Introduction to the Calculus of Variations

Transcription

1 Introduction to the Clculus of Vritions 1. Introduction. by Peter J. Olver University of Minnesot Minimiztion principles form one of the most wide-rnging mens of formulting mthemticl models governing the equilibrium configurtions of physicl systems. Moreover, mny populr numericl integrtion schemes such s the powerful finite element method re lso founded upon minimiztion prdigm. In these notes, we will develop the bsic mthemticl nlysis of nonliner minimiztion principles on infinite-dimensionl function spces subject known s the clculus of vritions, for resons tht will be explined s soon s we present the bsic ides. Clssicl solutions to minimiztion problems in the clculus of vritions re prescribed by boundry vlue problems involving certin types of differentil equtions, known s the ssocited Euler Lgrnge equtions. The mthemticl techniques tht hve been developed to hndle such optimiztion problems re fundmentl in mny res of mthemtics, physics, engineering, nd other pplictions. In this chpter, we will only hve room to scrtch the surfce of this wide rnging nd lively re of both clssicl nd contemporry reserch. The history of the clculus of vritions is tightly interwoven with the history of mthemtics, [9]. The field hs drwn the ttention of remrkble rnge of mthemticl luminries, beginning with Newton nd Leibniz, then initited s subject in its own right by the Bernoulli brothers Jkob nd Johnn. The first mjor developments ppered in the work of Euler, Lgrnge, nd Lplce. In the nineteenth century, Hmilton, Jcobi, Dirichlet, nd Hilbert re but few of the outstnding contributors. In modern times, the clculus of vritions hs continued to occupy center stge, witnessing mjor theoreticl dvnces, long with wide-rnging pplictions in physics, engineering nd ll brnches of mthemtics. Minimiztion problems tht cn be nlyzed by the clculus of vritions serve to chrcterize the equilibrium configurtions of lmost ll continuous physicl systems, rnging through elsticity, solid nd fluid mechnics, electro-mgnetism, grvittion, quntum mechnics, string theory, nd mny, mny others. Mny geometricl configurtions, such s miniml surfces, cn be conveniently formulted s optimiztion problems. Moreover, numericl pproximtions to the equilibrium solutions of such boundry vlue problems re bsed on nonliner finite element pproch tht reduces the infinite-dimensionl minimiztion problem to finite-dimensionl problem. See [14; Chpter 11] for full detils. Just s the vnishing of the grdient of function of severl vribles singles out the criticl points, mong which re the minim, both locl nd globl, so similr functionl grdient will distinguish the cndidte functions tht might be minimizers of the functionl. The finite-dimensionl clculus leds to system of lgebric equtions for the 1/17/16 1 c 2016 Peter J. Olver

2 Figure 1. The Shortest Pth is Stright Line. criticl points; the infinite-dimensionl functionl nlog results boundry vlue problem for nonliner ordinry or prtil differentil eqution whose solutions re the criticl functions for the vritionl problem. So, the pssge from finite to infinite dimensionl nonliner systems mirrors the trnsition from liner lgebric systems to boundry vlue problems. 2. Exmples of Vritionl Problems. The best wy to pprecite the clculus of vritions is by introducing few concrete exmples of both mthemticl nd prcticl importnce. Some of these minimiztion problems plyed key role in the historicl development of the subject. And they still serve s n excellent mens of lerning its bsic constructions. Miniml Curves, Optics, nd Geodesics The miniml curve problem is to find the shortest pth between two specified loctions. In its simplest mnifesttion, we re given two distinct points = (,α) nd b = (b,β) in the plne R 2, (2.1) nd our tsk is to find the curve of shortest length connecting them. Obviously, s you lern in childhood, the shortest route between two points is stright line; see Figure 1. Mthemticlly, then, the minimizing curve should be the grph of the prticulr ffine function y = cx+d = β α (x )+α (2.2) b tht psses through or interpoltes the two points. However, this commonly ccepted fct tht (2.2) is the solution to the minimiztion problem is, upon closer inspection, perhps not so immeditely obvious from rigorous mthemticl stndpoint. Let us see how we might formulte the miniml curve problem in mthemticlly precise wy. For simplicity, we ssume tht the miniml curve is given s the grph of We ssume tht b, i.e., the points,b do not lie on common verticl line. 1/17/16 2 c 2016 Peter J. Olver

3 smooth function y = u(x). Then, the length of the curve is given by the stndrd rc length integrl J[u] = 1+u (x) 2 dx, (2.3) where we bbrevite u = du/dx. The function u(x) is required to stisfy the boundry conditions u() = α, u(b) = β, (2.4) in order tht its grph pss through the two prescribed points (2.1). The miniml curve problem sks us to find the function y = u(x) tht minimizes the rc length functionl (2.3) mong ll resonble functions stisfying the prescribed boundry conditions. The reder might puse to meditte on whether it is nlyticlly obvious tht the ffine function (2.2) is the one tht minimizes the rc length integrl (2.3) subject to the given boundry conditions. One of the motivting tsks of the clculus of vritions, then, is to rigorously prove tht our everydy intuition is indeed correct. Indeed, the word resonble is importnt. For the rc length functionl (2.3) to be defined, the function u(x) should be t lest piecewise C 1, i.e., continuous with piecewise continuous derivtive. Indeed, if we were to llow discontinuous functions, then the stright line (2.2) does not, in most cses, give the minimizer. Moreover, continuous functions which re not piecewise C 1 need not hve well-defined rc length. The more seriously one thinks bout these issues, the less evident the obvious solution becomes. But before you get too worried, rest ssured tht the stright line (2.2) is indeed the true minimizer. However, fully rigorous proof of this fct requires creful development of the mthemticl mchinery of the clculus of vritions. A closely relted problem rises in geometricl optics. The underlying physicl principle, first formulted by the seventeenth century French mthemticin Pierre de Fermt, is tht, when light ry moves through n opticl medium, it trvels long pth tht minimizes the trvel time. As lwys, Nture seeks the most economicl solution. In n inhomogeneous plnr opticl medium, the speed of light, c(x, y), vries from point to point, depending on the opticl properties. Speed equls the time derivtive of distnce trveled, nmely, the rc length of the curve y = u(x) trced by the light ry. Thus, c(x,u(x)) = ds dt = 1+u (x) 2 dx dt. Integrting from strt to finish, we conclude tht the totl trvel time long the curve is equl to T dt b T[u] = dt = 0 dx dx = 1+u (x) 2 dx. (2.5) c(x,u(x)) Fermt s Principle sttes tht, to get from point = (.α) to point b = (b,β), the light ry follows the curve y = u(x) tht minimizes this functionl subject to the boundry conditions u() = α, u(b) = β, Assuming time = money! 1/17/16 3 c 2016 Peter J. Olver

4 b Figure 2. Geodesics on Cylinder. If the medium is homogeneous, e.g., vcuum, then c(x,y) c is constnt, nd T[u] is multiple of the rc length functionl (2.3), whose minimizers re the obvious stright lines trced by the light rys. In n inhomogeneous medium, the pth tken by the light ry is no longer evident, nd we re in need of systemtic method for solving the minimiztion problem. Indeed, ll of the known lws of geometric optics, lens design, focusing, refrction, berrtions, etc., will be consequences of the geometric nd nlytic properties of solutions to Fermt s minimiztion principle, [3]. Another minimiztion problem of similr ilk is to construct the geodesics on curved surfce, mening the curves of miniml length. Given two points,b lying on surfce S R 3, we seek the curve C S tht joins them nd hs the miniml possible length. For exmple, if S is circulr cylinder, then there re three possible types of geodesic curves: stright line segments prllel to the center line; rcs of circles orthogonl to the center line; nd spirl helices, the ltter illustrted in Figure 2. Similrly, the geodesics on sphere re rcs of gret circles. In eronutics, to minimize distnce flown, irplnes follow geodesic circumpolr pths round the globe. However, both of these clims re in need of mthemticl justifiction. In order to mthemticlly formulte the geodesic minimiztion problem, we suppose, for simplicity, tht our surfce S R 3 is relized s the grph of function z = F(x,y). We seek the geodesic curve C S tht joins the given points = (,α,f(,α)), nd b = (b,β,f(b,β)), lying on the surfce S. Let us ssume tht C cn be prmetrized by the x coordinte, in the form y = u(x), z = v(x) = F(x,u(x)), In the bsence of grvittionl effects due to generl reltivity. Cylinders re not grphs, but cn be plced within this frmework by pssing to cylindricl coordintes. Similrly, sphericl surfces re best treted in sphericl coordintes. In differentil geometry, [6], one extends these constructions to rbitrry prmetrized surfces nd higher dimensionl mnifolds. 1/17/16 4 c 2016 Peter J. Olver

5 where the lst eqution ensures tht it lies in the surfce S. In prticulr, this requires b. The length of the curve is supplied by the stndrd three-dimensionl rc length integrl. Thus, to find the geodesics, we must minimize the functionl b ( ) dy 2 ( ) dz 2 J[u] = 1+ + dx dx dx = 1+ ( ) du 2 + dx ( F x (x,u(x))+ F u ) du 2 (x,u(x)) dx, dx (2.6) subject to the boundry conditions u() = α, u(b) = β. For exmple, geodesics on the prboloid z = 1 2 x y2 (2.7) cn be found by minimizing the functionl Miniml Surfces J[u] = 1+(u ) 2 +(x+uu ) 2 dx. (2.8) The miniml surfce problem is nturl generliztion of the miniml curve or geodesicproblem. Initssimplest mnifesttion, weregivensimpleclosedcurvec R 3. The problem is to find the surfce of lest totl re mong ll those whose boundry is the curve C. Thus, we seek to minimize the surfce re integrl re S = ds over ll possible surfces S R 3 with the prescribed boundry curve S = C. Such n re minimizing surfce is known s miniml surfce for short. For exmple, if C is closed plne curve, e.g., circle, then the miniml surfce will just be the plnr region it encloses. But, if the curve C twists into the third dimension, then the shpe of the minimizing surfce is by no mens evident. Physiclly, if we bend wire in the shpe of the curve C nd then dip it into sopy wter, the surfce tension forces in the resulting sop film will cuse it to minimize surfce re,ndhencebeminimlsurfce. Sopfilmsndbubbleshvebeenthesourceofmuch fscintion, physicl, æstheticl nd mthemticl, over the centuries, [9]. The miniml surfce problem is lso known s Plteu s Problem, nmed fter the nineteenth century French physicist Joseph Plteu who conducted systemtic experiments on such sop films. A stisfctory mthemticl solution to even the simplest version of the miniml surfce problem ws only chieved in the mid twentieth century, [11, 12]. Miniml surfces nd S More ccurtely, the sop film will relize locl but not necessrily globl minimum for the surfce re functionl. Non-uniqueness of locl minimizers cn be relized in the physicl experiment the sme wire my support more thn one stble sop film. 1/17/16 5 c 2016 Peter J. Olver

6 C Figure 3. Miniml Surfce. relted vritionl problems remin n ctive re of contemporry reserch, nd re of importnce in engineering design, rchitecture, nd biology, including foms, domes, cell membrnes, nd so on. Let us mthemticlly formulte the serch for miniml surfce s problem in the clculus of vritions. For simplicity, we shll ssume tht the bounding curve C projects down to simple closed curve Γ = tht bounds n open domin R 2 in the (x,y) plne, s in Figure 3. The spce curve C R 3 is then given by z = g(x,y) for (x,y) Γ =. For resonble boundry curves C, we expect tht the miniml surfce S will be described s the grph of function z = u(x,y) prmetrized by (x,y). According to the bsic clculus, the surfce re of such grph is given by the double integrl J[u] = 1+ ( ) u 2 + x ( ) u 2 dxdy. (2.9) y To find the miniml surfce, then, we seek the function z = u(x,y) tht minimizes the surfce re integrl (2.9) when subject to the Dirichlet boundry conditions u(x,y) = g(x,y) for (x,y). (2.10) As we will see, (5.10), the solutions to this minimiztion problem stisfy complicted nonliner second order prtil differentil eqution. A simple version of the miniml surfce problem, tht still contins some interesting fetures, is to find miniml surfces with rottionl symmetry. A surfce of revolution is obtined by revolving plne curve bout n xis, which, for definiteness, we tke to be the x xis. Thus, given two points = (,α), b = (b,β) R 2, the gol is to find the curve y = u(x) joining them such tht the surfce of revolution obtined by revolving the curve round the x-xis hs the lest surfce re. Ech cross-section of the resulting surfce is circle centered on the x xis. The re of such surfce of revolution is given by J[u] = 2π u 1+(u ) 2 dx. (2.11) 1/17/16 6 c 2016 Peter J. Olver

7 We seek minimizer of this integrl mong ll functions u(x) tht stisfy the fixed boundry conditions u() = α, u(b) = β. The miniml surfce of revolution cn be physiclly relized by stretching sop film between two circulr wires, of respective rdius α nd β, tht re held distnce b prt. Symmetry considertions will require the minimizing surfce to be rottionlly symmetric. Interestingly, the revolutionry surfce re functionl (2.11) is exctly the sme s the opticl functionl (2.5) when the light speed t pointisinverselyproportionltoitsdistncefromthehorizontlxis: c(x,y) = 1/(2π y ). Isoperimetric Problems nd Constrints The simplest isoperimetric problem is to construct the simple closed plne curve of fixed lengthlthtencloses thedomin oflrgest re. Inother words, weseek to mximize re = dx dy subject to the constrint length = ds = l, over ll possible domins R 2. Of course, the obvious solution to this problem is tht the curve must be circle whose perimeter is l, whence the nme isoperimetric. Note tht the problem, s stted, does not hve unique solution, since if is mximizing domin, ny trnslted or rotted version of will lso mximize re subject to the length constrint. To mke progress on the isoperimetric problem, let us ssume tht the boundry curve is prmetrized by its rc length, so x(s) = (x(s),y(s)) with 0 s l, subject to the requirement tht ( ) 2 dx + ds ( ) 2 dy = 1. (2.12) ds We cn compute the re of the domin by line integrl round its boundry, re = xdy = l 0 x dy ds, (2.13) ds nd thus we seek to mximize the ltter integrl subject to the rc length constrint (2.12). We lso impose periodic boundry conditions x(0) = x(l), y(0) = y(l), (2.14) tht gurntee tht the curve x(s) closes up. (Techniclly, we should lso mke sure tht x(s) x(s ) for ny 0 s < s < l, ensuring tht the curve does not cross itself.) A simpler isoperimetric problem, but one with less evident solution, is the following. Among ll curves of length l in the upper hlf plne tht connect two points (,0) nd (,0), find the one tht, long with the intervl [,], encloses the region hving the lrgest re. Of course, we must tke l 2, s otherwise the curve will be too short to connect the points. In this cse, we ssume the curve is represented by the grph of non-negtive function y = u(x), nd we seek to mximize the functionl u dx subject to the constrint 1+u 2 dx = l. (2.15) 1/17/16 7 c 2016 Peter J. Olver

8 In the previous formultion (2.12), the rc length constrint ws imposed t every point, wheres here it is mnifested s n integrl constrint. Both types of constrints, pointwise nd integrl, pper in wide rnge of pplied nd geometricl problems. Such constrined vritionl problems cn profitbly be viewed s function spce versions of constrined optimiztion problems. Thus, not surprisingly, their nlyticl solution will require the introduction of suitble Lgrnge multipliers. 3. The Euler Lgrnge Eqution. Even the preceding limited collection of exmples of vritionl problems should lredy convince the reder of the tremendous prcticl utility of the clculus of vritions. Let us now discuss the most bsic nlyticl techniques for solving such minimiztion problems. We will exclusively del with clssicl techniques, leving more modern direct methods the function spce equivlent of grdient descent nd relted methods to more in depth tretment of the subject, [5]. Let us concentrte on the simplest clss of vritionl problems, in which the unknown is continuously differentible sclr function, nd the functionl to be minimized depends upon t most its first derivtive. The bsic minimiztion problem, then, is to determine suitble function y = u(x) C 1 [,b] tht minimizes the objective functionl J[u] = L(x,u,u )dx. (3.1) The integrnd is known s the Lgrngin for the vritionl problem, in honor of Lgrnge. We usully ssume tht the Lgrngin L(x, u, p) is resonbly smooth function of ll three of its (sclr) rguments x,u, nd p, which represents the derivtive u. For exmple, the rc length functionl (2.3) hs Lgrngin function L(x,u,p) = 1+p 2, wheres in the surfce of revolution problem (2.11), L(x,u,p) = 2π u 1+p 2. (In the ltter cse, the points where u = 0 re slightly problemtic, since L is not continuously differentible there.) In order to uniquely specify minimizing function, we must impose suitble boundry conditions. All of the usul suspects Dirichlet (fixed), Neumnn (free), s well s mixed nd periodic boundry conditions re lso relevnt here. In the interests of brevity, we shll concentrte on the Dirichlet boundry conditions u() = α, u(b) = β, (3.2) lthough some of the exercises will investigte other types of boundry conditions. The First Vrition The (locl) minimizers of (sufficiently nice) objective function defined on finitedimensionl vector spce re initilly chrcterized s criticl points, where the objective function s grdient vnishes. An nlogous construction pplies in the infinite-dimensionl context treted by the clculus of vritions. Every sufficiently nice minimizer of sufficiently nice functionl J[ u] is criticl function, Of course, not every criticl point turns out to be minimum mxim, sddles, nd mny degenerte points re lso criticl. 1/17/16 8 c 2016 Peter J. Olver

9 The chrcteriztion of nondegenerte criticl points s locl minim or mxim relies on the second derivtive test, whose functionl version, known s the second vrition, will be is the topic of the following Section 4. But we re getting hed of ourselves. The first order of business is to lern how to compute the grdient of functionl defined on n infinite-dimensionl function spce. The generl definition of the grdient requires tht we first impose n inner product u;v on the underlying function spce. The grdient J[u] of the functionl (3.1) will then be defined by the sme bsic directionl derivtive formul: J[u];v = d dε J[u+εv] ε=0. (3.3) Here v(x) is function tht prescribes the direction in which the derivtive is computed. Clssiclly, v is known s vrition in the function u, sometimes written v = δu, whence the term clculus of vritions. Similrly, the grdient opertor on functionls is often referred to s the vritionl derivtive, nd often written δj. The inner product used in (3.3) is usully tken (gin for simplicity) to be the stndrd L 2 inner product f ;g = f(x)g(x)dx (3.4) on function spce. Indeed, while the formul for the grdient will depend upon the underlying inner product, the chrcteriztion of criticl points does not, nd so the choice of inner product is not significnt here. Now, strting with (3.1), for ech fixed u nd v, we must compute the derivtive of the function h(ε) = J[u+εv] = L(x,u+εv,u +εv )dx. (3.5) Assuming sufficient smoothness of the integrnd llows us to bring the derivtive inside the integrl nd so, by the chin rule, h (ε) = d dε J[u+εv] = d dε L(x,u+εv,u +εv )dx = [ v u (x,u+εv,u +εv )+v p (x,u+εv,u +εv ) Therefore, setting ε = 0 in order to evlute (3.3), we find J[u];v = ] dx. [ v ] u (x,u,u )+v p (x,u,u ) dx. (3.6) The resulting integrl often referred to s the first vrition of the functionl J[u]. The condition J[u];v = 0 for minimizer is known s the wek form of the vritionl principle. 1/17/16 9 c 2016 Peter J. Olver

10 To obtin n explicit formul for J[u], the right hnd side of (3.6) needs to be written s n inner product, J[u];v = J[u]v dx = hvdx between some function h(x) = J[u] nd the vrition v. The first summnd hs this form, but the derivtive v ppering in the second summnd is problemtic. However, one cn esily move derivtives round inside n integrl through integrtion by prts. If we set r(x) p (x,u(x),u (x)), we cn rewrite the offending term s where, gin by the chin rule, r (x) = d ( ) dx p (x,u,u ) r(x)v (x)dx = [ r(b)v(b) r()v() ] r (x)v(x)dx, (3.7) = 2 L x p (x,u,u )+u 2 L u p (x,u,u )+u 2 L p 2 (x,u,u ). (3.8) So fr we hve not imposed ny conditions on our vrition v(x). We re only compring the vlues of J[ u] mong functions tht stisfy the prescribed boundry conditions, nmely u() = α, u(b) = β. Therefore, we must mke sure tht the vried function remins within this set of functions, nd so û(x) = u(x)+εv(x) û() = u()+εv() = α, û(b) = u(b)+εv(b) = β. For this to hold, the vrition v(x) must stisfy the corresponding homogeneous boundry conditions v() = 0, v(b) = 0. (3.9) As result, both boundry terms in our integrtion by prts formul (3.7) vnish, nd we cn write (3.6) s J[u];v = J[u]v dx = Since this holds for ll vritions v(x), we conclude tht J[u] = u (x,u,u ) d dx [ v u (x,u,u ) d ( )] dx p (x,u,u ) dx. ( p (x,u,u ) ). (3.10) 1/17/16 10 c 2016 Peter J. Olver

11 This is our explicit formul for the functionl grdient or vritionl derivtive of the functionl (3.1) with Lgrngin L(x,u,p). Observe tht the grdient J[u] of functionl is function. The criticl functions u(x) re, by definition, those for which the functionl grdient vnishes: stisfy J[u] = u (x,u,u ) d dx p (x,u,u ) = 0. (3.11) In view of (3.8), the criticl eqution (3.11) is, in fct, second order ordinry differentil eqution, E(x,u,u,u ) = u (x,u,u ) 2 L x p (x,u,u ) u 2 L u p (x,u,u ) u 2 L p 2 (x,u,u ) = 0, (3.12) known s the Euler Lgrnge eqution ssocited with the vritionl problem (3.1), in honor of two of the most importnt contributors to the subject. Any solution to the Euler Lgrnge eqution tht is subject to the ssumed boundry conditions forms criticl point for the functionl, nd hence is potentil cndidte for the desired minimizing function. And, in mny cses, the Euler Lgrnge eqution suffices to chrcterize the minimizer without further do. Theorem 3.1. Suppose the Lgrngin function is t lest twice continuously differentible: L(x,u,p) C 2. Then ny C 2 minimizer u(x) to the corresponding functionl J[u] = L(x,u,u )dx, subject to the selected boundry conditions, must stisfy the ssocited Euler Lgrnge eqution (3.11). Let us now investigte wht the Euler Lgrnge eqution tells us bout the exmples of vritionl problems presented t the beginning of this section. One word of cution: there do exist seemingly resonble functionls whose minimizers re not, in fct, C 2, nd hence do not solve the Euler Lgrnge eqution in the clssicl sense; see [2] for exmples. Fortuntely, in most vritionl problems tht rise in rel-world pplictions, such pthologies do not pper. Curves of Shortest Length Plnr Geodesics Let us return to the most elementry problem in the clculus of vritions: finding the curve of shortest length connecting two points = (,α), b = (b,β) R 2 in the plne. As we noted in Section 3, such plnr geodesics minimize the rc length integrl J[u] = subject to the boundry conditions 1+(u ) 2 dx with Lgrngin L(x,u,p) = 1+p 2, u() = α, u(b) = β. 1/17/16 11 c 2016 Peter J. Olver

12 Since u = 0, p = p 1+p 2, the Euler Lgrnge eqution (3.11) in this cse tkes the form 0 = d dx u 1+(u ) = u 2 (1+(u ) 2 ) 3/2. Since the denomintor does not vnish, this is the sme s the simplest second order ordinry differentil eqution u = 0. (3.13) We deduce tht the solutions to the Euler Lgrnge eqution re ll ffine functions, u = cx + d, whose grphs re stright lines. Since our solution must lso stisfy the boundry conditions, the only criticl function nd hence the sole cndidte for minimizer is the stright line y = β α b (x )+α (3.14) pssing through the two points. Thus, the Euler Lgrnge eqution helps to reconfirm our intuition tht stright lines minimize distnce. Be tht s it my, the fct tht function stisfies the Euler Lgrnge eqution nd the boundry conditions merely confirms its sttus s criticl function, nd does not gurntee tht it is the minimizer. Indeed, ny criticl function is lso cndidte for mximizing the vritionl problem, too. The nture of criticl function will be elucidted by the second derivtive test, nd requires some further work. Of course, for the minimum distnce problem, we know tht stright line cnnot mximize distnce, nd must be the minimizer. Nevertheless, the reder should hve smll ngging doubt tht we my not hve completely solved the problem t hnd... Miniml Surfce of Revolution Consider next the problem of finding the curve connecting two points tht genertes surfce of revolution of miniml surfce re. For simplicity, we ssume tht the curve is given by the grph of non-negtive function y = u(x) 0. According to (2.11), the required curve will minimize the functionl J[u] = u 1+(u ) 2 dx, with Lgrngin L(x,u,p) = u 1+p 2, (3.15) where we hve omitted n irrelevnt fctor of 2π nd used positivity to delete the bsolute vlue on u in the integrnd. Since u = 1+p 2, p = up 1+p 2, 1/17/16 12 c 2016 Peter J. Olver

13 the Euler Lgrnge eqution (3.11) is 1+(u ) 2 d dx uu 1+(u ) 2 = 1+(u ) 2 uu (1+(u ) 2 ) 3/2 = 0. (3.16) Therefore, to find the criticl functions, we need to solve nonliner second order ordinry differentil eqution nd not one in fmilir form. Fortuntely, there is little trick we cn use to find the solution. If we multiply the eqution by u, we cn then rewrite the result s n exct derivtive We conclude tht the quntity u [ 1+(u ) 2 uu (1+(u ) 2 ) 3/2 ] = d dx u 1+(u ) 2 = 0. u = c, (3.17) 1+(u ) 2 is constnt, nd so the left hnd side is first integrl for the differentil eqution. Solving for du dx = u2 c u = 2 c results in n utonomous first order ordinry differentil eqution, which we cn immeditely solve: c du u2 c = x+δ, 2 where δ is constnt of integrtion. The most useful form of the left hnd integrl is in terms of the inverse to the hyperbolic cosine function coshz = 1 2 (ez +e z ), whereby ( ) cosh 1 u x+δ = x+δ, nd hence u = c cosh. (3.18) c c In this mnner, we hve produced the generl solution to the Euler Lgrnge eqution (3.16). Any solution tht lso stisfies the boundry conditions provides criticl function for the surfce re functionl (3.15), nd hence is cndidte for the minimizer. The curve prescribed by the grph of hyperbolic cosine function (3.18) is known s ctenry. It is not prbol, even though to the untrined eye it looks similr. Owing to their minimizing properties, ctenries re quite common in engineering design for instnce hnging chin hs the shpe of ctenry, while the rch in St. Louis is n inverted ctenry. So fr, we hve not tken into ccount the boundry conditions It turns out tht there re three distinct possibilities, depending upon the configurtion of the boundry points: Actully, s withmny tricks, this isrellynindictiontht somethingprofound is goingon. Noether s Theorem, result of fundmentl importnce in modern physics tht reltes symmetries nd conservtion lws, [7, 13], underlies the integrtion method. The squre root is rel since, by (3.17), u c. 1/17/16 13 c 2016 Peter J. Olver

14 () There is precisely one vlue of the two integrtion constnts c, δ tht stisfies the two boundry conditions. In this cse, it cn be proved tht this ctenry is the unique curve tht minimizes the re of its ssocited surfce of revolution. (b) There re two different possible vlues of c, δ tht stisfy the boundry conditions. In this cse, one of these is the minimizer, nd the other is spurious solution one tht corresponds to sddle point for the surfce re functionl. (c) There re no vlues of c,δ tht llow (3.18) to stisfy the two boundry conditions. This occurs when the two boundry points,b re reltively fr prt. In this configurtion, the physicl sop film spnning the two circulr wires breks prt into two circulr disks, nd this defines the minimizer for the problem, i.e., unlike cses () nd (b), there is no surfce of revolution tht hs smller surfce re thn the two disks. However, the function tht minimizes this configurtion consists of two verticl lines from the boundry points to the x xis, long with tht segment on the xis lying between them. More precisely, we cn pproximte this function by sequence of genuine functions tht give progressively smller nd smller vlues to the surfce re functionl (2.11), but the ctul minimum is not ttined mong the clss of (smooth) functions. Thus, even in such resonbly simple exmple, number of the subtle complictions cn lredy be seen. Lck of spce precludes more detiled development of the subject, nd we refer the interested reder to more specilized books on the clculus of vritions, including [4, 7, 10]. The Brchistochrone Problem The most fmous clssicl vritionl principle is the so-clled brchistochrone problem. The compound Greek word brchistochrone mens miniml time. An experimenter lets bed slide down wire tht connects two fixed points. The gol is to shpe the wire in such wy tht, strting from rest, the bed slides from one end to the other in miniml time. Nïve guesses for the wire s optiml shpe, including stright line, prbol, circulr rc, or even ctenry re wrong. One cn do better through creful nlysis of the ssocited vritionl problem. The brchistochrone problem ws originlly posed by the Swiss mthemticin Johnn Bernoulli in 1696, nd served s n inspirtion for much of the subsequent development of the subject. We tke, without loss of generlity, the strting point of the bed to be t the origin: = (0,0). The wire will bend downwrds, nd so, to void distrcting minus signs in the subsequent formule, we tke the verticl y xis to point downwrds. The shpe of the wire will be given by the grph of function y = u(x) 0. The end point b = (b,β) is ssumed to lie below nd to the right, nd so b > 0 nd β > 0. The set-up is sketched in Figure 4. To mthemticlly formulte the problem, the first step is to find the formul for the trnsit time of the bed sliding long the wire. Arguing s in our derivtion of the optics Here function must be tken in very brod sense, s this one does not even correspond to generlized function! 1/17/16 14 c 2016 Peter J. Olver

15 Figure 4. The Brchistochrone Problem. functionl (2.5), if v(x) denotes the instntneous speed of descent of the bed when it reches position (x, u(x)), then the totl trvel time is l ds b T[u] = v = 1+(u ) 2 dx, (3.19) 0 v 0 where ds = 1+(u ) 2 dx is the usul rc length element, nd l is the overll length of the wire. We shll use conservtion of energy to determine formul for the speed v s function of the position long the wire. The kinetic energy of the bed is 1 2 mv2, where m is its mss. On the other hnd, due to our sign convention, the potentil energy of the bed when it is t height y = u(x) is mgu(x), where g the grvittionl constnt, nd we tke the initil height s the zero potentil energy level. The bed is initilly t rest, with 0 kinetic energy nd 0 potentil energy. Assuming tht frictionl forces re negligible, conservtion of energy implies tht the totl energy must remin equl to 0, nd hence 0 = 1 2 mv2 mgu. We cn solve this eqution to determine the bed s speed s function of its height: v = 2gu. (3.20) Substituting this expression into (3.19), we conclude tht the shpe y = u(x) of the wire is obtined by minimizing the functionl b 1+(u ) 2 T[u] = dx, (3.21) 2gu subjet to the boundry conditions The ssocited Lgrngin is 0 u(0) = 0, u(b) = β. (3.22) L(x,u,p) = 1+p 2 u, 1/17/16 15 c 2016 Peter J. Olver

16 where we omit n irrelevnt fctor of 2g (or dopt physicl units in which g = 1 2 ). We compute 1+p u = 2, 2u 3/2 p = p u(1+p2 ). Therefore, the Euler Lgrnge eqution for the brchistochrone functionl is 1+(u ) 2 2u 3/2 d dx u u(1+(u ) 2 ) = 2uu +(u ) u(1+(u ) 2 ) = 0. (3.23) Thus, the minimizing functions solve the nonliner second order ordinry differentil eqution 2uu +(u ) 2 +1 = 0. Rther thn try to solve this differentil eqution directly, we note tht the Lgrngin does not depend upon x, nd therefore we cn use the following result. Theorem 3.2. Suppose the Lgrngin L(x,u,p) = L(u,p) does not depend on x. Then the Hmiltonin function H(u,u ) = L(u,u ) u p (u,u ) (3.24) is first integrl for the Euler Lgrnge eqution, mening tht it is constnt on ech solution. Proof: Differentiting (3.24), we find d dx H(u,u ) = d [ L(u,u ) u ] ( dx p (u,u ) = u u (u,u ) d dx ) p (u,u ) = 0, which vnishes s consequence of the Euler Lgrnge eqution (3.11). This implies tht H(u,u ) = k, (3.25) where k is constnt, whose vlue cn depend upon the solution u(x). Eqution (3.25) hs the form of n implicitly defined first order ordinry differentil eqution which cn, in fct, be integrted. Indeed, solving for u = h(u,k) produces n utonomous first order differentil eqution, whose generl solution cn be obtined by integrtion. Q.E.D. Remrk: This result is specil cse of Noether s powerful Theorem, [13; Chpter 4], tht reltes symmetries of vritionl problems in this cse trnsltions in the x coordinte with first integrls,.k.. conservtion lws. 1/17/16 16 c 2016 Peter J. Olver

17 Figure 5. A Cycloid. In our cse, the Hmiltonin function (3.24) is defines first integrl. Thus, H(x,u,u ) = H(x,u,p) = L p p = 1 u(1+p2 ) 1 u(1+(u ) 2 ) = k, which we rewrite s u(1+(u ) 2 ) = c, where c = 1/k 2 is constnt. (This cn be checked by directly clculting dh/dx 0.) Solving for the derivtive u results in the first order utonomous ordinry differentil eqution du c u dx =. u This eqution cn be explicitly solved by seprtion of vribles, nd so u du = x+δ c u for some constnt δ. The left hnd integrtion relies on the trigonometric substitution whereby x+δ = c 2 u = 1 2 c(1 cosθ), 1 cosθ 1+cosθ sinθdθ = c 2 (1 cosθ)dθ = 1 c(θ sinθ). 2 The left hnd boundry condition implies δ = 0, nd so the solution to the Euler Lgrnge eqution re curves prmetrized by x = r(θ sinθ), u = r(1 cosθ). (3.26) With little more work, it cn be proved tht the prmeter r = 1 2c is uniquely prescribed by the right hnd boundry condition, nd moreover, the resulting curve supplies the globl minimizer of the brchistochrone functionl, [7]. The miniming curve is known s cycloid, which cn be visulized s the curve trced by point sitting on the edge of rolling wheel of rdius r, s plotted in Figure 5. Interestingly, in certin configurtions, nmely if β < 2b/π, the cycloid tht solves the brchistochrone problem dips below the right hnd endpoint b = (b,β), nd so the bed is moving upwrds when it reches the end of the wire. 1/17/16 17 c 2016 Peter J. Olver

18 4. The Second Vrition. The solutions to the Euler Lgrnge boundry vlue problem re the criticl functions for the vritionl principle, mening tht they cuse the functionl grdient to vnish. For finite-dimensionl optimiztion problems, being criticl point is only necessry condition for minimlity. One must impose dditionl conditions, bsed on the second derivtive of the objective function t the criticl point, in order to gurntee tht it is minimum nd not mximum or sddle point. Similrly, in the clculus of vritions, the solutions to the Euler Lgrnge eqution my lso include (locl) mxim, s well s other non-extreml criticl functions. To distinguish between the possibilities, we need to formulte second derivtive test for the objective functionl. In the clculus of vritions, the second derivtive of functionl is known s its second vrition, nd the gol of this section is to construct nd nlyze it in its simplest mnifesttion. For finite-dimensionl objective function F(u 1,...,u n ), the second derivtive test ws bsed on the positive definiteness of its Hessin mtrix. The justifiction ws bsed on the second order Tylor expnsion of the objective function t the criticl point. In n nlogous fshion, we expnd n objective functionl J[ u] ner the criticl function. Consider the sclr function h(ε) = J[u+εv], where the function v(x) represents vrition. The second order Tylor expnsion of h(ε) tkes the form h(ε) = J[u+εv] = J[u]+εK[u;v]+ 1 2 ε2 Q[u;v]+. The first order terms re liner in the vrition v, nd, ccording to our erlier clcultion, given by the inner product h (0) = K[u;v] = J[u];v between the vrition nd the functionl grdient. In prticulr, if u is criticl function, then the first order terms vnish, K[u;v] = J[u];v = 0, for ll llowble vritions v, mening those tht stisfy the homogeneous boundry conditions. Therefore, the nture of the criticl function u minimum, mximum, or neither will, in most cses, be determined by the second derivtive terms h (0) = Q[u;v]. Now, if u is minimizer, then Q[u;v] 0. Conversely, if Q[u;v] > 0 for ll v 0, i.e., the second vrition is positive definite, then the criticl function u will be strict locl minimizer. This forms the crux of the second derivtive test. Let us explicitly evlute the second vritionl for the simplest functionl (3.1). Consider the sclr function h(ε) = J[u+εv] = L(x,u+εv,u +εv )dx, 1/17/16 18 c 2016 Peter J. Olver

19 whose first derivtive h (0) ws lredy determined in (3.6); here we require the second vrition Q[u;v] = h [ (0) = Av 2 +2Bvv +C(v ) 2] dx, (4.1) where the coefficient functions A(x) = 2 L u 2 (x,u,u ), B(x) = 2 L u p (x,u,u ), C(x) = 2 L p 2 (x,u,u ), (4.2) re found by evluting certin second order derivtives of the Lgrngin t the criticl function u(x). In contrst to the first vrition, integrtion by prts will not eliminte ll of the derivtives on v in the qudrtic functionl(4.1), which cuses significnt complictions in the ensuing nlysis. The second derivtive test for minimizer relies on the positivity of the second vrition. So, in order to formulte conditions tht the criticl function be minimizer for the functionl, we need to estblish criteri gurnteeing the positive definiteness of such qudrtic functionl, mening tht Q[ u; v] > 0 for ll non-zero llowble vritions v(x) 0. Clerly, if the integrnd is positive definite t ech point, so A(x)v 2 +2B(x)vv +C(x)(v ) 2 > 0 whenever < x < b, nd v(x) 0, (4.3) then Q[u;v] > 0 is lso positive definite. Exmple 4.1. For the rc length minimiztion functionl (2.3), the Lgrngin is L(x,u,p) = 1+p 2. To nlyze the second vrition, we first compute 2 L u 2 = 0, For the criticl stright line function we find 2 L u p = 0, 2 L p 2 = 1 (1+p 2 ) 3/2. u(x) = β α b (x )+α, with p = u (x) = β α b, A(x) = 2 L u 2 = 0, B(x) = 2 L u p = 0, C(x) = 2 L p 2 = (b ) 3 [ (b )2 +(β α) 2] 3/2 c. Therefore, the second vrition functionl (4.1) is Q[u;v] = c(v ) 2 dx, where c > 0 is positive constnt. Thus, Q[u;v] = 0 vnishes if nd only if v(x) is constnt function. But the vrition v(x) is required to stisfy the homogeneous boundry conditions v() = v(b) = 0, nd hence Q[u;v] > 0 for ll llowble nonzero vritions. Therefore, we conclude tht the stright line is, indeed, (locl) minimizer for the rc length functionl. We hve t lst rigorously justified our intuition tht the shortest distnce between two points is stright line! 1/17/16 19 c 2016 Peter J. Olver

20 However, s the following exmple demonstrtes, the pointwise positivity condition (4.3) is overly restrictive. Exmple 4.2. Consider the qudrtic functionl Q[v] = 1 0 [ (v ) 2 v 2] dx. (4.4) We clim tht Q[v] > 0 for ll nonzero v 0 subject to homogeneous Dirichlet boundry conditions v(0) = 0 = v(1). This result is not trivil! Indeed, the boundry conditions ply n essentil role, since choosing v(x) c 0 to be ny constnt function will produce negtive vlue for the functionl: Q[v] = c 2. To prove the clim, consider the qudrtic functionl Q[v] = 1 0 (v +vtnx) 2 dx 0, which is clerly non-negtive, since the integrnd is everywhere 0. Moreover, by continuity, the integrl vnishes if nd only if v stisfies the first order liner ordinry differentil eqution v +vtnx = 0, for ll 0 x 1. The only solution tht lso stisfies boundry condition v(0) = 0 is the trivil one v 0. We conclude tht Q[v] = 0 if nd only if v 0, nd hence Q[v] is positive definite qudrtic functionl on the spce of llowble vritions. Let us expnd the ltter functionl, Q[v] = = [ (v ) 2 +2vv tnx+v 2 tn 2 x ] dx [ (v ) 2 v 2 (tnx) +v 2 tn 2 x ] 1 [ dx = (v ) 2 v 2] dx = Q[v]. In the second equlity, we integrted the middle term by prts, using (v 2 ) = 2vv, nd noting tht the boundry terms vnish owing to our imposed boundry conditions. Since Q[v] is positive definite, so is Q[v], justifying the previous clim. To pprecite how subtle this result is, consider the lmost identicl qudrtic functionl 4 [ Q[v] = (v ) 2 v 2] dx, (4.5) 0 the only difference being the upper limit of the integrl. A quick computtion shows tht the function v(x) = x(4 x) stisfies the boundry conditions v(0) = 0 = v(4), but Q[v] = 4 0 [ (4 2x) 2 x 2 (4 x) 2] dx = 64 5 < 0. Therefore, Q[v] is not positive definite. Our preceding nlysis does not pply becuse the function tnx becomes singulr t x = 1 2π, nd so the uxiliry integrl 4 0 (v +vtnx) 2 dx does not converge. 1/17/16 20 c 2016 Peter J. Olver 0

21 The complete nlysis of positive definiteness of qudrtic functionls is quite subtle. Indeed, the strnge ppernce of tnx in the preceding exmple turns out to be n importnt clue! In the interests of brevity, let us just stte without proof fundmentl theorem, nd refer the interested reder to [7] for full detils. Theorem 4.3. Let A(x),B(x),C(x) C 0 [,b] be continuous functions. The qudrtic functionl [ Q[v] = Av 2 +2Bvv +C(v ) 2] dx ispositivedefinite, so Q[v] > 0for llv 0stisfyingthehomogeneousDirichlet boundry conditions v() = v(b) = 0, provided () C(x) > 0 for ll x b, nd (b) For ny < c b, the only solution to its liner Euler Lgrnge boundry vlue problem (Cv ) +(A B )v = 0, v() = 0 = v(c), (4.6) is the trivil function v(x) 0. Remrk: A vlue c for which (4.6) hs nontrivil solution is known s conjugte point to. Thus, condition (b) cn be restted tht the vritionl problem hs no conjugte points in the intervl (,b]. Exmple 4.4. The qudrtic functionl hs Euler Lgrnge eqution Q[v] = 0 [ (v ) 2 v 2] dx (4.7) v v = 0. The solutions v(x) = ksinx stisfy the boundry condition v(0) = 0. The first conjugte point occurs t c = π where v(π) = 0. Therefore, Theorem 4.3 implies tht the qudrtic functionl (4.7) is positive definite provided the upper integrtion limit b < π. This explins why the originl qudrtic functionl (4.4) is positive definite, since there re no conjugte points on the intervl [0,1], while the modified version (4.5) is not, becuse the first conjugte point π lies on the intervl (0,4]. In the cse when the qudrtic functionl rises s the second vrition of functionl (3.1), the coefficient functions A,B,C re given in terms of the Lgrngin L(x,u,p) by formule (4.2). In this cse, the first condition in Theorem 4.3 requires 2 L p 2 (x,u,u ) > 0 (4.8) for the minimizer u(x). This is known s the Legendre condition. The second, conjugte point condition requires tht the so-clled liner vritionl eqution d ( 2 L dx p 2 (x,u,u ) dv ) ( 2 L + dx u 2 (x,u,u ) d 2 ) L dx u p (x,u,u ) v = 0 (4.9) 1/17/16 21 c 2016 Peter J. Olver

22 hs no nontrivil solutions v(x) 0 tht stisfy v() = 0 nd v(c) = 0 for < c b. In this wy, we hve rrived t rigorous form of the second derivtive test for the simplest functionl in the clculus of vritions. Theorem 4.5. If the function u(x) stisfies the Euler Lgrnge eqution (3.11), nd, in ddition, the Legendre condition (4.8) nd there re no conjugte points on the intervl, then u(x) is strict locl minimum for the functionl. 5. Multi-dimensionl Vritionl Problems. The clculus of vritions encompsses very brod rnge of mthemticl pplictions. The methods of vritionl nlysis cn be pplied to n enormous vriety of physicl systems, whose equilibrium configurtions inevitbly minimize suitble functionl, which, typiclly, represents the potentil energy of the system. Minimizing configurtions pper s criticl functions t which the functionl grdient vnishes. Following similr computtionl procedures s in the one-dimensionl clculus of vritions, we find tht the criticl functions re chrcterized s solutions to system of prtil differentil equtions, known s the Euler Lgrnge equtions ssocited with the vritionl principle. Ech solution to the boundry vlue problem specified by the Euler Lgrnge equtions subject to pproprite boundry conditions is, thus, cndidte minimizer for the vritionl problem. In mny pplictions, the Euler Lgrnge boundry vlue problem suffices to single out the physiclly relevnt solutions, nd one need not press on to the considerbly more difficult second vrition. Implementtion of the vritionl clculus for functionls in higher dimensions will be illustrted by looking t specific exmple first order vritionl problem involving single sclr function of two vribles. Once this is fully understood, generliztions nd extensions to higher dimensions nd higher order Lgrngins re redily pprent. Thus, we consider n objective functionl J[u] = L(x,y,u,u x,u y ) dxdy, (5.1) hving the form of double integrl over prescribed domin R 2. The Lgrngin L(x,y,u,p,q) is ssumed to be sufficiently smooth function of its five rguments. Our gol is to find the function(s) u = f(x,y) tht minimize the vlue of J[u] when subject to set of prescribed boundry conditions on, the most importnt being our usul Dirichlet, Neumnn, or mixed boundry conditions. For simplicity, we concentrte on the Dirichlet boundry vlue problem, nd require tht the minimizer stisfy The First Vrition u(x,y) = g(x,y) for (x,y). (5.2) The bsic necessry condition for n extremum (minimum or mximum) is obtined in precisely the sme mnner s in the one-dimensionl frmework. Consider the sclr function h(ε) J[u+εv] = L(x,y,u+εv,u x +εv x,u y +εv y ) dxdy 1/17/16 22 c 2016 Peter J. Olver

23 depending on ε R. The vrition v(x, y) is ssumed to stisfy homogeneous Dirichlet boundry conditions v(x,y) = 0 for (x,y), (5.3) to ensure tht u+εv stisfies the sme boundry conditions (5.2) s u itself. Under these conditions, if u is minimizer, then the sclr function h(ε) will hve minimum t ε = 0, nd hence h (0) = 0. When computing h (ε), we ssume tht the functions involved re sufficiently smooth so s to llow us to bring the derivtive inside the integrl, nd then pply the chin rule. At ε = 0, the result is h (0) = d ( dε ε=0 J[u+εv] = v ) u +v x p +v y dxdy, (5.4) q where the derivtives of L re ll evluted t x,y,u,u x,u y. To identify the functionl grdient, we need to rewrite this integrl in the form of n inner product: h (0) = J[u];v = h(x,y)v(x,y)dxdy, where h = J[u]. To convert (5.4) into this form, we need to remove the offending derivtives from v. In two dimensions, the requisite integrtion by prts formul is bsed on Green s Theorem: ( v x w 1 + v ) ( y w w1 2 dxdy = v( w 2 dx+w 1 dy) v x + w ) 2 dxdy, y (5.5) in which w 1,w 2 re rbitrry smooth functions. Setting w 1 = p, w 2 =, we find q ( ) v x p +v y dxdy = q [ ( ) v + x p y ( q )] dxdy, where the boundry integrl vnishes owing to the boundry conditions (5.3) tht we impose on the llowed vritions. Substituting this result bck into (5.4), we conclude tht [ h (0) = v u ( ) ( )] dxdy = J[u];v, (5.6) x p y q where J[u] = u ( ) ( ) x p y q is the desired first vrition or functionl grdient. Since the grdient vnishes t criticl function, we conclude tht the minimizer u(x, y) must stisfy the Euler Lgrnge eqution u (x,y,u,u x,u y ) ( ) x p (x,y,u,u x,u y ) ( ) y q (x,y,u,u x,u y ) = 0. (5.7) 1/17/16 23 c 2016 Peter J. Olver

24 Once we explicitly evlute the derivtives, the net result is second order prtil differentil eqution L u L xp L yq u x L up u y L uq u xx L pp 2u xy L pq u yy L qq = 0, (5.8) where we use subscripts to indicte derivtives of both u nd L, the ltter being evluted t x,y,u,u x,u y. Exmple 5.1. As first elementry exmple, consider the Dirichlet minimiztion problem ( ) 1 J[u] = 2 u 2 x +u 2 y dxdy. (5.9) In this cse, the ssocited Lgrngin is L = 1 2 (p2 +q 2 ), with u = 0, Therefore, the Euler Lgrnge eqution (5.7) becomes p = p = u x, x (u x ) y (u y ) = u xx u yy = u = 0, q = q = u y. which is the two-dimensionl Lplce eqution. Subject to the selected boundry conditions, the solutions, i.e., the hrmonic functions, re criticl functions for the Dirichlet vritionl principle. However, the clculus of vritions pproch, s developed so fr, leds to much weker result since it only singles out the hrmonic functions s cndidtes for minimizing the Dirichlet integrl; they could just s esily be mximizing functions or sddle points. When deling with qudrtic vritionl problem, the direct lgebric pproch is, when pplicble, the more powerful, since it ssures us tht the solutions to the Lplce eqution relly do minimize the integrl mong the spce of functions stisfying the pproprite boundry conditions. However, the direct method is restricted to qudrtic vritionl problems, whose Euler Lgrnge equtions re liner prtil differentil equtions. In nonliner cses, one relly does need to utilize the full power of the vritionl mchinery. Exmple 5.2. Let us derive the Euler Lgrnge eqution for the miniml surfce problem. From (2.9), the surfce re integrl J[u] = 1+u 2 x +u 2 y dxdy hs Lgrngin L = 1+p 2 +q 2. Note tht u = 0, p = p 1+p2 +q 2, q = q 1+p2 +q 2. Therefore, replcing p u x nd q u y nd then evluting the derivtives, the Euler Lgrnge eqution (5.7) becomes x u x 1+u 2 x +u 2 y y u y 1+u 2 x +u 2 y = (1+u2 y )u xx +2u x u y u xy (1+u2 x )u yy (1+u 2 x +u2 y )3/2 = 0. 1/17/16 24 c 2016 Peter J. Olver

25 Thus, surfce described by the grph of function u = f(x,y) is criticl function, nd hence cndidte for minimizing surfce re, provided it stisfies the miniml surfce eqution (1+u 2 y )u xx 2u x u y u xy +(1+u2 x )u yy = 0. (5.10) We re confronted with complicted, nonliner, second order prtil differentil eqution, which hs been the focus of some of the most sophisticted nd deep nlysis over the pst two centuries, with significnt progress on understnding its solution only within the pst 70 yers. In this book, we hve not developed the sophisticted nlyticl, geometricl, nd numericl techniques tht re required to hve nything of substnce to sy bout its solutions. We refer the interested reder to the dvnced texts [11, 12] for further developments in this fscinting problem. Exmple 5.3. The deformtions of n elstic body R n re described by the displcement field, u: R n. Ech mteril point x in the undeformed body will move to new position y = x+u(x) in the deformed body = {y = x+u(x) x }. The one-dimensionl cse governs brs, bems nd rods, two-dimensionl bodies include thin pltes nd shells, while n = 3 for fully three-dimensionl solid bodies. See [1,8] for detils nd physicl derivtions. For smll deformtions, we cn use liner theory to pproximte the much more complicted equtions of nonliner elsticity. The simplest cse is tht of homogeneous nd isotropic plnr body R 2. The equilibrium mechnics re described by the deformtion function u(x) = (u(x,y),v(x,y)). A detiled physicl nlysis of the constitutive ssumptions leds to minimiztion principle bsed on the following functionl: [ J[u,v] = 1 2 µ u (λ+µ)( u)2] dxdy (5.11) [( = 1 2 λ+µ) (u 2 x +v2 y )+ 1 2 µ(u2 y +v2 x )+(λ+µ)u x v ] y dxdy. The prmeters λ,µ re known s the Lmé moduli of the mteril, nd govern its intrinsic elstic properties. They re mesured by performing suitble experiments on smple of the mteril. Physiclly, (5.11) represents the stored (or potentil) energy in the body under the prescribed displcement. Nture, s lwys, seeks the displcement tht will minimize the totl energy. To compute the Euler Lgrnge equtions, we consider the functionl vrition h(ε) = J[u+εf,v +εg], in which the individul vritions f, g re rbitrry functions subject only to the given homogeneous boundry conditions. If u,v minimize J, then h(ε) hs minimum t ε = 0, nd so we re led to compute h (0) = J ;f = (f u J +g v J)dxdy, 1/17/16 25 c 2016 Peter J. Olver

26 which we write s n inner product (using the stndrd L 2 inner product between vector fields) between the vrition f nd the functionl grdient J = ( u J, v J ). For the prticulr functionl (5.11), we find [( ) h (0) = λ+2µ (ux f x +v y g y )+µ(u y f y +v x g x )+(λ+µ)(u x g y +v y f x ) ] dxdy. We use the integrtion by prts formul (5.5) to remove the derivtives from the vritions f, g. Discrding the boundry integrls, which re used to prescribe the llowble boundry conditions, we find ( [ ] ) (λ+2µ)uxx h +µu yy +(λ+µ)v xy f (0) = + [ ] dxdy. (λ+µ)u xy +µv xx +(λ+2µ)v yy g The two terms in brckets give the two components of the functionl grdient. Setting them equl to zero, we derive the second order liner system of Euler Lgrnge equtions (λ+2µ)u xx +µu yy +(λ+µ)v xy = 0, (λ+µ)u xy +µv xx +(λ+2µ)v yy = 0, known s Nvier s equtions, which cn be compctly written s (5.12) µ u+(µ+λ) ( u) = 0 (5.13) for the displcement vector u = (u,v). The solutions to re the criticl displcements tht, under pproprite boundry conditions, minimize the potentil energy functionl. Since we re deling with qudrtic functionl, more detiled lgebric nlysis will demonstrte tht the solutions to Nvier s equtions re the minimizers for the vritionl principle (5.11). Although only vlid in limited rnge of physicl nd kinemticl conditions, the solutions to the plnr Nvier s equtions nd its three-dimensionl counterprt re successfully used to model wide clss of elstic mterils. In generl, the solutions to the Euler Lgrnge boundry vlue problem re criticl functions for the vritionl problem, nd hence include ll (smooth) locl nd globl minimizers. Determintion of which solutions re genuine minim requires further nlysis of the positivity properties of the second vrition, which is beyond the scope of our introductory tretment. Indeed, complete nlysis of the positive definiteness of the second vrition of multi-dimensionl vritionl problems is quite complicted, nd still wits completely stisfctory resolution! 1/17/16 26 c 2016 Peter J. Olver

27 References [1] Antmn, S.S., Nonliner Problems of Elsticity, Appl. Mth. Sci., vol. 107, Springer Verlg, New York, [2] Bll, J.M., nd Mizel, V.J., One-dimensionl vritionl problem whose minimizers do not stisfy the Euler-Lgrnge eqution, Arch. Rt. Mech. Anl. 90 (1985), [3] Born, M., nd Wolf, E., Principles of Optics, Fourth Edition, Pergmon Press, New York, [4] Cournt, R., nd Hilbert, D., Methods of Mthemticl Physics, vol. I, Interscience Publ., New York, [5] Dcorogn, B., Introduction to the Clculus of Vritions, Imperil College Press, London, [6] do Crmo, M.P., Differentil Geometry of Curves nd Surfces, Prentice-Hll, Englewood Cliffs, N.J., [7] Gel fnd, I.M., nd Fomin, S.V., Clculus of Vritions, Prentice Hll, Inc., Englewood Cliffs, N.J., [8] Gurtin, M.E., An Introduction to Continuum Mechnics, Acdemic Press, New York, [9] Hildebrndt, S., nd Tromb, A., Mthemtics nd Optiml Form, Scientific Americn Books, New York, [10] Kot, M., A First Course in the Clculus of Vritions, Americn Mthemticl Society, Providence, R.I., [11] Morgn, F., Geometric Mesure Theory: Beginner s Guide, Acdemic Press, New York, [12] Nitsche, J.C.C., Lectures on Miniml Surfces, Cmbridge University Press, Cmbridge, [13] Olver, P.J., Applictions of Lie Groups to Differentil Equtions, 2nd ed., Grdute Texts in Mthemtics, vol. 107, Springer Verlg, New York, [14] Olver, P.J., Introduction to Prtil Differentil Equtions, Undergrdute Texts in Mthemtics, Springer Verlg, New York, to pper. 1/17/16 27 c 2016 Peter J. Olver