Variation calculus

It turns out that there is another branch of mathematics, closely related to tire calculus of variations, although historically the two fields grew up somewhat separately, known as optimal control theory (OCT). Although the boundary between these two fields is somewhat blurred, in practice one may view optimal control theory as the application of the calculus of variations to problems with differential equation constraints. OCT is used in chemical, electrical, and aeronautical engineering where the differential equation constraints may be chemical kinetic equations, electrical circuit equations, the Navier-Stokes equations for air flow, or Newton s equations. In our case, the differential equation constraint is the TDSE in the presence of the control, which is the electric field interacting with the dipole (pemianent or transition dipole moment) of the molecule [53, 54, 55 and 56]. From the point of view of control theory, this application presents many new features relative to conventional applications perhaps most interesting mathematically is the admission of a complex state variable and a complex control conceptually, the application of control teclmiques to steer the microscopic equations of motion is both a novel and potentially very important new direction. 

In this chapter, principal relations of solid mechanics, elements of convex analysis and calculus of variations, and methods of approximation are considered. 

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. 

Variations-fahigkelt, /. variability, -kurve, /. curve of variations, frequency curve, -rcch-ntmg, /. calculus of variations. 

The problems of operations research have stimulated new developments in several mathematical fields various aspects of game theory, stochastic processes, the calculus of variations, graph theory, and numerical analysis, to name a few. 

Can the calculus of variations be used to find the optimal temperature profile in Example 6.5 ... 

Some problems in functional optimization can be solved analytically. A topic known as the calculus of variations is included in most courses in advanced calculus. It provides ground rules for optimizing integral functionals. The ground rules are necessary conditions analogous to the derivative conditions (i.e., df jdx = 0) used in the optimization of ordinary functions. In principle, they allow an exact solution but the solution may only be implicit or not in a useful form. For problems involving Arrhenius temperature dependence, a numerical solution will be needed sooner or later. 

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. 

For optimization problems that are derived from (ordinary or partial) differential equation models, a number of advanced optimization strategies can be applied. Most of these problems are posed as NLPs, although recent work has also extended these models to MINLPs and global optimization formulations. For the optimization of profiles in time and space, indirect methods can be applied based on the optimality conditions of the infinite-dimensional problem using, for instance, the calculus of variations. However, these methods become difficult to apply if inequality constraints and discrete decisions are part of the optimization problem. Instead, current methods are based on NLP and MINLP formulations and can be divided into two classes ... 

The gradients of the molecular integrals with respect to the nonlinear variational parameters (i.e., the exponential parameters Ak and the orbital centers Sk) were derived using the methods of matrix differential calculus as introduced by Kinghom [116]. It was shown there that the energy gradient with respect to all nonlinear variational parameters can be written as... 

Renal stones Triamterene has been found in renal stones with other usual calculus components. Use cautiously in patients with histories of stone formation. Hematologic effects Triamterene is a weak folic acid antagonist. Because cirrhotics with splenomegaly may have marked variations in hematological status, it may contribute to the appearance of megaloblastosis in cases where folic acid stores have been depleted. Perform periodic blood studies in these patients. 

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

The electronic energy is a functional of the spin orbitals, and we want to minimize it subject to some set of constraints. This can be done using the calculus of variations applied to functionals. So lets look at a general example of functional variation applied to E, a functional of some trial wavefunction that can be linearly varied under a single constraint. 

Now, it is useful to keep in mind our objective. The variational principle instructs us that as we get closer and closer to the true one-electron ground-state wave function, we will obtain lower and lower energies from our guess. Thus, once wc have selected a basis set, we would like to choose the coefficients a, so as to minimize the energy for all possible linear combinations of our basis functions. From calculus, we know that a necessary condition for a function (i.e., the energy) to be at its minimum is that its derivatives with respect to all of its free variables (i.e., the coefficients a,) are zero. Notationally, that is... 

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... 

To give perspective to /( S ) when S is defined for the continuous p(x) considered before, we mention the following facts, provable by the elementary principles of convexity, calculus of variations, and moment theory. 

After manipulations systematically dropping higher order terms in x, the problem is reduced to one in classical calculus of variations. In taking the variations of, Q , certain dependencies exist. Thus Pax is proportional to the kinetic energy part of E. Our final end product will be explicit functional dependencies of Pap, Qa, on p,ua,E, whose approximations are the classical macroscopic relations and the Navier-Stokes equations. 

As explained in Section VI, we must find functions S.p, Qx which minimize J Eq. (5). What we shall do is solve the modified problem containing the fourth moment condition M, and only at the end of the work allow M to approach its canonical value M. Furthermore, we shall use the formalism of the calculus of variations. 

The formalism used to calculate the pulse shape that maximizes J is optimal control theory. This formalism can be considered to be an extension of the calculus of variations to the case where the constraints include differential equations. In general, the constraints expressed in the form of differential equations express the restriction that the amplitude must always satisfy the Schrodinger equation. In addition, there can be a variety of other constraints, such as a restriction on the total energy in the pulse or on the shape of the pulse. These constraints are accounted for by the method of Lagrange multipliers, which modify the objective functional (4.6) and thereby permit the calculation of the unconstrained maximum of the modified objective functional. When the only constraints are satisfaction of the Schrodinger equation and limitation of the pulse energy, the modified objective functional can be written in the form... 

I.M. Gelfand and S.V. Fomin. Calculus of Variations. Prentice-Hall, Englewood Cliffs, NJ, 1963. 

Some readers will recognize this development as the calculus of variations [7]. A functional is a function of a function in this case, T takes the function (r,t0) and maps it to a scalar value that is numerically equal to the total free energy of the system. 

J.M. Ball. The calculus of variations and materials science. Quart. Appl. Math., 56(4) 719-740, 1998. 

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. 

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