Calculus of variation

The calculus of variations is concerned primarily with determining maxima and minima of quantities that depend on functions. It can be effectively used for optimization problems (Denn, 1969). We present here a brief overview of the variational calculus followed by a simple illustrative example. 

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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. 

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. 

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 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. 

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. 

R. Weinstock, in Calculus of Variations, McGraw-Hill, New York, Toronto, London, 1952, Chapter 3. 

The calculus of variations is applied to H(i) by introducing the variable coefficient A. When the variation method is applied to H(t) as... 

Under these conditions, the form of the probability density distribution function p(t) for the maximum value of information entropy H(i)mdX is investigated. By using the calculus of variations, it is clarified that the information entropy H( t) takes the maximum value as follows ... 

By using the calculus of variations, it is clarified that the information entropy H(t) takes the maximum value as follows ... 

Dieudonnd, J. (1981). History of Functional Analysis (North-Holland, Amsterdam). [147] Goldstine, H.H. (1980). A History of the Calculus of Variations from the 17th through the 19th Century (Springer-Verlag, Berlin). 

Pars, L.A. (1962). An Introduction to the Calculus of Variations (Wiley, New York). [436] Yourgrau, W. and Mandelstam, S. (1968). Variational Principles in Dynamics and Quantum Theory, 3rd edition (Dover, New York). 

Derivation of a ray path for the geometrical optics of an inhomogeneous medium, given v(r) as a function of position, requires a development of mathematics beyond the calculus of Newton and Leibniz. The elapsed time becomes a functional T [x(f)] of the path x(r), which is to be determined so that ST = 0 for variations Sx(t) with fixed end-points Sxp = Sxq = 0. Problems of this kind are considered in the calculus of variations [5, 322], proposed originally by Johann Bernoulli (1696), and extended to a full mathematical theory by Euler (1744). In its simplest form, the concept of the variation Sx(t) reduces to consideration of a modified function xf (t) = x(f) + rw(f) in the limit e — 0. The function w(f) must satisfy conditions of continuity that are compatible with those of x(r). Then Sx(i) = w(l)dc and the variation of the derivative function is Sx (l) = w (f) de. 

The calculus of variations [5,322] is concerned with problems in which a function is determined by a stationary variational principle. In its simplest form, the problem is to find a function v(x) with specified values at end-points xo, x such that the integral J = /(x, y, y )dx is stationary. The variational solution is derived... 

Blanchard and Briining [26] bring the history of the calculus of variations into the twentieth century, as the source of contemporary developments in pure mathematics. A search for existence and uniqueness theorems for variational problems engendered deep studies of the continuity and compactness of mathematical entities... 

Variational principles for classical mechanics originated in modem times with the principle of least action, formulated first imprecisely by Maupertuis and then as an example of the new calculus of variations by Euler (1744) [436], Although not stated explicitly by either Maupertuis or Euler, stationary action is valid only for motion in which energy is conserved. With this proviso, in modem notation for generalized coordinates,... 

Hestenes, M.R. (1966). Calculus of Variations and Optimal Control Theory (Wiley, New York). 

Caratheodory, C. (1967). Calculus of Variations and Partial Differential Equations of the First Order, Part II, tr. R.B. Dean and J.J. Brandstatter (Holden-Day,... 

Elsgolc, L.E. (1962). Calculus of Variations (Pergamon Press, London). 

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