The variational calculus

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 

This is a simple example of the general form of Euler s equation (1744), derived directly from a variational expression. 

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 

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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... 

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... 

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. 

The notation used here is adopted from the variational calculus it is defined Appendix G. Since the system is initially at equilibrium, S is a maximum. Such max ima have 8S = 0, and they have... 

Variational calculus is concerned with finding extrema for example, what is the shortest distance between two points on the surface of a parabolic cylinder In ordinary, garden-variety calculus, we deal wiihfunctions, which are objects whose values depend on the values of numerical quantities. But in the variational calculus, the focus of attention is on functionals, which are objects whose values depend on functions. For example, we may interpret the entropy as a functional because its value depends on other thermodynamic functions, such as temperature, pressure, and composition. Since the functionals differ from functions, we sometimes find it convenient to use a notation for operators on functionals that differs somewhat from the notation for operators on functions. For our purposes, the most important notational distinction occurs for differential operators. 

Since the quantities /of interest to us form exact differentials, the second-order variation in (G.0.5) is invariant under an exchange of the indices i and j. We need not proceed further into the variational calculus here because in this book we use the variations 8/and 8 /merely as a notational convenience relations (G.0.4) and (G.0.5) define the notation we use. 

The variational calculus approach to classical mechanics is based on minimizing the action Al over the class G of parameterized curves. This is normally referred to as the principle of least action . It is difficult to provide a physical motivation for this concept, but it is normally taken as a foundation stone for classical mechanics. 

In the variational calculus the equation for the optimum 4>, or the conditional minimum of a functional e, is called the Euler equation. As one can see in this case the Euler equation is identical with the Schrodinger one. 

To apply the variational calculus methods, one should know when a gyroscopic system is Lagrangian, i.e is described by the Lagrange equations with the Lagrange function L on TAf. For this, it is necessary and sufficient that the form of the gyroscopic forces be exact, that is, F = dA where A is the differential 1-form on Af. 

It is well known from the variational calculus that the mathematical character and the necessary condition of a extremmn are invariant to the transformations of the Langrangian... 

When the variational calculus is carried out to minimize the energy in the molecule [as in approximation (iii)] different values are obtained for and for instance, for nitrogen in Ng it is found that Uva — 2-2524 and 1.9090 while the Slater s values for the free atom are = 1-95. Such a... 

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 ... 

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... 

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. 

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. 

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... 

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,... 

THE VARIATIONAL PRINCIPLES OF MECHANICS, Cornelius Lanzcos. Graduate level coverage of calculus of variations, equations of motion, relativistic mechanics, more. First inexpensive paperbound edition of classic treatise. Index. Bibliography. 418pp. 55 x 8(4. 65067-7 Pa. 10.95... 

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) ... 

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