Mathematical Background of Variational Calculus

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1 A Mthemticl Bckground of Vritionl Clculus A 1

2 Apendix A: MATHEMATICAL BACKGROUND OF VARIATIONAL CALCULUS TABLE OF CONTENTS Pge A.1. Vector Spces A 3 A.1.1. Vector Spce Properties A 3 A.1.2. Vector Spce Exmples A 3 A.1.3. Normed Vector Spces A 4 A.1.4. One-Forms A 4 A.1.5. Function Spces In 1D A 5 A.1.6. More Generl Function Spces A 7 A.2. Functionls: Bsic Concepts A 7 A.2.1. Functionl Continuity A 7 A.2.2. Continuous Liner Functionls A 8 A.2.3. Fundmentl Lemms of Vritionl Clculus A 9 A.3. Functionls: First Vrition A 10 A.3.1. Increments nd Differentibility A 10 A.3.2. First Vrition Properties A 10 A.4. Functionls: Second Vrition A 11 A.4.1. Biliner nd Qudrtic Functionls A 11 A.4.2. Second Vrition Definition A 12 A.4.3. Second Vrition Differentil Forms A 14 A.4.4. Legendre s Condition A 14 A 2

3 A.1 VECTOR SPACES This Appendix outlines key portions of the mthemticl pprtus tht supports Chpters 1 5 of AVMM. The exposition omits proofs but gives references to where those cn be found. A rich source for this kind of bckground mteril is Chpter 1 of Gelfnd nd Fomin [274]. 1 A.1. Vector Spces A vector spce V is nonempty collection of objects clled vectors, which my be dded together nd multiplied by numbers clled sclrs. Generic vectors re denoted by lower cse bold symbols such s x, y, etc. This nottion grees with tht used for the ordinry vectors of liner lgebr. If x belongs to V, we write x V, nd sy tht x is member 2 of V. Sclrs tht my pper in the vector-scling opertion defined below re denoted by c, d, etc. In the ensuing description these will be rel numbers: c, d R, in which s usul R denotes the set of rel numbers. (Generliztions to complex sclrs, etc., re possible but unnecessry here.) A.1.1. Vector Spce Properties At minimum, two opertions re defined in vector spce: vector ddition, denoted x + y, nd sclr multipliction, denoted c x. These obey the following properties, sometimes clled xioms 3 Closure Properties: C1. Closure Under Addition: x + y V whenever x V nd y V. C2. Closure Under Sclr Multipliction: c x V whenever x V nd c R. Addition Properties: A1. Additive Identity: There is zero vector, denoted 0 V, such tht x + 0 = x for ll x V. A2. Additive Negtion: For ll x V there is vector x V such tht x + ( x) = 0. This x is clled the negtive of x. A3. Associtivity: (x + y) + z = x + (y + z) for ll x, y, z V. A4. Commuttivity: x + y = y + x for ll x, y V. Sclr Multipliction Properties: M1. Sclr Multiplictive Identity: 1x = x for ech x V, in which 1 is the rel number 1. M2. First Distributive Property: c (x + y) = c x + c y for ll x, y V nd ll c R. M3. Second Distributive Property. (c + d) x = c x + d x for ll x V nd ll c, d R. M4. Associtivity: c (d x) = (cd) x for ll x V nd ll c, d R. The two closure properties cn be checked t once by verifying the following property: C0. Closure Under Liner Combintion: c x + d y V whenever x, y V nd c, d R. A vector subspce W of vector spce V is nonempty subset of V tht is itself vector spce set of vectors tht stisfy the foregoing 10 conditions. 1 Unfortuntely the Russin to English trnsltion therein is occsionlly outdted. E.g., liner spce should be vector spce. (The former term is imprecise s it is presently used for other things.) Such infelicities re silently emended. 2 Some texts, such s [274], use element of vector spce insted of member. Tht term will be voided here to reduce the risk of confusion with finite elements. 3 Trnscribed from [746] with minor corrections nd dditions. A 3

4 Apendix A: MATHEMATICAL BACKGROUND OF VARIATIONAL CALCULUS Symbol Dim Description Tble A.1. Vector Spce Exmples R or R 1 1 Spce of ll rel numbers R 2 2 Spce of ll ordered rel number pirs (.k.. 2-vectors) R 3 3 Spce of ll ordered rel number triples (.k.. 3-vectors) R n n Spce of ll ordered n-tuples of rel numbers (.k.. n-vectors) C or C 1 1 Spce of ll complex numbers C n n Spce of ll ordered n-tuples of complex numbers P Spce of ll polynomils P n n+1 Spce of ll polynomils of degree n M mn mn Spce of ll m n mtrices F Generic function spce C 0 (I ) Spce of ll continuous functions on the intervl I (which my be open or closed, finite or infinite) C n (I ) Spce of ll functions on the intervl I tht hve n continuous derivtives (I s bove) Spce dimensionlity: crdinlity of bsis. Dimensionlity of C n is 2n if the bsis consists of rel numbers. A.1.2. Vector Spce Exmples Some importnt exmples of vector spces re listed in Tble A.1. Note tht dimensionlity my be finite, s in the ordinry n-vectors nd m n mtrices, or infinite, s in function spces. A.1.3. Normed Vector Spces A vector spce V is sid to ne normed if ech member x V cn be ssigned non-negtive number x clled the norm of x, which obeys the following rules: N1. Uniqueness of Zero Norm: x = 0 if nd only if x = 0. N2. Positive Homogeneity Under Scling: c x = c x for ll x V nd ll c R. N3. Tringulr Inequlity. x + y x + y for ll x, y V. If V is normed vector spce the distnce between two vectors x nd y tht belong to V is defined s the nonnegtive quntity x y. A.1.4. One-Forms The principl vector spce studied in liner lgebr is R n, the set of ll ordered n-tuples. Its member objects re n-dimensionl column vectors, which for n = 3 reduce to the ordinry vectors (visulizble s rrowed segments) of 3D spce. 4 A liner mp from R n to sclr field is clled one-form (often written 1-form) or covector. See Figure A.1() for slot mchine representtion. For exposition simplicity it is ssumed tht the 4 The R n spce is ctully the source of the vector spce concept, which generlizes liner lgebr rules nd opertions to more complex mthemticl objects. A 4

5 A.1 VECTOR SPACES Finitedimensionl vector () (b) (c) 1-FORM or COVECTOR Liner Mp Number Function FUNCTIONAL Generl Mp Number Function LINEAR FUNCTIONAL Liner Mp Number Figure A.1. Slot mchine visuliztion of the mpping of members of vector spce onto numbers (sclrs): () A one-form, lso spelled 1-form, receives finite dimensionl vector nd mps it linerly to number; (b) A functionl (without qulifier) receives function nd mps it to number; (c) A liner functionl receives function nd linerly mps it to number. finite-dimensionl vector spce is R n, the set of rel n-vectors. Its members re denoted x, y, etc. A 1-form mp from tht spce to sclr must comply with the following linerity rules. L1. Degree-One Homogeneity: F[α x] = α F[x] for ny x R n nd α R. L2. Liner Superposition: F[x + y] = F[x] + F[y] for ny x, y R n. For the spce R n 1-form cn be expediciously built on premultiplying by row n-vector, since the dot product produces sclr. For exmple, suppose tht n = 3sox = [ x 1 x 2 x 3 ] T R 3 is 3-vector. The vector entry sum is provided by the 1-form [ ] x1 S(x) = e3 T x = [ 1 1 1] x 2 = x 1 + x 2 + x 3. (A.1) x 3 in which e k denotes the k-vector of ones. Here re dditionl 1-form exmples: Algebriclly lrgest entry of x R n is the 1-form mxi=1 n x i (but not mxi=1 n x i, s rule L2 would be violted). Projection length of x long the unit direction vector d, in which d T d = 1, is the 1-form d T x, s long s d is independent of x. Let g R 3 be the 3-vector grdient of sptil field φ(x 1, x 2, x 3 ), tht is, g = [ φ/ x 1 φ/ x 2 φ/ x 3 ] T. The directionl grdient vlue long unit direction d is the 1-form g = d T g. The trce of n n mtrix A with entries ij : trce(a) = n i=1 ii, is 1-form. On the other hnd, the squred length of x, given by the dot product L 2 (x) = x 2 2 = xt x = [ x 1 x 2 x 3 ] [ x1 x 2 x 3 ] = x1 2 + x x 3 2. (A.2) is not 1-form since it violtes rules L1 nd L2. Actully (A.2) is n instnce of 2-form, lso clled biliner or qudrtic forms. Those re considered in the cse of functionls in A.4.1 Wht if the vector spce becomes n function spce? Its members re now functions nd the spce hs infinite dimension. Then 1-forms become liner functionls. Those re covered in A.2.2 A 5

6 Apendix A: MATHEMATICAL BACKGROUND OF VARIATIONAL CALCULUS A.1.5. Function Spces In 1D Normed vector spces whose members re functions re clled function vector spces or function spces for short. Those re importnt in the study of vritionl methods. For description simplicity the members considered below re rel-vlued functions y(x) of rel vrible x, defined over closed intervl x [, b] with b, of the rel xis. They will be clssified on the bses of function continuity s follows. Spce C 0 (, b) of continuous functions. This vector spce consists of ll continuous functions y(x) defined in [, b]. The ddition nd multipliction-by-sclr opertions re defined by the ordinry ddition of functions nd multipliction of function by numbers, respectively. The norm is defined s the mximum of the bsolute vlue: y 0 = mx x b y(x). (A.3) The zero subscript in y indictes tht tking the norm only involves the function itself. The number y 0 is usully clled the supremum norm or lest upper bound of y(x) over [, b]. Thus mny uthors cll (A.3) the supremum norm. Spce C 1 (, b) of continuously differentible functions. This vector spce consists of ll functions y(x) defined in [, b] tht re continuous nd possess continuous first derivtives. The ddition nd multipliction-by-sclr opertions re the sme s in C 0 (, b). The norm is defined by the formul y 1 = mx y(x) + mx x b x b y (x). (A.4) From this definition it follows tht two functions in C 1 (, b), sy y(x) nd z(x), re viewed s close if both the functions themselves nd their first derivtives re close together, since if y z <ɛ implies y(x) z(x) <ɛnd y (x) z (x) <ɛfor ll x [, b]. Consequently the norm (A.4) gives more weight to regulrity thn the supremum norm. Spce C n (, b) of n-times continuously differentible functions. This vector spce consists of ll functions y(x) defined in [, b] tht re continuous nd possess continuous derivtives up to order n inclusive, in which n is positive integer. The ddition nd multipliction-by-sclr opertions re the sme s the preding cses. The norm is defined by the formul y n = n mx x b y(k) (x) k=0 (A.5) in which y(k)(x) denotes the k th derivtive of y(x). The previously introduced vector spces C 0 (, b) nd C 1 (, b) correspond to the cses n = 0 nd n = 1, respectively, of C n (, b). Obviously C k (, b) is included in C k 1 (, b) for k 1. Remrk A.1. The cse n is of specil interest for some pplictions. It gives the clss of functions C (, b) tht re infinitely smooth nd possess continuous derivtives of ll orders. For exmple, polynomils. Tht spce will not be used explicitly here. A 6

7 A.2 FUNCTIONALS: BASIC CONCEPTS Remrk A.2. The foregoing norm choices do not exhust ll possibilities. One problem with (A.3) (A.5) is their extreme sensitivity to tiny locl chnces in the functions, especilly if derivtives re included. To sugrcot tht undesirble feture numerous globl norms hve been devised. These involve integrtion over the intervl [, b]. For exmple, the widely used L p norms for the C 0 (, b) spce, with p positive integer, re given by y L p = y(x) p dx, p = 1, 2..., y(x) C 0 (, b). (A.6) This is clled the Eucliden norm for p = 2. The supremum norm (A.3) is recovered in the limit p, whence the lterntive nme infinity norm. Remrk A.3. Another difficulty with norms tht combine function vlues nd derivtives, s in (A.5) for k 1, is improper scling when pplied to physicl problems. An exmple in dynmics: if y(t) is displcement response in time t, ẏ(t) = dy(t)/dt is velocity, so tking the norm y 1 of (A.4) would dd pples to ornges. This nonsense cn be resolved by pplying pproprite weights tht restore the correct physicl dimensions (e.g., multiplying ẏ(t) by chrcteristic length), so one is effectively dding pples to pples. A.1.6. More Generl Function Spces The function spces introduced bove my be generlized in severl directions, for exmple Multiple Dimensions. The spce of sclr functions tht depend on multiple independent vribles. For exmple y = y(x 1, x 2, x 3 ) in 3D spce or y = y(x 1, x 2, x 3, t) in spcetime. Multiple Functions. The spce of m multiple functions tht depend on n multiple independent vribles. These re usully rrnged s n ordinry m-vector tht (s usul) is identified by bold symbol. For exmple, if m = 2 nd n = 3: [ ] [ ] x1 y1 (x y(x) = y(x 1, x 2, x 3 ) = 1, x 2, x 3 ), in which x = x y 2 (x 1, x 2, x 3 ) 2. (A.7) x 3 Such generliztions re well covered in textbooks devoted to functionl nlysis, s well s dvnced tretment of vritionl clculus. A.2. Functionls: Bsic Concepts The foregoing concepts re now pplied to the study of functionls. As noted in Chpter 1 functionl is function of function. More precisely, it receives function nd produces number. Figure A.1(b) gives visul representtion of functionl s slot mchine. In this Section we only consider functionls J[y] tht depend on single function y(x) in which x, y nd J re rel, nd defined over the closed intervl x [, b]. The function y(x) is viewed s point of its vector spce, which will be clled y V y. A.2.1. Functionl Continuity The functionl J[y] is sid to be continuous t the point y R if for ny ɛ>0there is δ>0 such tht J[y] J[y ] <ɛ, (A.8) provided tht y y <δ, in which is the norm defined for V y. A 7

8 Apendix A: MATHEMATICAL BACKGROUND OF VARIATIONAL CALCULUS Remrk A.4. The inequlity (A.8) is equivlent to two inequlities: J[y] J[y ] > ɛ, (A.9) nd J[y] J[y ] <ɛ, (A.10) If in the definition of continuity, (A.8) is replced by (A.9), J[y] is sid to be lower semicontinuous t y.if insted (A.8) is replced by (A.10), J[y] is sid to be upper semicontinuous t y. Remrk A.5. Which spce V y is pproprite for y(x)? The decision depends primrily on the order of y- derivtives tht pper in J[y]. For exmple, in the cse of the simplest vritionl problem, which involves functionls of the form J[y] = F(x, y, y ) dx (A.11) the pproprite function spce for y(x) would be C 1 (, b), the spce of continuous differentible functions. Selecting the lrger spce C 0 (, b) of continuous functions my led to problems since we could select two functions y(x) nd y (x) rbitrrily close in the sense of the supremum norm (A.3), but whose derivtives differ by ny given mount. The continuity test (A.8) would then fil. Bottom line: one should pick function spce in which the continuity of the functionl under study holds. Remrk A.6. In mny vritionl problems we del with functionls defined on sets of dmissible functions tht do not form vector spce. For exmple, the dmissible functions for the simplest functionl (A.11) under the essentil boundry conditions y() =ŷ, y(b) =ŷ b, (A.12) re the continuously differentible plne curves y = y(x) tht pss through the points A(, ŷ ) nd B(b, ŷ b ). But the sum of two dmissible functions will not pss through those points unless ŷ =ŷ b = 0. The concept of normed vector spce s well s relted notions of distnce nd continuity still ply, however, n importnt role. It is generlly possible to modify the given vritionl problem through vrious techniques so tht the pproprite tools cn be rigorously pplied. A.2.2. Continuous Liner Functionls The generliztion of the 1-form discussed in A.1.4 to function spces is cler fits the definition of functionl. But crrying over the linerity conditions L1 nd L2 of A.1.4 mens tht we get restricted kind, clled liner functionl. See Figure A.1(c). This cn be further specilized by requiring tht the continuity property (A.8) hold. The end product of these speciliztions is continuous liner functionl, or CLF. These figure prominently in vritionl clculus. For simplicity they re introduced next for the one-dimensionl cse. Consider one-dimensionl, normed function vector spce F(, b). Its members re functions h(x) F with domin x [, b]. 5 with domin x [, b]. The functionl φ[h] is clled continuous liner functionl, or CLF for short, if it stisfies the following conditions. CL1. φ[α h] = αφ[h] for ny h F nd α R. CL2. φ[h 1 + h 2 ] = φ[h 1 ] + φ[h 2 ] for ny h 1, h 2 F. CL3. φ[h] is continuous for ll h F. 5 We devite here from the use of y(x) s generic member function so s to merge semlessly with?, which introduces the concept of vrition. In tht subsection, h(x) is the increment of y(x). A 8

9 A.2 FUNCTIONALS: BASIC CONCEPTS Exmple A.1. The function h(x) C 0 (, b), the spce of continuous functions in x [, b]. Define the functionl φ[h] s the vlue tken t point c [, b]: φ[h] = h(c). (A.13) This φ[h] is CLF on [, b]. Exmple A.2. Agin h(x) C 0 (, b). The re integrl φ[h] = h(x) dx, (A.14) defines CLF in C 0 (, b) Exmple A.3. Agin h(x) C 0 (, b) while α(x) denotes given function lso in C 0 (, b). The integrl φ[h] = α(x) h(x) dx, (A.15) defines CLF on the function spce C 0 (, b). Exmple A.4. More generlly, tke h(x) C n (, b), with n positive integer wheres α i (x), i = 0, 1,...n re fixed functions in C(, b). the integrl φ[h] = defines CLF on the function spce C n (, b). A.2.3. Fundmentl Lemms of Vritionl Clculus [ α0 (x) h(x) + α 1 (x) h (x) α n (x) h ( n)(x) ] dx (A.16) Suppose tht the CLF (A.16) stted in Exmple A.4 vnishes for ll h(x) pertining to certin function spce. Wht cn be sid bout the functions α i (x)? The following results re typicl of the so-clled fundmentl lemm of vritionl clculus. 6 Lemm A.1. Ifα(x) C 0 (, b) nd if α(x) h(x) dx = 0, (A.17) for every h(x) C 0 (, b) such tht h() = h(b) = 0, then α(x) = 0 for ll x [, b]. Proof: see [274, p. 9]. It is shown there tht the result lso holds if C 0 (, b) is replced by C n (, b). Lemm A.2. Ifα(x) C 0 (, b) nd if α(x) h (x) dx = 0, (A.18) 6 Strictly speking, tht nme pplies to Lemm A.1, when h(x) is tken s the dmissible vrition of the input to functionl. Lemms A.2 A.4 re directly deduced from it fter minor gyrtions, nd so re lrgely vritions on theme. A 9

10 Apendix A: MATHEMATICAL BACKGROUND OF VARIATIONAL CALCULUS for every h(x) C 1 (, b) such tht h() = h(b) = 0, then α(x) = c for ll x [, b], in which c is constnt. Proof: see [274, p. 10]. Lemm A.3. Ifα(x) C 0 (, b) nd if α(x) h (x) dx = 0, (A.19) for every h(x) C 2 (, b) such tht h() = h(b) = 0 nd h () = h (b) = 0, then α(x) = c 0 x +c 1 for ll x [, b], in which c 0 nd c 1 re constnts. Proof: see [274, p. 10]. Lemm A.4. Ifα(x) C 0 (, b) nd β(x) C 0 (, b) nd if [ α(x) h(x) + β(x) h (x) ] dx = 0, (A.20) for every h(x) C 1 (, b) such tht h() = h(b) = 0, then β(x) is differentible nd β (x) = α(x) for ll x [, b]. Proof: see [274, p. 11]. A.3. Functionls: First Vrition The concept of vrition of functionl is nlogous to tht of differentil of function of n vribles. Its min ppliction is to find extrem of functionls. Those re input function(s), known s extremls, for which the functionl becomes sttionry with respect to smll vritions. As usul we restrict the ensuing exposition to the one-dimensionl cse. A.3.1. Increments nd Differentibility Let J[y] be functionl defined on some normed function vector spce with domin x [, b]. Suppose tht n dmissible input function y = y(x) is incremented by h = h(x), while tking cre tht y + h = y(x) + h(x) remins dmissible. (For exmple, if y(x) hs to stisfy the essentil boundry conditions y() =ŷ nd y(b) =ŷ b, plinly h() = h(b) = 0.) The increment of the functionl is defined s J[h] = J[y + h] J[y]. (A.21) If y is fixed, J[y + h] J[h] is functionl of h = h(x). For rbitrry h this will be generlly nonliner functionl. If the increment J cn be expressed s J[h] = φ[h] + ɛ h (A.22) in which φ[h] is CLF nd ɛ 0s h 0, the functionl J[y] is sid to be differentible t tht fixed y. If (A.22) holds, the functionl φ[h] is clled the principl liner prt of the increment J[h]. If J[y] is differentible t ech dmissible y, the functionl is genericlly clled differentible. A 10

11 A.3.2. First Vrition Properties A.4FUNCTIONALS: SECOND VARIATION For nottionl convenience, s well s smooth linkge to ordinry differentil clculus, the increment φ[h] is renmed the first vrition of J[y], nd denoted by δ J = δ J[h]. 7 Two importnt first-vrition theorems pertining to differentible functionls follow. Theorem A.1. The first vrition of differentible functionl is unique. Proof: see [274, p. 12]. Theorem A.2. Anecessry condition for differentible functionl J[y] to hve n extremum t y = y is tht its first vrition vnish there; tht is δ J[y] = 0, (A.23) for y = y nd ll dmissible h. Proof: see [274, p. 13]. Functions y = y(x) t which δ J vnishes re clled extremls. Remrk A.7. Theorem A.2 is the nlog of well known theorem of ordinry clculus. If the one-dimensionl differentible function f (x) hs n extremum t x = x, its differentil must vnish there: df(x ) = 0, which my lso be expressed s y (x ) = 0. Tht is, the function must be sttionry t x. Now in ordinry clculus n extremum cn be locl mximum, locl minimum, or locl inflexion point (neither mximum nor minimum). If one is looking for ctul (locl) mxim or minim, the condition df(x) = 0 is only necessry. Sufficient conditions depend on exmintion of the sign of the second differentil d 2 f (x) or, lterntively, y (x) in the vicinity of x = x. The nlogous sufficiency conditions for functionls re fr more elborted; in fct rigorous results took over century (from Euler nd Lgrnge to Weierstrss) to be worked out. A.4. Functionls: Second Vrition Theorem 2 bove gives only necessry conditions for n extremum of J[y]. An extremum cn be mximum, minimum or just sttionry point. As hinted in Remrk A.7, sufficient conditions for mximum or minimum require exmintion of the functionl-level equivlent of the second differentil of ordinry clculus. Tht equivlent object is clled second vrition. Before defining it, some uxiliry concepts re introduced. A.4.1. Biliner nd Qudrtic Functionls An obvious generliztion of the liner functionls introduced in A.2.2 is the concept of biliner functionl. This becomes qudrtic functionl s specil cse. A functionl B[y, z] tht depends on two input functions y = y(x) nd z = z(x) tht belong to normed function vector spce F, is sid to be biliner if it is liner functionl of y for fixed z, nd liner functionl of z for fixed y. More specificlly, if it stifies the following bilinerity conditions B1. Homogeneity in y: B[α y, z] = αb[y, z] for ll y, z F nd α R. 7 Of course the first vrition is ctully functionl of y = y(x) (the bseline input function) nd h = h(x) (its increment). So in some contexts it my help to write J[y, h] = δ J[y, h] + ɛ h. A 11

12 Apendix A: MATHEMATICAL BACKGROUND OF VARIATIONAL CALCULUS B2. Associtivity in y: B[y 1 + y 2, z] = B[y 1, z] + B[y 2, z] for ll y 1, y 2, z F. B3. Homogeneity in z: B[y,αz] = αb[y, z] for ll y, z F nd α R. B4. Associtivity in z: B[y, z 1 + z 2 ] = B[y, z 1 ] + B[y, z 2 ] for ll y, z 1, z 2 F If we set y = z in biliner functionl, we obtin n expression clled qudrtic functionl A qudrtic functionl A[y] = B[y, y] is sid to be positive definite A[y] > 0 for ll y F, y 0. (A.24) If the > is replced by in (A.24) the functionl is sid to be nonnegtive. A qudrtic functionl A[y] = B[y, y] is sid to be strongly positive definite if there exists constnt k > 0 such tht A[y] k y. (A.25) In ll of the folloing exmples, x, y(x), z(x), α(x), β(x), γ (x) R, nd x [, b]. Exmple A.5. The expression B[y, z] = y(x) z(x) dx, (A.26) in which y, z C 0 (, b), is biliner functionl. The corresponding qudrtic functionl is obtined by mking z(x) = y(x): The qudrtic functionl (A.27) is positive definite. A[y] = (y(x)) 2 dx y(x)y(x) 2 dx (A.27) Exmple A.6. The expression B[y, z] = α(x) y(x) z(x) dx, (A.28) in which y, z C 0 (, b) nd α(x) is given function, is biliner functionl. The corresponding qudrtic functionl is A[y] = α(x)(y(x)) 2 dx = α(x) y(x)y(x) dx (A.29) If α(x) >0 for ll x [, b] the functionl (A.29) is positive definite. Exmple A.7. The expression A[y] = [ α(x) y(x)y(x) + β(x) y(x) y (x) + γ(x) y (x)y (x) ] dx, (A.30) in which y C 1 [, b] nd α(x), β(x), nd γ(x) re given functions, is qudrtic functionl. A 12

13 A.4FUNCTIONALS: SECOND VARIATION A.4.2. Second Vrition Definition We retke the developments of A.3.1 nd? to proceed beyond the first vrition. To briefly recpitulte, let J[y] be functionl defined on some normed function vector spce F with domin x [, b]. Suppose tht n dmissible input function y = y(x) is incremented by h = h(x), while tking cre tht y + h = y(x) + h(x) remins dmissible. If the increment of the functionl cn be expressed s J[h] = J[y + h] J[y] = δ J[h] + ɛ h (A.31) in which φ[h] is CLF nd ɛ 0s h 0, then: (i) δ J ws clled the first vrition of J[y] t the given y, nd (ii) J[y] ws sid to be differentible t y. If (ii) holds for ny y, the functionl J[y] ws genericlly clled differentible. Continuing on this theme, we will sy tht J[y]istwice differentible if its increment cn be written s J[h] = J[y + h] J[y] = φ 1 [h] + φ 2 [h] + ɛ h 2, (A.32) in which φ 1 [h]isliner functionl, φ 2 [h] isqudrtic functionl nd ɛ 0s h 0. The φ 1 [h] term tht ppers in (A.32) ws clled the first vrition in?. Accordingly we will renme φ 2 [h] the second vrition, nd denote it by δ 2 J[h]. 8 On replcing φ 1 δ J nd φ 2 δ 2 J, the increment (A.32) my be recst s J[h] = J[y + h] J[y] = δ J + δ 2 J + ɛ h 2. (A.33) If (A.33) holds for ny y nd h, we will genericlly stte tht J[y] is twice differentible. Two importnt second-vrition theorems pertining to twice differentible functionls follow. (These correspond to those stted for the first vrition in?.) Theorem A.3. The second vrition of twice differentible functionl is unique. Proof: see [274, p. 99]. Theorem A.4. Anecessry condition for twice differentible functionl J[y] tohve minimum t y = y is tht δ J[y] = 0, δ 2 J[y] 0, (A.34) for y = y nd ll dmissible h. For mximum, replce the sign in the second of (A.34) by. Proof: see [274, p. 99]. Note tht condition (A.34), or its counterprt with, isnot sufficient to gurntee minimum or mximum. To chieve sufficiency we need to recll the concept of strong positivity, which ws defined for qudrtic functionl in (A.25). Theorem A.5. Asufficient condition for twice differentible functionl J[y]tohveminimum t y = y is tht is second vrition δ 2 J[h] be strongly positive for y = y. Proof: see [274, p. 100]. For mximum, replce strongly positive by strongly negtive, or chnge the sign of δ 2 J[h]. 8 The comment mde in previous footnote lso pplies here: δ 2 J is ctully function of y nd h. In some contexts it might be convenient to denote it s δ 2 J[y, h]. A 13

14 Apendix A: MATHEMATICAL BACKGROUND OF VARIATIONAL CALCULUS Remrk A.8. In finite dimensionl normed vector spce, strong positivity of qudrtic form is equivlent to positive definiteness of tht form. Therefore twice differentible function f (x) of finite number of vribles collected in vector x = [ x 1 x 2... x n ] T, hs minimum t point P = P(x ) where its first differentil vnishes, if its second differentil is positive definite t P. In more generl cse of functionl, however, strong positivity is stronger condition thn positive definiteness. A.4.3. Second Vrition Differentil Forms We now find n expression for the second vrition of the simplest vritionl problem, ssocited with functionls of the form (A.11), in which input functions y = y(x) stisfy priori the fixedpoint conditions (A.12). As regrds smoothness we ssume tht the integrnd of (A.11), which hs the form F(x, y, z), hs continuous prtil derivtives up to the necessry order with respect to ll its rguments. Increment y(x) by function h(x) such tht y + h stisfies ll dmissibility conditions; in prticulr h() = h(b) = 0. The corresponding increment J[h] = J[y + h] J[y] of the functionl is expnded in Tylor s series J[h] = (F y h + F y h ) dx + 1 [ 2 F yy h 2 + F yy hh + F y y (h ) 2] dx. (A.35) in which the overbr indicted tht the corresponding derivtives re evluted long certin intermedite curves; for exmple ( F yy = F yy x, y + θ h, y + θh ), θ [0, 1], (A.36) nd similrly for F yy nd F y y. If F yy, F yy nd F y y re replced by the derivtives F yy, F yy nd F y y, respectivelyt, evluted t (x, y(x), y (x) the expnsion (A.35) cn be rewritten with reminder term s J[h] = (F y h + F y h [ ) dx ɛ1 h 2 + ɛ 2 hh + ɛ 3 (h ) 2] dx + ɛ, (A.37) in which ɛ = [ Fyy h 2 + F yy hh + F y y (h ) 2] dx+ (A.38) Becuse of the bssumed continuity of F yy, F yy nd F y y, it follows tht ɛ 1, ɛ 2 nd ɛ 3 tend to zero s h 1 0. Hence ɛ is n infinitesiml of order higher thn 2 with respect to h 1.Bydefinition the first nd second terms of (A.37) re the first vrition δ J[h] nd δ 2 J[h], respectively, of (A.11). Therefore the ltter cn be expressed s δ 2 [ J[h] = Fyy h 2 + F yy hh + F y y (h ) 2] dx. (A.39) Integrting the middle term by prts nd noting tht h() = h(b) = 0 we obtin F yy hh dx = (df yy /dx)h2 dx, whence (A.39) cn be trnsformed into the more compct form δ 2 [ J[h] = Qh 2 + P (h ) 2] ( dx, P = 1 2 F y y, Q = 1 F 2 yy df ) yy dx. (A.40) dx Here P = P(x) nd Q = Q(x). This is the most convenient expression for further investigtions. A 14

15 A.4FUNCTIONALS: SECOND VARIATION A.4.4. Legendre s Condition (TBC) A 15

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