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Chapter 21 the Calculus of Variations

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Chapter 21 the Calculus of Variations Chapter 21 The Calculus of Variations We have already had ample encounters with Nature's propensity to optimize. Min- imization principles form one of the most powerful tools for formulating mathematical models governing the equilibrium con¯gurations of physical systems. Moreover, the de- sign of numerical integration schemes such as the powerful ¯nite element method are also founded upon a minimization paradigm. This chapter is devoted to the mathematical analysis of minimization principles on in¯nite-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. Solutions to minimization problems in the calculus of variations lead to boundary value problems for ordinary and partial di®erential equations. Numerical solutions are primarily based upon a nonlinear version of the ¯nite element method. The methods developed to handle such problems prove to be fundamental in many areas of mathematics, physics, engineering, and other applications. The history of the calculus of variations is tightly interwoven with the history of calcu- lus, and has merited the attention of a remarkable range of mathematicians, beginning with Newton, then developed as a ¯eld of mathematics in its own right by the Bernoulli family. The ¯rst 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 in research, including major theoretical advances, along with wide-ranging applica- tions in physics, engineering and all branches of mathematics. In this chapter, we will only have time to scratch the surface of the vast area of classical and contemporary research. Minimization problems amenable to the methods of the calculus of variations serve to characterize the equilibrium con¯gurations of almost all continuous physical systems, rang- ing through elasticity, solid and fluid mechanics, electro-magnetism, gravitation, quantum mechanics, and many, many others. Many geometrical systems, such as minimal surfaces, can be conveniently formulated as optimization problems. Moreover, numerical approxima- tions to the equilibrium solutions of such boundary value problems are based on a nonlinear ¯nite element approach that reduced the in¯nite-dimensional minimization problem to a ¯nite-dimensional problem, to which we can apply the optimization techniques learned in Section 19.3. We have already treated the simplest problems in the calculus of variations. As we learned in Chapters 11 and 15, minimization of a quadratic functional requires solving an associated boundary value problem for a linear di®erential equation. Just as the vanishing of the gradient of a function of several variables singles out the critical points, among which are the minima, both local and global, so a similar \functional gradient" will distinguish 2/25/04 925 c 2004 Peter J. Olver ° Figure 21.1. The Shortest Path is a Straight Line. the candidate functions that might be minimizers of the functional. The ¯nite-dimensional gradient leads to a system of algebraic equations; the functional gradient leads to a bound- ary value problem for a nonlinear ordinary or partial di®erential equation. Thus, the passage from ¯nite to in¯nite dimensional nonlinear systems mirrors the transition from linear algebraic systems to boundary value problems. 21.1. Examples of Variational Problems. The best way to introduce the subject is to introduce some concrete examples of both mathematical and practical importance. These particular minimization problems played a key role in the historical development of the calculus of variations. And they still serve as an excellent motivation for learning its basic constructions. Minimal Curves and Geodesics The minimal curve problem is to ¯nd the shortest path connecting two points. In its simplest manifestation, we are given two distinct points a = (a; ®) and b = (b; ¯) in the plane R2. (21:1) Our goal is to ¯nd the curve of shortest length connecting them. \Obviously", as you learn in childhood, the shortest path between two points is a straight line; see Figure 21.1. Mathematically, then, the minimizing curve we are after should be given as the graph of the particular a±ne functiony ¯ ® y = c x + d = ¡ (x a) + ® (21:2) b a ¡ ¡ passing through the two points. However, this commonly accepted \fact" that (21.2) is the solution to the minimization problem is, upon closer inspection, perhaps not so immediately obvious from a rigorous mathematical standpoint. y We assume that a =6 b, i.e., the points a; b do not lie on a common vertical line. 2/25/04 926 c 2004 Peter J. Olver ° Let us see how we might properly formulate the minimal curve problem. Let us assume that the minimal curve is given as the graph of a smooth function y = u(x). Then, according to (A.27), the length of the curve is given by the standard arc length integral b J[u] = 1 + (u0)2 dx; (21:3) a Z p where we abbreviate u0 = du=dx. The function is required to satisfy the boundary condi- tions u(a) = ®; u(b) = ¯; (21:4) in order that its graph pass through the two prescribed points (21.1). The minimal curve problem requires us to ¯nd the function y = u(x) that minimizes the arc length functional (21.3) among all reasonable functions satisfying the prescribed boundary conditions. The student should pause to reflect on whether it is mathematically obvious that the a±ne function (21.2) is the one that minimizes the arc length integral (21.3) subject to the given boundary conditions. One of the motivating tasks of the calculus of variations, then, is to rigorously prove that our childhood intuition is indeed correct. Indeed, the word \reasonable" is important. For the arc length functional to be de¯ned, the function u(x) should be at least piecewise C1, i.e., continuous with a piecewise continuous derivative. If we allow discontinuous functions, then the straight line (21.2) does not, in most cases, give the minimizer; see Exercise . Moreover, continuous functions which are not piecewise C1 may not have a well-de¯ned length. The more seriously one thinks about these issues, the less evident the solution becomes. But, rest assured that the \obvious" solution (21.2) does indeed turn out to be the true minimizer. However, a fully rigorous mathematical proof of this fact requires a proper development of the calculus of variations machinery. A closely related problem arises in optics. The general principle, ¯rst formulated by the seventeenth century French mathematician Pierre de Fermat, is that when a light ray moves through an optical medium, e.g., a vacuum, it travels along a path that will minimize the travel time. As always, Nature seeks the most economical solution! Let c(x; y) denote the speed of light at each point in the mediumy. Speed is equal to the time derivative of distance traveled, namely, the arc length (21.3) of the curve y = u(x) traced by the light ray. Thus, ds dx c(x; u(x)) = = 1 + u0(x)2 : dt dt Integrating from start to ¯nish, we conclude pthat the total travel time of the light ray is equal to T b dt b 1 + u0(x)2 T [u] = dt = dx = dx: (21:5) dx c(x; u(x)) Z0 Za Za p Fermat's Principle states that, to get from one point to another, the light ray follows the curve y = u(x) that minimizes this functional. If the medium is homogeneous, then y For simplicity, we only consider the two-dimensional case here. 2/25/04 927 c 2004 Peter J. Olver ° Figure 21.2. Geodesics on a Cylinder. c(x; y) c is constant, and T [u] equals a multiple of the arc length functional, whose minimizers´ are the \obvious" straight lines. In an inhomogeneous medium, the path taken by the light ray is no longer evident, and we are in need of a systematic method for solving the minimization problem. All of the known laws of optics and lens design, governing focusing, refraction, etc., all follow as consequences of the minimization principle, [optics]. Another problem of a similar ilk is to construct the geodesics on a curved surface, meaning the curves of minimal length. In other words, given two points a; b on a surface S R3, we seek the curve C S that joins them and has the minimal possible length. For½example, if S is a circular½cylinder, then the geodesic curves turn out to be straight lines parallel to the center line, circles orthogonal to the center line, and spiral helices; see Figure 21.2 for an illustration. Similarly, the geodesics on a sphere are arcs of great circles; these include the circumpolar paths followed by airplanes around the globe. However, both of these claims are in need of rigorous justi¯cation. In order to mathematically formulate the geodesic problem, we suppose, for simplicity, that our surface S R3 is realized as the graphy of a function z = F (x; y). We seek the geodesic curve C ½S that joins the given points ½ a = (a; ®; F (a; ®)); and b = (b; ¯; F (b; ¯)); on the surface S: Let us assume that C can be parametrized by the x coordinate, in the form y = u(x); z = F (x; u(x)): In particular, this requires a = b. The length of the curve is given by the usual arc length integral (B.17), and so we must6 minimize the functional b dy 2 dz 2 J[u] = 1 + + dx dx dx Za s µ ¶ µ ¶ b du 2 @F @F du 2 = 1 + + (x; u(x)) + (x; u(x)) dx; dx @x @u dx Za s µ ¶ µ ¶ y Cylinders are not graphs, but can be placed within this framework by passing to cylindrical coordinates.
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