Time-dependent system

This paper will discuss the formulation of the simulator for the filament winding process which describes the temperature and extent of cure in a cross-section of a composite part. The model consists of two parts the kinetic model to predict the curing kinetics of the polymeric system and the heat transfer model which incorporates the kinetic model. A Galerkin finite element code was written to solve the specially and time dependent system. The program was implemented on a microcomputer to minimize computer costs. 

Because energy is not conserved in a time-dependent system, it is not meaningful to ascribe a certain energy to a TS trajectory in the way that the fixed point and the NHIM in an autonomous system exist at different energies. Instead, there is typically a single TS trajectory that is uniquely defined by the... 

Runge E, Gross EKU (1984) Density-functional theory for time-dependent systems. Phys Rev Lett 52 997... 

Runge, E., and E. K. U. Gross. 1984. Density Functional Theory for Time-Dependent Systems. Phys. Rev. Lett. 52, 997. 

When the ys have their steady-state values, the right-hand sides of these equations are zero, so the solution will yield zero for all of the delys. The solution of the time-dependent system is related to the solution of the steady-state system. 

To extend TFD to the time-dependent system Ha, we introduce a fictitious Hamiltonian... 

Dynamical chaos in periodically driven systems has become attractive topic in many areas of contemporary physics such as atomic, molecular, nuclear and particle physics. Dynamical systems which can exhibit chaotic dynamics can be divided into two classes time independent and time-dependent systems. Billiards, atoms in a constant magnetic field, celestial systems with chaotic dynamics are time independent systems, whose dynamics can be chaotic. 

Barthomieu, B. and Diaz, M. (1991) Modeling and verification of time dependent systems using time Petri nets. IEEE Trans. Software Eng., 17(3), 259-273. 

The scheme we employ uses a Cartesian laboratory system of coordinates which avoids the spurious small kinetic and Coriolis energy terms that arise when center of mass coordinates are used. However, the overall translational and rotational degrees of freedom are still present. The unconstrained coupled dynamics of all participating electrons and atomic nuclei is considered explicitly. The particles move under the influence of the instantaneous forces derived from the Coulombic potentials of the system Hamiltonian and the time-dependent system wave function. The time-dependent variational principle is used to derive the dynamical equations for a given form of time-dependent system wave function. The choice of wave function ansatz and of sets of atomic basis functions are the limiting approximations of the method. Wave function parameters, such as molecular orbital coefficients, z,(f), average nuclear positions and momenta, and Pfe(0, etc., carry the time dependence and serve as the dynamical variables of the method. Therefore, the parameterization of the system wave function is important, and we have found that wave functions expressed as generalized coherent states are particularly useful. A minimal implementation of the method [16,17] employs a wave function of the form ... 

In the development of the aforementioned time-dependent systems, care has to be taken to ensure a homogeneous coating. If the coat is not homogenous, its rigidity will be affected, possibly leading to undesirable infiltration of the aqueous medium and, in turn, imdesired alteration of the lag time before which the drug is supposed to be released. 

Among the most striking changes brought about by fractional dynamics is the substitution of the traditionally obtained exponential system equilibration of time-dependent system quantities by the Mittag-Leffler pattern [44-46]... 

In this chapter we discuss the close relationship between the Born-Oppenheimer treatment of molecular systems and field theory as applied to elementary particles. The theory is based on the Born-Oppenheimer non-adiabatic coupling terms which are known to behave as vector potentials in electromagnetic dynamics. Treating the time-dependent Schrodinger equation for the electrons and the nuclei we show that enforcing diabatization produces for non-Abelian time-dependent systems the four-component Curl equation as obtained by Yang and Mills (Phys. Rev. 95, 631 (1954)). 

This expression will be used in the following for time-independent as well as time-dependent system Hamiltonians. 

For time-dependent systems again the purely exponential time-dependence of the correlation function allows the derivation of a set of differential equations for the auxiliary operators... 

Here, there appear the Jacobian Fc, which is in fact J as defined above in (9.60), the function F itself, applying at partly augmented T and C values, and, in case of time-dependent systems, the time derivative Ft, written in short form, as it is applied to the present T and C. This last term is often zero, if the system does not include functions of time. 

The same procedure, both for the steady state and the time dependent system, can be extended to the channel with two bands, in generator-collector mode, as shown in Fig. f3.7. There are more boundary conditions, but they are straight-forward to apply. For details, the reader is referred to a series of articles by the Compton group [40,171,174,175,244,463] (citing just a selection of a large opus). 

Instability of an autonomous system is strictly for time-dependent systems that would display growth of disturbances in time. This may also mean that either we are stud3ung the stability of a flow at a fixed spatial location or the full system displays identical variation in time for each... 

Fourier and Laplace transforms are linear transforms and are very often used for analyzing problems in various branches of science and engineering. Since receptivity is studied with respect to onset of instability, it is quite natural that these transform techniques will be the tool of choice for such studies. Fourier transform provides an approach wherein the differential equation of a time dependent system is solved in the transformed plane as. 

The vector (di, 2) is called the Melnikov vector. For systems of more than two degrees of freedom, as well as for time-dependent systems of two degrees of freedom, we need the Melnikov vectors to investigate intersections between the stable and unstable manifolds of NHlMs. 

Not explicitly time dependent systems axe called autonomous. For autonomous systems dH/dt = 0 and we have H — E = const, i.e. the total energy of the system is conserved. Clearly the system of equations (3.1.21) is more symmetric than the set (3.1.6) of second order dilferential equations obtained from the Lagrangian formalism. 

Third, GPC of very dilute solutions of asphaltenes and their fractions clearly confirm the above implications and demonstrate that these materials represent a dynamic, time-dependent system that changes to MW entities of the size of polyaromatic hydrocarbons and similar materials. This in turn indicates that the forces involved in the apparent high MW species formation are not fully understood and questions the tenability of formulations based on (r-bonded polymers to explain the apparent high MW of the aggregates. 

In addition, there is interest in further extending the discussion to a variety of situations, that have recently gained much attention in the nonrelativistic case, as time-dependent systems [49], excited states [45] or finite temperature ensembles [110]. As an example of work along these lines we mention the gradient expansion of the noninteracting, relativistic free energy [110], leading to a temperature-dependent relativistic extended Thomas-Fermi model. 

To proceed beyond eqn (5.68) the procedure is always the same one rids the expression of 5 ij/ (or 5>j/ for a time-dependent system) using an integration by parts to transform the integrand into a quantity multiplied only by dij/. Setting this quantity equal to zero yields the Euler equation. 

The second-order expansion given in eqn (6.96) recovers all of the physical quantities needed to describe a quantum system and determine its properties the charge density p and its gradient vector field Vp define atoms and determine many of their properties in a stationary state the current density determines the system s magnetic properties and the change in p in a time-dependent system and, finally, the stress tensor determines the local and average mechanical properties of the system. Thus, one does not need all the... 

---