are considered for a finite number of biased patterns. The existence and stability properties of the Mattis states and symmetric states are discussed at zero temperature for all q and arbitrary bias and near the “critical” temperature for q = 3 and two classes of representative bias parameters. Some of these properties are illustrated through the solution of the dynamical equations governing the time evolution of the macroscopic overlap between the learned patterns and the instantaneous microscopic state of the network.

We consider an equation of the Bethe-Salpeter type, with arbitrary potential and kernel, respectively, for space-like momentum transfer. The invariance group of the equation is then the Lorentz-group in three dimensions, the O(1, 2) group. The standard procedure for the diagonalization of such equations (valid for square integrable solutions only) is generalized to include the case of power bounded solutions, by means of a generalized O(1, 2) expansion formalism. The result is a two-dimensional integral equation for the O(1, 2) expansion coefficients. The right-most l-plane singularities of these determine the asymptotic behaviour of the amplitudes as in ordinary Regge theory. The formalism can be applied to other dynamical equations possessing O(1, 2) symmetry.

It has been argued that Bose-Einstein diffusion is a conceptual improvement in the treatment of radiative transport in supernovae. First and foremost, it represents a two-parameter description of the radiation field, the two parameters being µ and Tγ whose associated integrals, the total photon number and the total photon energy, can be budgeted independently. As the first iteration in a bootstrap process to achieve an internally consistent SN model and to develop general techniques of handling radiative transport at all optical depths, it seems clear from the above discussion that such a two-parameter fit is a step toward these goals. Secondly, the microphysics of the relevant photon/matter interactions in the SN strongly suggests that the appropriate two-parameter function is indeed Bose-Einstein, since it gives the correct physical limits of photon equilibrium whenever the appropriate conditions as dictated by the laws of statistical mechanics apply. A major difficulty is the explicit formulation of the interaction terms in the dynamical equations. These integrals over frequency must be made computationally tractable and require more knowledge of the energy spectrum of the radiation. The simplest approach would be to employ the nondegenerate BE energy distribution; however, it should be kept in mind that although the frequency integrated moments of a Maxwell-Boltzmann gas can approximate the degenerate limit with an 8 % error, the corresponding energy distribution is a poor approximation of the Rayleigh-Jeans portion of the blackbody spectrum. Nevertheless, such an approach may be preferable in a first set of calculations.

The time-dependent two-Hilbert-space formulation of the scattering theory is first reviewed and the transition to time-independent theory made (Secs. 1-6). Dynamical equations for the transition operator Tare then written down (Sec. 7). In these equations the cluster-(or channel-) projection operators are incorporated into the transition operator, a feature characteristic of our two-Hilbert-space~formulation. The next step (Sec. 8) is the introduction of a new system of N-body equations, which we call M-operator equations. These equations have a unique solution and are strongly approximation-solvable. The theory of strong approximation-solvability (Sec. 9) with projection type approximations leads to the conclusion that the Tand M-operator equations are also stable. There are other advantages of our equations, when compared (Sec. 10) with, for example, the Faddeev-Yakubovskii equations, that are especially apparent when the number of particles is large but the number of interesting channels is small. Still another advantage is provided by the fact that the solutions to the approximate equations can be related to approximate scattering systems (Sec. 11). This permits insight into the faithfulness with which the approximations represent the physics of the scattering process. The basic development is then brought to a close with a discussion (Secs. 12 and 13) of how to include effects of particle indistinguishability and of Coulomb interactions.

The dynamical equations which are basic for the description of the dynamics of quantum felds in arbitrary space--time geometries, can be derived from the requirements of a unique deterministic evolution of the quantum fields, the superposition principle, a finite propagation speed, and probability conservation. We suggest and describe observations and experiments which are able to test the unique deterministic evolution and analyze given experimental data from which restrictions of anomalous terms violating this basic principle can be concluded. One important point is, that such anomalous terms are predicted from loop gravity as well as from string theories. Most accurate data can be obtained from future astrophysical observations. Also, laboratory tests like spectroscopy give constraints on the anomalous terms.

We discuss the generalization of the Drinfeld-Sokolov formalism to superspace in the case of supersymmetric integrable systems. We bring out the connection between this and the zero curvature formulations, for these systems, in components. Various examples are worked out in detail. We show how the symmetric space techniques can be profitably used to solve the zero curvature conditions and to obtain the hierarchy of equations in the case of sAKNS hierarchy. The dynamical equations are formulated also as the Cartan-Maurer equations in superspace and we show how the zero curvature condition can be solved, alternately, by expanding potentials according to the grading charge. We also discuss the sTB-B hierarchy to bring out some open questions.

The Lateral Leg Spring model (LLS) was developed by Schmitt and Holmes to model the horizontal-plane dynamics of a running cockroach. The model captures several salient features of real insect locomotion, and demonstrates that horizontal plane locomotion can be passively stabilized by a well-tuned mechanical system, thus requiring minimal neural reflexes. We propose two enhancements to the LLS model. First, we derive the dynamical equations for a more flexible placement of the center of pressure (COP), which enables the model to capture the phase relationship between the body orientation and center-of-mass (COM) heading in a simpler manner than previously possible. Second, we propose a reduced LLS "plant model" and biologically inspired control law that enables the model to follow along a virtual wall, much like antenna-based wall following in cockroaches.

The purpose of this article is numerical verification of the thory of weak turbulence. We performed numerical simulation of an ensemble of nonlinearly interacting free gravity waves (swell) by two different methods: solution of primordial dynamical equations describing potential flow of the ideal fluid with a free surface and, solution of the kinetic Hasselmann equation, describing the wave ensemble in the framework of the theory of weak turbulence. Comparison of the results demonstrates pretty good applicability of the weak turbulent approach. In both cases we observed effects predicted by this theory: frequency downshift, angular spreading as well as formation of Zakharov-Filonenko spectrum I ω ∼ ω −4. To achieve quantitative coincidence of the results obtained by different methods one has to accomplish the Hasselmann kinetic equation by an empirical dissipation term S diss modeling the coherent effects of white-capping. Adding of the standard dissipation terms used in the industrial wave predicting model (WAM) leads to significant improvement but not resolve the discrepancy completely, leaving the question about optimal choice of S diss open.

Onsager's point vortex model of two dimensional turbulence is extended by the inclusion of time dependent vortex circulations. If the time dependence of the circulations is governed by statistically independent Onstein-Uhlenbeck processes we observe the emergence of scaling regimes for the structure functions of the Eulerian and the Lagrangian velocity increments. Fully developed turbulent flows are flux equilibrium systems leading to selfsimilarity and scaling behaviour of correlation functions. The probability distributions corresponding to the simplest case of idealized, i.e. homogeneous, isotropic, and stationary fully developed flows are unknown although the underlying fluid dynamical equations and its statistical counterparts are well-established [1]. Fluid motions can be treated either from an Eulerian or a Lagrangian point of view. Most analytical theories have been formulated in the Eulerian framework. Point vortex models (see e.g. [2]), which have been extensively investigated especially for the case of two dimensional flows, essentially make use of a Lagrangian description. Since the point vortex equations of an ideal two dimensional fluid, which is not stirred, are of Hamiltonian nature, a statistical treatment based on the microcanonical ensemble can be established. This has been discussed for the first time by Onsager [3]. The purpose of the present Letter is to show that an extension of Onsager's point vortex model, which allows for fluctuating circulations of the point vortices, leads to a state which shows scaling behaviour of the structure functions.

This paper presents a computational procedure for deriving and solving the governing dynamical equations of multi-body human systems subjected to impact. The procedure is developed on the basis of the assumption that the duration of the impact is very short, during which the configuration of the system remain the same, although the velocities of the system have undergone significant changes. These assumptions lead to a set of linear algebraic equations for the velocity increments. These equations may then be solved to obtain initial conditions for the analysis of the subsequent motion of the multi-body human system. A calculating task of ejecting seat problem is presented to illustrate and validate the procedure.

The new theory and new method of the analysis of intelligent structures are studied in the paper. With piezoelectric intelligent beam as a example: the spline dispersing dynamical equations of the piezoelectric intelligent beam is given by the spline finite-point method, from the displacement modal, basing on intelligent constitutive relation theory, instantaneous variational principle and spline dispersing. Then the control equation of the active vibration control analysis of the piezoelectric intelligent beam is get.

In this paper a novel 3-D Finite Element (FE) particle based approach is presented to investigate the ion flow across ionic channels. This consistent model foresees direct integration of the dynamical equations of ions subject to electromagnetic forces inside membrane channels, considering ion-ion interactions and taking into account explicitly the effects of molecular friction and thermal noise. The simulation results presented show that the mechanism of opening and closing of the membrane channels (Ca) as a function of the membrane voltage can be correctly reproduced by a particle model.

This paper reports on the results of combining the Maple packages Dynaflex and RifSimp. The Dynaflex package has been developed to generate the governing dynamical equations for mechanical systems; the RifSimp package has been developed for the symbolic analysis of differential equations. We show that the output equations from Dynaflex can be converted into a form which can be analyzed by RifSimp. Of particular interest is the ability of RifSimp to split a set of differential equations into different cases; each case corresponds to a different set of assumptions, and under some sets of assumptions there are significant simplifications. In order to allow RifSimp to conduct its analysis, the governing equations must be converted from trigonometric form into a polynomial form. After this is done, RifSimp can analyze the system and present its results either graphically, or in list form. The mechanical systems considered are restricted to open-loop systems, because at present, closed-loop systems require too much computation by RifSimp to permit analysis.

The RifSimp package in Maple transforms a set of differential equations to Reduced Involutive Form. This paper describes the application of RifSimp to challenging real-world problems found in engineering design and modelling. RifSimp was applied to sets of equations arising in the dynamical studies of multibody systems. The equations were generated by the Maple package Dynaflex, which takes as input a graph-like description of a multibody mechanical system and generates a set of differential equations with algebraic constraints. Application of the standard RifSimp procedure to such Differential Algebraic Equations can require large amounts of computer memory and time, and can fail to finish its computations on larger problems. We discuss the origin of these difficulties and propose an Implicit Reduced Involutive Form to assist in alleviating such problems. This form is related to RifSimp form by the symbolic inversion of a matrix. For many applications such as numerically integrating the multibody dynamical equations, the extra cost of symbolically inverting the matrix to obtain explicit RifSimp form can be impractical while Implicit Reduced Involutive Form is sufficient. An approach to alleviating expression swell involving a hybrid analytic polynomial computation is discussed. This can avoid the excessive expression swell due to the usual method of transforming the entire input analytic differential system to polynomial form, by only applying this method in intermediate computations when it is required.

This paper is about guaranteed nonlinear parameter and state estimation. Sets are computed that contain all possible values of the parameter (or state) vector given bounds on the acceptable errors. The main requirement is that the dynamical equations describing the evolution of the model can be bounded between cooperatives models, i.e., models such that the off-diagonal entries of their Jacobian matrix remain positive. The performances and limitations of the techniques proposed are illustrated on a nonlinear compartmental model.

We have seen in the earlier chapters that dynamical equations of typical nonlinear systems are not amenable to exact solutions in terms of known functions, except for isolated cases like the equations of motion of the undamped and unforced Duffing oscillator, the pendulum, the Kepler particle and so on. Generally speaking, finite dimensional nonlinear systems of the type (3.1), for n > 2, are not explicitly solvable, and they are often chaotic depending upon the values of the control parameters. Integrating the equations of motion completely, obtaining analytic solutions and finding acceptable constants of motion/integrals of motion/invariants for such nonlinear systems seem to be rare. Yet they do exist in isolated, but a large number of cases; one says that they are measure zero systems among the totality of nonlinear systems.

One strategy for searching for effects due to quantum gravity (QG) is to employ the Einstein Equivalence Principle (EEP) as an operational tool. The EEP, consisting of the Universality of Free Fall, the Universality of the Gravitational Redshift, and Local Lorentz Invariance, implies on the one hand that gravity has to be described by a space-time metric (pseudo-Riemannian Geometry) and on the other hand that the equations of motion for particles or quantum fields possess a specific structure compatible with the metrical structure of gravity. Therefore, any deviation from that structure of the dynamical equations of matter is related to a deviation from the metrical structure of gravity. As one consequence, any “new physics” will be accompanied by a breakdown of the validity of the EEP. We now take the EEP as guiding principle for the search for possible QG effects. Various proposed QG effects are then classified according to the various violations of the EEP, which is then used for a comparison with the present experimental status.

The organization of the paper is as follows. First, we obtain the dynamical equations of the pole-cart system by applying the Lagrange equations for the two variables of interest: the cart position and the deviation angle of the pendulum with respect to the vertical line. The corresponding system of two differential equations is highly nonlinear, with a rather restricted analytical treatment. After obtaining the approximate, linear equations of the pole-cart couple, we discuss its control and stabilization throughout an external force that depends on the deviation angle of the pendulum. The type of feedback control schemes that stabilize the pendulum on its unstable, vertical position are discussed in the paper. We also illustrate such stabilization with several simulated cases, for both the complete, nonlinear version and the approximate, linear version. Conventional P and PD control schemes are applied, with excellent results. Finally, we approach the evaluation of the computer-based implementation by means of computer simulations and obtain the critical design parameters for the control and the stabilization of the polecart system.

In the past ten-fifteen years, stochastic models of continuous wave function collapse were being proposed to describe the continuous emergence of classicality from quantum. We advocate that the hybrid dynamics of canonically coupled quantum and classical systems is a more basic concept. Continuous collapse formalisms are obtained as special cases. To illustrate our claim we show how von Neumann collapse follows from hybrid dynamical equations.

In this paper we give an overview of a recently developed theory [1, 2] which allows for calculating finite size corrections to the dynamical equations describing the dynamics of separable Neural Networks, away from saturation. According to this theory, finite size effects are described by a linear-noise Fokker Planck equation for the fluctuations (corresponding to an Ornstein-Uhlenbeck process), whose solution is characterized by the first two moments. The theory is applied to a particular problem in which detailed balance does not hold.

In the preceding chapter we discussed the microscopic foundations of extended irreversible thermodynamics through fluctuation theory. The present chapter deals with more general methods of non-equilibrium statistical mechanics. First, we start from the Liouville equation and introduce the projection operator technique to relate the memory function to the time correlation function of the fluctuations of the fluxes, a key result in modern non-equilibrium statistical mechanics. This result is of special interest in EIT, because it emphasizes the role played by the evolution of the fluctuations of the thermodynamic fluxes. Furthermore, the method of projection operators is useful, since it provides an explicit method for formulating the dynamical equations of the basic variables. Classically, the set of variables consists of the conserved slow variables (energy, linear momentum, mass) and some order parameters characterizing second-order phase transitions. Here we shall present a generalization of this method by adding the dissipative fast fluxes to the basic set of variables.

The strategy and algorithms for reconstructing dynamical equations directly from observed time series are discussed. A short review of the research literature is presented along with a technique for reconstructing the autonomous dynamical system in a given class of ordinary differential equations; these consist of a set of first-order differential equations with polynomial nonlinearity. The coefficients of these nonlinear terms are determined by the condition of fitting the best equations to the experimental time series. An important feature of the suggested algorithm is that it is able to restore hidden (not observed) variables from one-dimensional time series. The applications of this technique are illustrated by numerical examples for several dynamical systems, among them the Lorenz and Rossler attractors. The limiting accuracy of the differential equation reconstruction depends strongly on the noise level. As a rule, however, the reconstructed dynamical system manifests predictable behavior on time scales exceeding the linear correlation time of the observed process.

The presence of relaxation terms and higher-order gradients in the dynamical equations makes it very difficult to solve problems because they require the introduction of supplementary boundary and initial conditions compared to classical thermo-hydrodynamics. However, instead of trying to solve the whole mathematical problem, it is more fruitful to concentrate on the analysis of waves, which is, furthermore, a situation of great experimental interest.

Derived from epistemological fundamentals, a new concept of process is developed and applied to biological evolution. The new conception is based on a disctinction between external and internal views of processes. Dynamical equations are associated with the external view, whereas the observation of processes is shown to require an internal approach. Conventional models of dynamics are found to show properties (such as time-independence) that cannot be grounded for an important class of systems viewed internally. The processes that require a new approach are argued to include evolutionary phenomena and Whiteheadian instances of change. Problems of motion such as Zeno’s paradoxes are recognized as issues of externally represented static systems versus internally represented changing systems. The basic notions of an ontology for internal processes of a system are developed. Evolutionary models are discussed in the light of these concepts, and a new simulation framework, based on internally viewed processes that can alter themselves, is presented.

We give some examples where dynamical equations of certain physical systems take the form of Backlund transformations. By this way we discover structural relations between several equations of nonlinear optics.

A three-velocity one-dimensional Lattice Boltzmann Equation is presented. The stability of the equilibrium distribution is studied and the dynamical equations are given. It is shown that Galilean invariance can be recovered for the proper choice of the collision operator. This particular model is then used to compare the results of numerical simulations to the behavior of a shock tube.

In most computer codes developed for the dynamic analysis of mechanisms, such as the serial manipulators used in Robotics, differential equations of motion are derived from the hypothesis that the links are rigid while flexibilities, if any, are located at the joints. This simplification is widely used since it correctly models the sturdy members found in present robots while limiting the number of unknown parameters and of the related kinematical and dynamical equations. Indeed, a rigid body shows six degrees of freedom, and any connection between two adjacent members removes up to five possibilities of relative motion: the total number of independent position parameters of any complex mechanism remains small, and elaborating the kinematical models for the end effector is then a straightforward procedure (the geometrical model relates its position and orientation to the joint parameters, while the differential model expresses its linear and angular velocities in terms of the time derivatives of joint parameters), [1].

The picosecond vibrational excitation of polyatomic molecules is discussed in terms of solutions of the quantum dynamical equations of motion for simple model systems and realistic spectral structures of an asymmetric top molecule.

This paper illustrates the application of symbolic computation in connection with three aspects of mechanical systems: 1. The derivation of dynamical equations by Lagrangian methods. 2. The analysis and synthesis of kinematic mechanisms. 3. A robot manipulator arm.

Much attention has been given in recent years to universal features characterizing instabilities and transition phenomena in equilibrium and non-equilibrium systems [1]. Laser transitions [2], hydrodynamical instabilities [3], cooperative phenomena in data processing [4] and various examples of chemical and biochemical instabilities [5] are only a few of the many non-equilibrium transition phenomena that were studied in analogy to equilibrium phase transitions and critical phenomena. It has been pointed out [2–5] that enhancement of fluctuations, long range order and critical slowing down are common to both equilibrium and non-equilibrium critical phenomena. All these features are interrelated and in the non-equilibrium case, where dynamical equations of motion (kinetic equations) are the starting point of all theoretical treatments, can be all traced to the existence of a long lifetime mode (or modes) near the transition point. Enhancement of the fluctuation amplitude follows from the non-equilibrium equivalent of the fluctuation dissipation theorem [6]. The possibility for the appearance of long range order is seen from the emergence of the quantity \(\sqrt {\tau D} \), D being a characteristic diffusion coefficient and τ is the characteristic long lifetime, as a large scale length.

When the amount of stretch in a taut fiber varies rapidly with distance z along the fiber axis, the tension T on a cross section normal to the axis is not determined by only the temporal history of the local stretch, but is affected by the variation of stretch with z. Here a study is made of the implications of a constitutive relation giving the tension in a polymeric fiber as a function of the local stretch, λ, the spatial derivative, vz, of the velocity, and the first two spatial derivatives of the stretch, λz and λzz. When this relation is placed in the equation for balance of momentum in the axial direction, dynamical equations are obtained whose equilibrium solutions for λ as a function of z can be rendered explicit. The solutions describe necks, bulges, drawing configurations, and periodic striations. The assumption that motions resulting from slow changes in applied tension or length are homotopies formed from these equilibrium solutions is compatible with many of the observed properties of tension-induced necking and cold drawing in polymeric fibers. Properties of non-equilibrium solutions of the dynamical equations are also discussed here. Lyapunov functions appropriate to several types of boundary conditions, including elongation under a dead load, are constructed. It is shown that the presence of viscous stresses has a strong effect on the types of traveling waves that can occur in long fibers. A detailed discussion is given of a class of traveling waves, called "steady draws," that correspond to the continuous drawing processes frequently employed to improve the stiffness and tenacity of synthetic fibers.

As they do in many other fields, numerical techniques can play an important role in the development of our knowledge of chemical systems evolving far from equilibrium. When the macroscopic aspect of their behavior is the only one under consideration, classical numerical integration techniques give us the solution of dynamical equations. Indeed, concentrations are basically the variables that describe the state of the system and systems of nonlinear differential equations result as a complete description of the dynamics of the system.

A new symbolical formalism is introduced to derive the nonlinear control laws involved in controlled mechanisms. It is shown that the approach called the Direct Nonlinear Decoupling Method (DNDM) allows a direct derivation of the explicit form of control laws from the Lagrangian itself so that dynamical equations of mechanisms are not needed in the analysis and a reduction of the amount of calculations. Finally because the method works for many systems in giving the differential equations of motion, the DNDM is a good candidate for the design of computer codes involved in CAD graphics systems.

The theory of turbulence, based upon the Reynolds equations of mean motion and the dynamical equations of the velocity correlations of successive orders, derived from the equations of the turbulent velocity fluctuation, by using the condition of pseudo-similarity and the hypotheses on the viscous dissipation terms in the correlation equations, is developed by a method of successive approximation. As examples in the first order approximation, we have solved the turbulent flows through a channel, in a plane wake and in jets by using the equations of mean motion and of the double correlation, while the terms in the triple correlation have been neglected. The agreements between the calculated values and the experiments are satisfactory.