Euler-Lagrange minimization

Kohn-Sham functional makes the task of approximating the total energy functional easier but also offers a possibility to perform practical calculations. Euler-Lagrange minimization of the functional EKS[(pi, (p2,. [Pg.7]

We turn back to the energy minimization of Ss (Eq. 24). Euler-Lagrange minimization taking into account the conditions of orthogonality for each set ... 

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... 

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

The entropy production is a function of the temperature field. Then, the minimization problem is to obtain the temperature distribution T x) corresponding to a minimum entropy production using the following Euler-Lagrange equation ... 

The minimization of Wlost (=Elost) entails the minimization of the integral in Eq. (4.182) with the constraint of constant production rate J (mol/h) and use of the Euler-Lagrange method... 

If we now look back at Vcin der Waals treatment, sec. 2.5. several steps can be recognized. In a sense he anticipated the present method. His function to be minimized is [2.5.19] he eliminated boundary condition [A1.3] by working grand canonically and [2.5.25] is his Euler-Lagrange equation. From this. F could be written as [2.5.30] ). 

Equation (92) is simpler than (60) but otherwise structurally identical to it. In particular, the Euler-Lagrange equation associated with minimizing (92) has the form of a backward Kolmogorov equation ... 

Our interest is in determining the energy minimizing configuration of the bowed-out segment. To do so, we note that this has become a simple problem in variational calculus, with the relevant Euler-Lagrange equation being... 

Hi = [i(Z = 0), and h jy is then the rescaled surface field Hi, gjy the rescaled coefficient —R2k l/d. Now the Euler-Lagrange equations describing the solution that minimizes eq. (241) read... 

The curve that minimizes the surface area is given by the following Euler-Lagrange equation (GeTfand and Fomin, 2000) ... 

It is assumed that Ec is so defined that the functional E is minimized for ground states. Ground-state orbital functions and the density function are determined by Euler-Lagrange equations expressed in terms of functional derivatives of E. For any density or orbital functional, with fixed n, infinitesimal orbital variations determine the functional variation... 

We have assumed here that the variations are performed within the domain of normalized densities. Alternatively, the minimization can be performed using the Euler-Lagrange procedure. Then the densities are allowed to vary also outside the normalization domain. This we shall do by relaxing the normalization constraint of the wavefunctions and by using the definition (3) of the density also in the extended domain. The normalization constraint is enforced by means of a Lagrange multiplier (pi),... 

Above we have assumed that the minimization is carried out within the domain of normalized of densities. Alternatively, we can perform the minimization, using the Euler-Lagrange procedure. Then we use the extension of the functionals valid also outside the normalization domain and enforce the normalization constraint by a Lagrange multiplier.5 For the Levy-Lieb energy functional (70) this leads to... 

In order to obtain the maximum Z( Na ), we minimize Equation 2.19 with respect to the orientation distribution function while retaining the normalization condition 2.20. The Euler-Lagrange equation is... 

The chemical potential p, = 6J f6p enters the respective Euler-Lagrange equation obtained by minimizing the grand ensemble thermodynamic potential — p J pd a , which defines the equilibrium particle density distribution... 

The variational integral can be minimized whilst subject to the normalizing constraint (Arfken, 1970) by introducing a Lagrangian multiplier. The resulting Euler-Lagrange equation is... 

Example Minimal Surfaces — Euler-Lagrange Equations... 

To solve for the profile that minimizes F, one must solve the Euler-Lagrange equation taking into account the variations in x (x, y,z — h x, y)) in three dimensions ... 

From the discussion of the Euler-Lagrange equations of Chapter 2, this functional is minimized if f obeys... 

The Euler-Lagrange equation arising from the minimization of fg with respect to the density profile, n(z), is... 

But, the Euler-Lagrange equation, corresponding to the minimization of fs with respect to s yields... 

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