Approximate equation of motion

The right-hand side of the equation of motion is a quadratic expression in w, which is plotted in Fig. 4 for the same parameter values as in Fig. 2. 

The curvature of all three is moderate, which suggests that it might be possible to approximate this acceleration by a straight line over the interval over which the first integral is non-negative, to a function of the form 

Modern mathematical software, such as Mathematica, allows us to compute symbolically the mean square deviation of this approximation from the exact acceleration, integrated over the feasible region, differentiate the resulting expression symbolically with respect to the parameters a and b, set the results to zero and solve the equations symbolically, and simplify the whole lot to find the following remarkably simple expressions 

The value of a is always negative, so that the solutions will always be oscillatory and expressible in terms of the normal circular functions, with a frequency equal to the square root of — a, namely 

To our knowledge, this is the first analytical estimate of the frequency of the spherical pendulum to have been published. 

---

A similar approximation should be applied to the components of the equation of motion and the significant terms (with respect to ) consistent with the expanded constitutive equation identified. This analy.sis shows that only FI and A appear in the zero-order terms and hence should be evaluated up to the second order. Furthermore, all of the remaining terms in Equation (5.29), except for S, appear only in second-order terms of the approximate equations of motion and only their leading zero-order terms need to be evaluated to preserve the consistency of the governing equations. The term E, which only appears in the higlier-order terms of the expanded equations of motion, can be evaluated approximately using only the viscous terms. Therefore the final set of the extra stress components used in conjunction with the components of the equation of motion are... 

Fig. 7. The horizontal projection of the trajectories for the exact solution (the left curve) and the solution of the approximate equation of motion in terms of Mathieu functions (the right curve) iox n = I, k = 2.

For some problems, such as the motion of heavy particles in aqueous solvent (e.g., conformational transitions of exposed amino acid sidechains, the diffusional encounter of an enzyme-substrate pair), either inertial effects are unimportant or specific details of the dynamics are not of interest e.g., the solvent damping is so large that inertial memory is lost in a very short time. The relevant approximate equation of motion that is applicable to these cases is called the Brownian equation of motion,... 

The final form of the approximate equation of motion is therefore... 

Substituting Eq. [17] into [16], which is, again, the exact equation of motion for the perturbation q, we find the approximate equation of motion... 

In many situations, bubbles move through nonuniform flows. The flows may be either laminar or turbulent. For bubbles in sufficiently strong flows, the buoyancy of the bubble may be negligible or have a small effect on the motion of the bubble. In general, such problems are extremely difficult because one must account for bubble deformation and the disturbance created by the bubble and one must make use of empiricism to develop an approximate equation governing the motion of a bubble. However, for sufficiently small bubbles, approximate equations of motion have been developed that account for the deviation of the bubble trajectory from the path of a fluid particle due to buoyancy, fluid inertia, finite size effects, and related phenomena. We will first consider this regime. 

In many industrial applications such as bubble columns and stirred tank reactors, it is of interest to know the local concentration of bubbles. In general, this is an extremely difficult problem because the bubbles modify the flow and one must compute shape and velocity of the bubbles simultaneously with the motion of the liquid phase. However, for dilute flows, one may be able to obtain some progress with the so-called one-way coupling approximation. In this approach, one ignores the effect of the bubbles on the motion of the liquid. This is reasonable provided that the gas volume fraction is very small and if the bubbles are smaller than the energy-containing eddies so that the turbulence created by the bubbles is unimportant. One then integrates an approximate equation of motion for the bubble. 

The technique of normal mode analysis has been described as a relatively simple procedure for obtaining an exact solution to the approximate equations of motion for a chemical system. Despite its severe approximation (that the dynamics of a system can be represented by the sum of harmonic terms that are only strictly valid for small displacements), the normal mode technique has proven to perform well at predicting many experimentally observed properties. The preceding applications have illustrated the variety of ways in which normal modes can serve to define the dynamic structure and eneiget-ics of small molecules, proteins, and nucleic acids and to aid in the interpretation and refinement of experimental data. This technique is likely to see increased use in the future. 

In this minimal END approximation, the electronic basis functions are centered on the average nuclear positions, which are dynamical variables. In the limit of classical nuclei, these are conventional basis functions used in moleculai electronic structure theoiy, and they follow the dynamically changing nuclear positions. As can be seen from the equations of motion discussed above the evolution of the nuclear positions and momenta is governed by Newton-like equations with Hellman-Feynman forces, while the electronic dynamical variables are complex molecular orbital coefficients that follow equations that look like those of the time-dependent Hartree-Fock (TDHF) approximation [24]. The coupling terms in the dynamical metric are the well-known nonadiabatic terms due to the fact that the basis moves with the dynamically changing nuclear positions. 

A different approach is to represent the wavepacket by one or more Gaussian functions. When using a local harmonic approximation to the trae PES, that is, expanding the PES to second-order around the center of the function, the parameters for the Gaussians are found to evolve using classical equations of motion [22-26], Detailed reviews of Gaussian wavepacket methods are found in [27-29]. 

One drawback is that, as a result of the time-dependent potential due to the LHA, the energy is not conserved. Approaches to correct for this approximation, which is valid when the Gaussian wavepacket is narrow with respect to the width of the potential, include that of Coalson and Karplus [149], who use a variational principle to derive the equations of motion. This results in replacing the function values and derivatives at the central point, V, V, and V" in Eq. (41), by values averaged over the wavepacket. 

Related to the previous method, a simulation scheme was recently derived from the Onsager-Machlup action that combines atomistic simulations with a reaction path approach ([Oleander and Elber 1996]). Here, time steps up to 100 times larger than in standard molecular dynamics simulations were used to produce approximate trajectories by the following equations of motion ... 

We further discuss how quantities typically measured in the experiment (such as a rate constant) can be computed with the new formalism. The computations are based on stochastic path integral formulation [6]. Two different sources for stochasticity are considered. The first (A) is randomness that is part of the mathematical modeling and is built into the differential equations of motion (e.g. the Langevin equation, or Brownian dynamics). The second (B) is the uncertainty in the approximate numerical solution of the exact equations of motion. 

The two sources of stochasticity are conceptually and computationally quite distinct. In (A) we do not know the exact equations of motion and we solve instead phenomenological equations. There is no systematic way in which we can approach the exact equations of motion. For example, rarely in the Langevin approach the friction and the random force are extracted from a microscopic model. This makes it necessary to use a rather arbitrary selection of parameters, such as the amplitude of the random force or the friction coefficient. On the other hand, the equations in (B) are based on atomic information and it is the solution that is approximate. For ejcample, to compute a trajectory we make the ad-hoc assumption of a Gaussian distribution of numerical errors. In the present article we also argue that because of practical reasons it is not possible to ignore the numerical errors, even in approach (A). 

The last approximation is for finite At. When the equations of motions are solved exactly, the model provides the correct answer (cr = 0). When the time step is sufficiently large we argue below that equation (10) is still reasonable. The essential assumption is for the intermediate range of time steps for which the errors may maintain correlation. We do not consider instabilities of the numerical solution which are easy to detect, and in which the errors are clearly correlated even for large separation in time. Calculation of the correlation of the errors (as defined in equation (9)) can further test the assumption of no correlation of Q t)Q t )). 

One property of the exact trajectory for a conservative system is that the total energy is a constant of the motion. [12] Finite difference integrators provide approximate solutions to the equations of motion and for trajectories generated numerically the total energy is not strictly conserved. The exact trajectory will move on a constant energy surface in the 61V dimensional phase space of the system defined by. 

Among the main theoretical methods of investigation of the dynamic properties of macromolecules are molecular dynamics (MD) simulations and harmonic analysis. MD simulation is a technique in which the classical equation of motion for all atoms of a molecule is integrated over a finite period of time. Harmonic analysis is a direct way of analyzing vibrational motions. Harmonicity of the potential function is a basic assumption in the normal mode approximation used in harmonic analysis. This is known to be inadequate in the case of biological macromolecules, such as proteins, because anharmonic effects, which MD has shown to be important in protein motion, are neglected [1, 2, 3]. 

We will study the equations of motion that result from inserting all this in the full Schrodinger equation, Eq. (1). However, we would like to remind the reader that not the derivation of these equations of motion is the main topic here but the question of the quality of the underlying approximations. 

Notice that the solution is not identical to J but an approximation of it. The evolution of a and S in time may conveniently be described via the following classical Newtonian equations of motion Given the initial values... 

There are many algorithms for integrating the equations of motion using finite difference methods, several of which are commonly used in molecular dynamics calculations. All algorithms assume that the positions and dynamic properties (velocities, accelerations, etc.) can be approximated as Taylor series expansions ... 

The first finite element schemes for differential viscoelastic models that yielded numerically stable results for non-zero Weissenberg numbers appeared less than two decades ago. These schemes were later improved and shown that for some benchmark viscoelastic problems, such as flow through a two-dimensional section with an abrupt contraction (usually a width reduction of four to one), they can generate simulations that were qualitatively comparable with the experimental evidence. A notable example was the coupled scheme developed by Marchal and Crochet (1987) for the solution of Maxwell and Oldroyd constitutive equations. To achieve stability they used element subdivision for the stress approximations and applied inconsistent streamline upwinding to the stress terms in the discretized equations. In another attempt, Luo and Tanner (1989) developed a typical decoupled scheme that started with the solution of the constitutive equation for a fixed-flow field (e.g. obtained by initially assuming non-elastic fluid behaviour). The extra stress found at this step was subsequently inserted into the equation of motion as a pseudo-body force and the flow field was updated. These authors also used inconsistent streamline upwinding to maintain the stability of the scheme. 

In Chapter 4 the development of axisymmetric models in which the radial and axial components of flow field variables remain constant in the circumferential direction is discussed. In situations where deviation from such a perfect symmetry is small it may still be possible to decouple components of the equation of motion and analyse the flow regime as a combination of one- and two-dimensional systems. To provide an illustrative example for this type of approximation, in this section we consider the modelling of the flow field inside a cone-and-plate viscometer. 

Subdivision or discretization of the flow domain into cells or elements. There are methods, called boundary element methods, in which the surface of the flow domain, rather than the volume, is discretized, but the vast majority of CFD work uses volume discretization. Discretization produces a set of grid lines or cuives which define a mesh and a set of nodes at which the flow variables are to be calculated. The equations of motion are solved approximately on a domain defined by the grid. Curvilinear or body-fitted coordinate system grids may be used to ensure that the discretized domain accurately represents the true problem domain. 

The temporal behavior of molecules, which are quantum mechanical entities, is best described by the quantum mechanical equation of motion, i.e., the time-dependent Schrdd-inger equation. However, because this equation is extremely difficult to solve for large systems, a simpler classical mechanical description is often used to approximate the motion executed by the molecule s heavy atoms. Thus, in most computational studies of biomolecules, it is the classical mechanics Newtonian equation of motion that is being solved rather than the quantum mechanical equation. 

---