Technology & Engineering
Variational Method
The variational method is a mathematical technique used to approximate the lowest energy state of a physical system. It involves formulating an approximate trial wave function and using optimization methods to minimize the energy. This method is widely used in quantum mechanics, computational chemistry, and materials science to estimate properties of complex systems.
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6 Key excerpts on "Variational Method"
eBook - PDF- Jagdish S. Rustagi(Author)
- 2014(Publication Date)
- Academic Press(Publisher)
8 Variational Methods 8.1. INTRODUCTION Many statistical problems involve optimization of a functional defined over a specified function space. The theory of the calculus of variations plays a central role in functional analysis similar to that of maxima and minima in differential calculus. Some of the important concepts in functional analysis and other modern Variational Methods are introduced in Chapter 11. The classical theory of optimization of functionals was mainly concerned with optimization of integrals involving unknown functions and their deriva-tives. In statistics, optimization of many other functionals such as the gen-eralized variance of a random vector involving determinants of matrices and products of unknown functions resulting from likelihood of a sample are needed. The area of functional estimation has recently evolved as an important area of study in statistics. The classical theory of the calculus of variations was developed for solv-ing problems in applications to mechanics. One of the earliest problems leading to Variational Methods was considered by Sir Isaac Newton, who 209 210 Optimization Techniques in Statistics wanted to determine the shape of a ship's hull to allow minimum drag of water. Another important problem considered in the early development of the calculus of variations was to find the path of a particle traveling from one point in a vertical plane to another under the force of gravity in minimum time. Many famous mathematicians like Euler and Lagrange made fundamental contributions to the early development of the calculus of variations. During the past two centuries the calculus of variations has evolved as an important branch of mathematics. Recent applications of the calculus of variations have been made to many areas of science and technology such as economics, control theory, physics, and statistics.
- Kevin W. Cassel(Author)
- 2013(Publication Date)
- Cambridge University Press(Publisher)
It is remarkable that this mathematical framework, marked by an enduring elegance, is capable of encapsulating many phys- ical and optimization principles of which its inventors could have never conceived. This is the “miracle of the appropriateness of the language of mathematics to the formulation of the laws of physics” that Physics Nobel Laureate Eugene Wigner spoke of in 1960. The “miracle” of Variational Methods is stronger than ever today, in particular as formal optimization techniques permeate more and more areas of science, engineering, and economics. As such, the calculus of variations has come full circle; originally conceived to formally address a limited set of early optimization problems, Variational Methods now provide the mathematical foundation and frame- work to treat modern, large-scale optimization problems. Every day we optimize our decisions, activities, processes, and designs; therefore, Variational Methods are proving to be an indispensable tool in providing a formal basis for optimization 1.2 Introduction 7 along with providing the foundation for an impressive array of physical principles that encompass classical mechanics and modern physics. 1.2 Introduction With respect to its strong ties to applications, this treatment of the calculus of variations will emphasize three themes. First, formulating many problems in the physical sciences and engineering from first principles 3 leads naturally to variational forms. Second, these variational forms often provide additional physical insight that is not readily apparent from the equivalent, typically more familiar, differential formulation. Third, the variational approach to optimization and control supplies a general and formal framework within which to apply such principles to a broad spectrum of diverse fields.
eBook - PDF- Singiresu S. Rao(Author)
- 2019(Publication Date)
- Wiley(Publisher)
4 Derivation of Equations: Variational Approach 4.1 INTRODUCTION As stated earlier, vibration problems can be formulated using an equilibrium, a varia- tional, or an integral equation approach. The variational approach is considered in this chapter. In the variational approach, the conditions of the extremization of a functional are used to derive the equations of motion. The Variational Methods offer the following advantages: 1. Forces that do no work, such as forces of constraint on masses, need not be considered. 2. Accelerations of masses need not be considered; only velocities are needed. 3. Mathematical operations are to be performed on scalars, not on vectors, in deriv- ing the equations of motion. Since the Variational Methods make use of the principles of calculus of variations, the basic concepts of calculus of variations are presented. However, a brief review of the cal- culus of a single variable is given first to indicate the similarity of the concepts. 4.2 CALCULUS OF A SINGLE VARIABLE To understand the principles of calculus of variations, we start with the extremization of a function of a single variable from elementary calculus [2]. For this, consider a continuous and differentiable function of one variable, defined in the interval ( 1 , 2 ), with extreme points at a, b, and c as shown in Fig. 4.1. In Fig. 4.1, the point = denotes a local minimum with () ≤ () for all x in the neighborhood of a. Similarly, the point = represents a local maximum with () ≥ () for all x in the neighborhood of b. The point = indicates a stationary or inflection point with ( ) ≤ () on one side and ( ) ≥ () on the other side of the neighborhood of c.
eBook - PDFVariational Principles in Mathematical Physics, Geometry, and Economics
Qualitative Analysis of Nonlinear Equations and Unilateral Problems
- Alexandru Kristály, Csaba Varga, Vicenţiu D. Rădulescu(Authors)
- 2010(Publication Date)
- Cambridge University Press(Publisher)
Part I Variational principles in mathematical physics 1 Variational principles A man is like a fraction whose numerator is what he is and whose denominator is what he thinks of himself. The larger the denominator the smaller the fraction. Leo Tolstoy (1828–1910) Variational principles are very powerful techniques that exist at the interface between nonlinear analysis, calculus of variations, and mathematical physics. They have been inspired by and have deep applications in modern research fields such as geometrical analysis, constructive quantum field theory, gauge theory, superconductivity, etc. In this chapter we briefly recall the main variational principles which will be used in the rest of the book, such as Ekeland and Borwein–Preiss variational principles, minimax- and minimization-type principles (the mountain pass theorem, Ricceri-type multiplicity theorems, the Brezis–Nirenberg minimization technique), the principle of symmetric criticality for nonsmooth Szulkin-type functionals, as well as Pohozaev’s fibering method. 1.1 Minimization techniques and Ekeland’s variational principle Many phenomena arising in applications such as geodesics or minimal surfaces can be understood in terms of the minimization of an energy functional over an appropriate class of objects. For the problems of mathematical physics, phase transitions, elastic instability, and diffraction of light are among the phenomena that can be studied from this point of view. A central problem in many nonlinear phenomena is whether a bounded from below and lower semi-continuous functional f attains its infimum. A simple function for which the above statement clearly fails is f : R → R defined by f (s) = e −s . Nev- ertheless, further assumptions either on f or on its domain may give a satisfactory answer. In the following chapters we present two useful forms of the well-known Weierstrass theorem.
eBook - ePub- O. C. Zienkiewicz, K. Morgan, K. Morgan(Authors)
- 2013(Publication Date)
- Dover Publications(Publisher)
CHAPTER SIXVariational Methods6.1. INTRODUCTIONIt has been seen in the previous chapters that the analysis of many physical problems frequently requires the determination of a function which is the solution of a differential equation governing the behavior of the system under consideration. In some circumstances, however, it may be possible to determine a naturai variational principle for the particular problem of interest, and then an alternative solution approach may be adopted, which consists of determining the function which makes a certain integral statement (or functional ) stationary. Clearly, if such variational principles can be found, then immediately we have new methods available for constructing approximate solutions, for we can use the trial function or finite element methods of the previous chapters and attempt to make the functional stationary with respect to variations in the unknown parameters.Some physical problems can be stated directly in the form of a variational principle—an obvious example being the requirement of minimization of total potential energy for the equilibrium of a mechanical system. However, the form of the natural variational principle is not always obvious and, indeed, such a principle does not exist for many continuous problems for which well-defined differential equations may be formulated. We shall therefore begin by considering under what circumstances a natural variational principle can be derived from a differential equation, and we shall then investigate how special contrived variational forms can be constructed if no natural variational principle exists. These special forms will be seen to use either the standard Lagrange multiplier approach, which introduces additional variables into the analysis, the penalty function method, or the method of least squares.6.2. VARIATIONAL PRINCIPLESSuppose we are given a functional Π in the integral formwhere F and G are functions of (x ,... ) and its derivatives, and Γ is the curve bounding the closed region Ω. We will now attempt to make Π stationary with respect to variations in among the admissible set of functions satisfying general boundary conditions, of the type introduced in Eq. (2.25
eBook - PDF- Lawrence Turyn(Author)
- 2013(Publication Date)
- CRC Press(Publisher)
14 Calculus of Variations 14.1 Minimization Problems The Calculus of variations, as a systematic way of modeling and solving physical problems, is historically relevant but was also reinvigorated in the twentieth century in the subjects of both control theory and finite element methods. Very natural contexts for Calculus of variations include engineering mechanics and elec-tromagnetism where we use knowledge of the energy in the system. For example, it makes sense that if we deform a solid object then its new equilibrium shape should minimize its potential energy. There are many physical problems that are modeled by a Calculus of variations problem of finding an “admissible” function y ( x ) so as to ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ Minimize J [ y ] b a F ( x , y ( x ) , y ( x ) ) dx Subject to y ( a ) = y a y ( b ) = y b ⎫ ⎪ ⎪ ⎪ ⎪ ⎪ ⎬ ⎪ ⎪ ⎪ ⎪ ⎪ ⎭ (14.1) or subject to conditions on or including y ( a ) , y ( b ) . Also, the integral may involve higher order derivatives of y . Often the problem has a natural interpretation in terms of energy in a system. In Chapter 13 we studied minimization of a function that depends on several inde-pendent variables, that is, unknowns to be solved for. In Chapter 14 we will minimize an integral which depends on a function which is the unknown to be solved for. That is inherently a more mind boggling problem, but results from Chapter 13 will still be relevant. A function is admissible if it is continuous and piecewise continuously differentiable on the interval [ a , b ] . If higher order derivatives are in the integrand then the class of admissible functions may be further restricted to involve higher order differentiability. A functional is a mapping from a vector space to scalar values. For example, in (14.1) the mapping y → b a F ( x , y ( x ) , y ( x ) ) dx is a functional. 1081
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