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Unformatted text preview: Chapter 21 The Calculus of Variations We have already had ample opportunity to exploit Nature’s propensity to minimize. Minimization principles form one of the most wide-ranging means of formulating mathe- matical models governing the equilibrium configurations of physical systems. Moreover, many popular numerical integration schemes such as the powerful finite element method are also founded upon a minimization paradigm. In this chapter, we will develop the basic mathematical analysis of nonlinear minimization principles on infinite-dimensional function spaces — a subject known as the “calculus of variations”, for reasons that will be explained as soon as we present the basic ideas. Classical solutions to minimization problems in the calculus of variations are prescribed by boundary value problems involving certain types of differential equations, known as the associated Euler–Lagrange equations. The math- ematical techniques that have been developed to handle such optimization problems are fundamental in many areas of mathematics, physics, engineering, and other applications. In this chapter, we will only have room to scratch the surface of this wide ranging and lively area of both classical and contemporary research. The history of the calculus of variations is tightly interwoven with the history of mathematics. The field has drawn the attention of a remarkable range of mathematical luminaries, beginning with Newton, then initiated as a subject in its own right by the Bernoulli family. The first major developments appeared in the work of Euler, Lagrange and Laplace. In the nineteenth century, Hamilton, Dirichlet and Hilbert are but a few of the outstanding contributors. In modern times, the calculus of variations has continued to occupy center stage, witnessing major theoretical advances, along with wide-ranging applications in physics, engineering and all branches of mathematics. Minimization problems that can be analyzed by the calculus of variations serve to char- acterize the equilibrium configurations of almost all continuous physical systems, ranging through elasticity, solid and fluid mechanics, electro-magnetism, gravitation, quantum me- chanics, string theory, and many, many others. Many geometrical configurations, such as minimal surfaces, can be conveniently formulated as optimization problems. Moreover, nu- merical approximations to the equilibrium solutions of such boundary value problems are based on a nonlinear finite element approach that reduced the infinite-dimensional mini- mization problem to a finite-dimensional problem, to which we can apply the optimization techniques learned in Section 19.3; however, we will not pursue this direction here....
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This note was uploaded on 02/10/2012 for the course MATH 5485 taught by Professor Olver during the Fall '09 term at University of Central Florida.
- Fall '09