Variational calculus

Roubicek T. (1997) Relaxation in optimization theory and variational calculus. W de Gruyter, Berlin. 

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. 

In Section II, the basic equations of OCT are developed using the methods of variational calculus. Methods for solving the resulting equations are discussed in Section III. Section IV is devoted to a discussion of the Electric Nuclear Bom-Oppenhermer (ENBO) approximation [41, 42]. This approximation provides a practical way of including polarization effects in coherent control calculations of molecular dynamics. In general, such effects are important as high electric fields often occur in the laser pulses used experimentally or predicted theoretically for such processes. The limits of validity of the ENBO approximation are also discussed in this section. 

Variational calculus with this Lagrangian density leads [17] to the field equation ... 

As noted above, however, the Hamiltonian defined by Eqs. (4.32) and (4.33) does not include interelectronic repulsion, computation of which is vexing because it depends not on one electron, but instead on all possible (simultaneous) pairwise interactions. We may ask, however, how useful is the Hartree-product wave function in computing energies from the correct Hamiltonian That is, we wish to find orbitals that minimize (4 hp H I hp). By applying variational calculus, one can show that each such orbital i/f, is an eigenfunction of its own operator hi defined by... 

Equation 11 can be transformed by applying variational calculus analogously to the transformation of Equation 6. This procedure results in the following expression ... 

The pancake theory today is perceived by mathematicians as a chapter contributed by Ya.B. to the general mathematical theory of singularities, bifurcations and catastrophes which may be applied not only to the theory of large-scale structure formation of the Universe, but also to optics, the general theory of wave propagation, variational calculus, the theory of partial differential equations, differential geometry, topology, and other areas of mathematics. 

Structural analysis, initially developed on an intuitive basis, later became identified with variational calculus, in which the Ritz procedure is used to minimize a functional derived mathematically or arrived at directly from physical principles. By substituting the final solutions into the variational statement of the problem and minimizing the latter, the FEM equations are obtained. Example 15.2 gives a very simple demonstration of this procedure. 

It is well known t.hat the best way to solve an optimization iiroblern for conventional functions is based on difi cicntiating the functions and cejuating the resulting derivatives to zero. A similar approach can be apjdied in ])rinciple to functionals. However, we have to use an analog of calcuhis for functionals and operators, which is called variational calculus. This generalization has been discussed in Appendix D. 

The problem of minimization of the misfit functional (3.7) can be solved using variational calculus. Let us calculate the first variation of / (m) ... 

Note that, similar to the basic formulae of calculus for conventional functions, we can obtain simple rules and operations of the variational calculus. Actually, the variational operator acts like a differential operator. For example, let us consider the operators... 

Density functionals are functions of functions. Although this technique Is powerful In describing heterogeneous systems and often applied in advanced papers, we shall not go so far In this book. Chapter 9 of Davis book (1996), mentioned in sec. 2.13, Introduces the topic. However, in appendix 3 some aspects of variational calculus will be outlined. 

Because of the landmark nature of van der Waals work we shall now discuss some important aspects of his theory. In doing so a selection has to be made (the German version of van der Waals paper runs to over a hundred pages ). We shall use FICS-nomenclature and follow as much as possible van der Waals own arguments and derivations, although parts of the latter can nowadays be carried out more efficiently. For instance, the minimization of the Helmholtz energy as a function of the profile shape can nowadays be elegantly done by variational calculus, the principles of which will be outlined in appendix 3. 

Functionals are functions of functions. In this Volume we met functionals in van der Waals theory for the interfacial tension (sec. 2.5a) and in the mean field theory for the surface pressure of polymeric monolayers (sec. 3.4e). In these two cases equations were derived in which the excess interfacial Helmholtz energy had to be minimized as a function of a density distribution across the interface and of the spatial derivative of this profile [density Junctionals). The technique of finding the function that minimizes the Helmholtz energy is called variational calculus, or calculus of variations. 

Variational calculus is an extension of common calculus. In the latter, the minimum of a function of a variable is found by setting the derivative with respect to that VEirlable equal to zero. In variational calculus this should In principle be done by searching through the full set of functions, consistent with the boundary conditions, and look for that function which minimizes the functional. This procedure is virtually impossible. Fortunately, there are some general principles that can be applied to simplify the issue. We shall now discuss some of these, taking... 

The study of minimal surfaces arose naturally in the development of the calculus of variations. The problem of finding the surface forming the smallest area for a given perimeter was first posed by Lagrange in 1762, in the appendix of a famous paper that established the variational calculus [8]. He showed that a necessary condition for the existence of such a surface is the equation... 

Moreover, in [31] has been applied variational calculus showing that the saving function in [17] and modified ecological criteria are equivalent. In this section, internal irreversibilities are taken into account to obtain Equation (4), replacing (r2 + r)/2f instead (,< 2 +, < )/2 in case of a non-endoreversible Curzon and Ahlbom cycle. The procedure in [5] is combined with the cyclic model in [16] to obtain the form of power output function and of ecological function. 

A standard variational calculus, extended slightly as compared to that described in Sect. 2.1, is applied to find the profile ( )(z) and its surface value, minimizing Eq. (24). It yields a differential equation describing the profile <]>(z) (identical to Eq. 8b) ... 

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... 

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