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Introduction
Published in Desineni Subbaram Naidu, Optimal Control Systems, 2018
The calculus of variations is that branch of mathematics that deals with finding a function which is an extremum (maximum or minimum) of a functional. A functional is loosely defined as a function of a function. The theory of finding maxima and minima of functions is quite old and can be traced back to the isoperimetric problems considered by Greek mathematicians such as Zenodorus (495–435 B.C.) and by Poppus (c. 300 A.D.). But we will start with the works of Bernoulli. In 1699 Johannes Bernoulli (1667–1748) posed the brachistochrone problem: the problem of finding the path of quickest descent between two points not in the same horizontal or vertical line. This problem which was first posed by Galileo (1564–1642) in 1638, was solved by John, his brother Jacob (1654–1705), by Gottfried Leibniz (1646–1716), and anonymously by Isaac Newton (1642–1727). Leonard Euler (1707–1783) joined John Bernoulli and made some remarkable contributions, which influenc d Joseph-Louis Lagrange (1736–1813), who finally gave an elegant way of solving these types of problems by using the method of (first) variations. This led Euler to coin the phrase calculus of variations. Later this necessary condition for extrema of a functional was called the Euler - the Lagrange equation. Lagrange went on to treat variable end - point problems introducing the multiplier method, which later became one of the most powerful tools-Lagrange (or Euler-Lagrange) multiplier method-in optimization.
Calculus of Variations
Published in K.T. Chau, Theory of Differential Equations in Engineering and Mechanics, 2017
The application of the calculus of variations is mainly used in searching an optimum solution of problems. For example, these problems include what is the shortest distance between two points in space, what is the shape of the strongest column (proposed by Lagrange in 1773), what is the shape of the column strongest against torsion (St. Venant problem solved by George Polya in 1948), what is the shape of a drum of minimized tone for a given area (Rayleigh conjecture solved by Courant, Faber, and Krahn in 1920s), what is the shortest curve between two points on a curved surface (geodesics problem first considered Euler in 1755), what is the shape of a simply connected electric capacitor that maximizes capacity (solved by Poincare and Szego), what is the shape of a soap film form between two metal circular rings (catenoid problem), what is the profile of a wire for a frictionless sliding bead giving the shortest travel time (brachistochrone problem posed by Johann Bernoulli), what is the least-perimeter of a soap bubble enclosing a given volume of air (solved by Schwarz in 1884 as sphere bubble), what solid with 3-D shape minimizes heat loss (Polya’s cat curling problem), what is the shape of a rocket nose that minimizes the air resistance in supersonic flight (Newton1s minimal resistance problem), and what is the shape of a closed curve of fixed length giving the greatest enclosed area on a surface (isoperimetric problem or Dido problem).
Concepts from Functional Analysis
Published in Karan S. Surana, J. N. Reddy, The Finite Element Method for Boundary Value Problems, 2016
In an earlier section we introduced the concept of ‘functionals’. In the abstract sense, we mean a mapping or a correspondence that assigns a definite real number to each function belonging to some class or space. Functionals are variable quantities and play a very important role in mathematical physics, sciences, and engineering. Calculus of variations is a branch of applied mathematics that deals with extrema of functionals, that is, maximum, saddle points, and minimum. First, we need to establish a connection between the differential equations and their solutions, which is the subject of interest to us, and the functionals and their extrema, that is calculus of variations. There are four basic lemmas that are important in this regard. We state these and provide their proofs.
Calculus of variations as a basic tool for modelling of reaction paths and localisation of stationary points on potential energy surfaces
Published in Molecular Physics, 2020
Josep Maria Bofill, Wolfgang Quapp
An important subfield of optimisation theory is the calculus of variations. The calculus of variations is concerned with the problem of determining maximums or minimums or, in general, stationary values of functionals by seeking that argument function in the given domain of admissible functions for which the functional assumes the stationary value in question. In short, one seeks to optimise a functional over some space to a stationary value by varying a function of the coordinates.