Euler-Lagrange

Within this contimiiim approach Calm and Flilliard [48] have studied the universal properties of interfaces. While their elegant scheme is applicable to arbitrary free-energy fiinctionals with a square gradient fomi we illustrate it here for the important special case of the Ginzburg-Landau fomi. For an ideally planar mterface the profile depends only on the distance z from the interfacial plane. In mean field approximation, the profile m(z) minimizes the free-energy fiinctional (B3.6.11). This yields the Euler-Lagrange equation... 

Variational calculus, Dreyfus (1962), may be employed to obtain a set of differential equations with certain boundary condition properties, known as the Euler-Lagrange equations. The maximum principle of Pontryagin (1962) can also be applied to provide the same boundary conditions by using a Hamiltonian function. 

An alternative procedure is the dynamic programming method of Bellman (1957) which is based on the principle of optimality and the imbedding approach. The principle of optimality yields the Hamilton-Jacobi partial differential equation, whose solution results in an optimal control policy. Euler-Lagrange and Pontrya-gin s equations are applicable to systems with non-linear, time-varying state equations and non-quadratic, time varying performance criteria. The Hamilton-Jacobi equation is usually solved for the important and special case of the linear time-invariant plant with quadratic performance criterion (called the performance index), which takes the form of the matrix Riccati (1724) equation. This produces an optimal control law as a linear function of the state vector components which is always stable, providing the system is controllable. 

Writing the Euler-Lagrange equations in terms of the single-particle wave functions (tpi) the variation principle finally leads to the effective singleelectron equation, well-known as the Kohn-Sham (KS) equation ... 

Moreau, M., B. Bedat, and O. Simonin, From Euler-Lagrange to Euler-Euler large eddy simulation approaches for gas-particle turbulent flows, in ASME Fluids Engineering Summer Conference, Houston. 2005, ASME FED. 

Simulations of multiphase flow are, in general, very poor, with a few exceptions. Basically, there are three different kinds of multiphase models Euler-Lagrange, Euler-Euler, and volume of fluid (VOF) or level-set methods. The Euler-Lagrange and Euler-Euler models require that the particles (solid or fluid) are smaller than the computational grid and a finer resolution below that limit will not give a... 

Euler-Lagrange equations, electron nuclear dynamics (END), time-dependent variational principle (TDVP) basic ansatz, 330-333 free electrons, 333-334 Evans-Dewar-Zimmerman approach, phase-change rule, 435... 

On the analogy of simulating the process of adding blobs of a miscible liquid, two-phase flow in stirred tanks in a RANS context may be treated in two ways Euler-Lagrange or Euler-Euler, with the second, dispersed phase treated according to a Lagrangian approach and from a Eulerian point of view, respectively. 

Next we impose the orthonormality constraint on the wave functions by means of Lagrange multipliers, sy, and obtain the n one-electron Euler-Lagrange equations ... 

For a family of trajectories all starting at the value X(to) and at t=t all arriving at X(t), there is one trajectory that renders the action stationary. The classical mechanical trajectory of a given dynamical system is the one for which 5S=0, i.e. the action becomes stationary. The equation of motion is obtained from this variational principle [59], The corresponding Euler-Lagrange equations are obtained d(3L/3vk)/dt = 9L/dXk. In Cartesian coordinates these equations become Newton s equations of motion for each nucleus of mass Mk ... 

By functional differentiation, Equation 4.22 leads us to the Euler-Lagrange deterministic equation for the electron density, viz.,... 

For further details with respect to notation and the derivation of the corresponding Euler-Lagrange equations we refer to [11], By replacing the meson fields by their expectation values one obtains an effective Dirac equation for... 

In the framework of the Euler-Lagrange formalism, we write the equation of motion for the displacements of the atoms as ... 

The Euler-Lagrange equations can he formed for the dynamical variables q—Rji, Pji, Zph, Zph and collected into a matrix equation which, when solved, yields the wave function for the compound system at each time step. 

It is of interest also to notice that the solution of Eq. (6) minimizes the following Euler-Lagrange functional with respect to variations of... 

Gear, C. W., Leimkuhler, B., and Gupta, G. K., Automatic integration Euler-Lagrange equations with constraints, Journal of Computational and Applied Mathematics 12 and 13, 77-90 (1985). 

Euler-Lagrange Equation for Intra-Orbit Optimization of p(r) 206... 

Euler-Lagrange Equations for the Intra-Orbit Optimization of... 

As indicated in Fig. 7, the next step after either an explicit or an implicit energy density functional orbit optimization procedure. For this purpose, one introduces the auxiliary functional Q[p(r) made up of the energy functional [p(r) 9 ]. plus the auxiliary conditions which must be imposed on the variational magnitudes. Notice that there are many ways of carrying out this variation, but that - in general - one obtains Euler-Lagrange equations by setting W[p(r) = 0. 

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