Lagrangian formulations

Note that the shape functions used in the above discretization preserv c their originally defined forms. This i.s in contrast to the Lagrangian formulations in which the shape functions need to be modified (Donea and Qiuirtapellc, 1992). 

The relationship between H and vibrational frequencies can be made clear by recalling the classical equations of motion in the Lagrangian formulation ... 

The first approach is based on introducing simple velocity or position rescaling into the standard Newtonian MD. The second approach has a dynamic origin and is based on a refonnulation of the Lagrangian equations of motion for the system (so-called extended Lagrangian formulation.) In this section, we discuss several of the most widely used constant-temperature or constant-pressure schemes. 

The equations of motion follow directly from the Lagrangian formulation containing all constraints. The result is... 

A variant of this method [32] was also developed which uses an extended Lagrangian formulation of the form... 

We use the Lagrangian formulation since the constraint = o can be naturally expressed in this setting. The Lagrangian Jz F is associated with... 

Waldvogel and Poulikakos1501 extended the model and numerical techniques of Zhao et al.13681 by incorporating solidification and droplet-substrate contact resistance in the heat transfer model. They conducted both theoretical and experimental studies on the impact and solidification of molten solder droplets on a multilayer substrate. The theoretical model was based on the Lagrangian formulation, and accounted for a host of thermal-fluid phenomena,... 

Based on the manner of derivation of the Gaussian equations in Section III, we see that the dispersion parameters a-y and are originally defined for an instantaneous release and are functions of travel time from release. Since the puff equations depend on the travel time of individual puffs or releases, the dispersion coefficients depend on this time, i.e., these coefficients describe the growth of each puff about its own center. This is basically a Lagrangian formulation. 

This implies that, in general, temporal ordering parameters cannot be identified directly with physical time—they merely share one essential characteristic. This situation is identical to that encountered in the Lagrangian formulation of general relativity there, the situation is resolved by defining the concept of particle proper time. In the present case, this is not an option because the notion of particle proper time involves the prior definition of a system of observer s clocks—so that some notion of clock time is factored into the prior assumptions on which general relativity is built. 

A Smooth Lagrangian Formulation of the PCM Free Energy Functional... 

The strategy to obtain a Lagrangian formulation of PCM is to consider the PCM apparent charges as a set of dynamic variables, exactly as the solute nuclear coordinates. The algorithm proposed in the present chapter is applied within the MM framework, since it allows a simplified notation and faster calculations. However, we point out that it can be straightforwardly extended to QM calculations. 

Figure 1.9 Total and potential energy (au) of formaldehyde in water, (a) with the PCM charges equilibrated at each time step and (b) with the PCM extended Lagrangian formulation.

Moreover the shifts in the frequencies passing from the gas phase to the solution are qualitatively correct (we did not consider any anharmonicities in the analytical calculations). Thus also in the case of a larger test molecule, the extended Lagrangian formulation of CPCM is successful in describing the solvation effect. 

In the other two sections of the chapter two further generalizations of PCM models are presented to spatially and dynamically nonlocal media (Basilevsky Chuev) and to a Lagrangian formulation which includes the polarization of the medium as a dynamical variable (Caricato, Scalmani Frisch), respectively. In the first case, the goal is to account for the discreteness of molecular liquids still within a continuum description of... 

The Lagrangian Formulation and the Kuhn-Tucker Conditions. Formulate the Lagrangian function for the problem (Pi),... 

An advantage of the Lagrangian formulation is that it permits the identification of conserved quantities by studying the invariances of the action S. The fundamental result behind this statement is Noether s theorem, which identifies conserved currents associated with invariance of S (and hence L) under infinitesimal continuous transformation. 

The Lagrangian formulation is equally effective for discussing the idea of spontaneous breakdown of symmetry and the effect thereof can be demonstrated in terms of a simple example [18, 19]. 

Sensitivity Studies on 1969 Trajectories with the Expanded Model. Based on the semi-Lagrangian formulation of the photochemical/diffusion model, the computed endpoint composition of the air masses depends on initial conditions, flux from the ground along the trajectories, and reaction rates. For our tests we concentrate on El Monte data because much of the polluted air there comes from somewhere else. This is believed to be a more severe test of the model than that at Huntington Park. The initial conditions are based on measurements insofar as possible. The principal initial values for the 1030 trajectory are as follows for 0730 PST (given in parts per hundred million) ... 

Besides the mathematical improvements, the atmospheric model has been adapted to a semi-Lagrangian formulation. By following selected air masses, we avoid commitments of large quantities of memory and the incursions of artificial diffusion errors. Most important, we do not end up with stacks of computer printout that relate to regions where there are no measurements. Also, predictive calculations will become more useful, but our present levels of resources and sophistication demand that effort be concentrated on verification. Only in this way can the confidence be built that is needed for applying modeling techniques to implementation planning. 

Within a Lagrangian formulation of the mechanics of a field as used here one may define an energy-momentum tensor whose components summarize the principal properties of the field (Morse and Feshbach 1953 Landau and Lifshitz 1975). We show that the divergence equations satisfied by the spatial components of this tensor for the Schrodinger field yield the differential form of the atomic force law, eqn (8.175) (Bader 1980). 

In 1933 Dirac published a paper entitled The Lagrangian in quantum mechanics . After presenting a discussion as to why the Lagrangian formulation of classical mechanics could be considered to be more fundamental than the approach based on the Hamiltonian theory, Dirac went on to say For... 

G. R. Gruber, Quantization in generalized coordinates III—Lagrangian formulation. Int. 

The END theory was proposed in 1988 [11] as a general approach to deal with time-dependent non-adiabatic processes in quantum chemistry. We have applied the END method to the study of time-dependent processes in energy loss [12-16]. The END method takes advantage of a coherent state representation of the molecular wave function. A quantum mechanical Lagrangian formulation is employed to approximate the Schrodinger equation, via the time-dependent variational principle, by a set of coupled first-order differential equations in time to describe the END. 

The dependence of the average concentration on the Damkohler number can also be interpreted within the Lagrangian formulation. For example, the logistic growth function of the plankton population dynamics (Eq. (6.7)) is concave near the steady state P = K, i.e. the plankton population reacts more quickly when the carrying capacity is below the actual plankton density, than in the opposite case when higher carrying capacity allows for increase of the plankton concentration. Due to this asymmetric nonlinear response the... 

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