Variational methods Euler-Lagrange equations

The computational effort of solving orbital Euler-Lagrange (OEL) equations is significantly reduced if the generally nonlocal exchange-correlation potential vxc can be replaced or approximated by a local potential vxc(r). A variationally defined optimal local potential is determined using the optimized effective potential (OEP) method [380, 398]. This method can be applied to any theory in which the model... 

The approximate solution of Euler-Lagrange equation (3.11) can be found with the help of direct variational method, as it has been proposed in Refs. [14-16]. 

The application of direct variational method for solution of Euler-Lagrange equation is similar to that in Sect. 3.2.2.1. It ailows to obtain the free energy in the form of polarization power series with the coefficients dependent on average particles radius and the parameters of Euler-Lagrange equation (3.45). In particular, surface polarization Pd in the boundary conditions leads to appearance of built-in field Ecyi(R), which can be written as... 

Solving Euler-Lagrange equations by direct variational method with their subsequent substitution into Eq. (3.56), one can obtain ... 

The equilibrium density profile p z) is the one that minimises the surface tension functional thus by the standard methods of calculus of variations we can write down an Euler-Lagrange equation that the density profile must satisfy ... 

Classical methods of calculus of variations are attractive from the point of view of the opportunity to obtain solutions in analytical form. But this is feasible in simple cases, which often are far from the demands of the state-of-art practice. In complicated cases, at a large number of optimization parameters, numerical approaches are used to solve the appropriate Euler-Lagrange equations. The main obstacle arising here is related to the fact that the numerical solution of the system of differential equations may turn out to be more complicated than the solution fi-om the very beginning of the optimization problem by numerical methods of mathematical programming. 

Make a free variation-task from the partially restricted variation (131) by Lagrange s method of multipliers. Give the Euler-Lagrange equations. 

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