Euler-Lagrange variation procedure

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. 

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),... 

The example of the variational procedure considered in this section was very simple, because we operated only with one independent variable (angle (p). Sometime one needs to minimize the energy with respect to two variables in fact, we met this case in Section 6.3.3 for an infinite medium. For two variables, the system of two Euler equations can be constructed using the same procedure as earlier. However, very often one deals with some constraints as, for example, in the case of the director that has three projections satisfying the constraint + tiy- - n =. In such cases the Lagrange multipliers are introduced to solve the variational problem, however this, more general Euler - Lagrange approach will not be used in this book. 

Newton and Leibnitz. The foundations of calculus of variations were laid by Bernoulli, Euler, Lagrange and Weierstrass. The optimization of constrained problems, which involves the addition of unknown multipliers, became known by the name of its inventor Lagrange. Cauchy made the first application of the steepest descent method to solve unconstrained minimization problems. In spite of these early contributions, very little progress was made until the middle of the 20th century, when high-speed digital computers made the implementation of the optimization procedures possible and stimulated further research in new methods. 

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