Generalizations of the Basic Functional

3 Generliztions of the Bsic Functionl 3 1

Chpter 3: GENERALIZATIONS OF THE BASIC FUNCTIONAL TABLE OF CONTENTS Pge 3.1 Functionls with Higher Order Derivtives.......... 3 3 3.2 Severl Dependent Vribles............... 3 3 3.3 Functionls in Prmetric Form.............. 3 5 3.4Severl Independent Vribles.............. 3 5 3.5 Green-Guss Theorem in m Dimensions........... 3 6 3. Exercises...................... 3 8 3 2

3.2 SEVERAL DEPENDENT VARIABLES This Chpter provides some generliztions of the bsic functionl studied in Chpters 1-2. 3.1. Functionls with Higher Order Derivtives The function is still y = y(x) (one dependent nd one independent vrible) but the presence of the second derivtive is lllowed. Consider the functionl J = b F(y, y, y, x) dx, (3.1) where s usul primes denote derivtives with respect to x. Vrying the dependent vrible y nd its derivtives y nd y by h, h nd h, respectively, gives b ( h + h + ) h dx. (3.2) Integrting twice by prts yields where b Ehdx + [( d dx ) E = d + d2 dx dx 2 h + ] b h, (3.3). (3.4) The Euler eqution E = 0 is generlly fourth-order ODE whose generl solution hs four rbitrry constnts. These constnts re determined by four boundry conditions, two from ech end. For exmple, the conditions y = y() =ŷ, y () =ŷ (3.5) t the left end x = re essentil-essentil or EE. On the other hnd, the conditions d dx = 0, = 0, (3.6) re nturl-nturl or NN. Other possibilities re EN nd NE. Considering the right end x = b there re 2 4 = 16 possible combintions. Lnczos on pge 68 hs good discussion of the plne bem problem, which leds to functionl of the form (7.1). If F contins third derivtive, triple integrtion by prts (omitting boundry terms) gives s Euler eqution E = d + d + d (3.7) dx dx 2 dx 3 nd so on. The generl expression for E for rbitrry number of derivtives is given in Gelfnd nd Fomin, Sec. 11. 3 3

Chpter 3: GENERALIZATIONS OF THE BASIC FUNCTIONAL 3.2. Severl Dependent Vribles Next we generlize the bsic functionl to n 1 dependent vribles y 1 = y 1 (x), y 2 = y 2 (x),...y n = y n (x) whose first derivtives pper in F: J = b F(y 1,...y n, y 1,...y n, x) dx (3.8) where the y i = y i (x) re C 1 functions in [, b] nd primes denote derivtives with respect to x. It is often convenient to use vector nottion y 1 y 1 J = F(y, y, x) dx, y =., y =. (3.9) y n The first vrition of J in component nottion is ( d ) [ b i dx i h i dx + i h i], i = 1,...n. (3.10) where the summtion convention hs been used. In vector nottion b ( ) [ T ( ) F T b h dx + h] (3.11) It is convenient to cll f i = i, p i = i, i = 1,...n. (3.12). In nlyticl dynmics f i nd p i often hve the interprettion of generlized forces nd moment, respectively, nd we shll often cll them tht even in other pplictions. The n-vectors f nd p re defined in the usul mnner. Then the preceding eqution my be bbrevited to b ( f dp ) T h dx + [ p T h ] (3.13) dx b The Euler equtions nd boundry conditions re dp = f, (3.14) dx p i = 0orδy i = h i = 0tx =, i = 1,...n (3.15) p i = 0orδy i = h i = 0tx = b, i = 1,...n From the bove expressions it is seen tht there re 2 n possible combintions of essentil nd nturl boundry conditions, number tht grows rpidly with n. For exmple if n = 20 there re bout one million possible combintions. For detils on the derivtion of these results see Chpter 9 of Gelfnd nd Fomin. y n 3 4

3.4 SEVERAL INDEPENDENT VARIABLES Exmple 3.1. Dynmicl system of n linerly intercting prticles. The ction integrl is t2 t2 S = Ldt = (T U P) dt t 1 = t1 t 0 t 1 [ 1 2 m i u 2 i 1 2 k ij(u i u j ) 2 + f i u i ] dt (3.16) where T, U nd P denote the kinemtic energy, the internl energy nd the potentil of pplied forces, respectively, t is time, u i = u i (t) re the prticle motions bout equilibrium positions, m i the prticle msses, k ij the stiffness coefficients tht chrcterize interction forces within prticles, f i re pplied forces, nd superposed dot denotes temporl derivtive. Hmilton s principle sttes tht δs = 0. Since S fits the formt of the functionl (7.9), the Euler equtions re re m i ü 2 k ij (u i u j ) + f i = 0, (3.17) or, in mtrix form nd reorgnzing terms Mü + Ku = f. (3.18) This is the fundmentl equilibrium eqution of undmped liner mss-discrete dynmics. 3.3. Functionls in Prmetric Form Consider J = t1 t 0 (y, ẏ, x, ẋ) dt, where x = x(t), y = y(t), ( ) d/dt. These functionls re sid to be in prmetric form. They often occur in dynmics where t is time nd the pir x(t), y(t) denotes trjectory in (x, y) spce. Prmetric functionls hve mny fcets in common with the cse treted in the previous section. They re studied in some detil in Gelfnd nd Fomin, Sec. 10, where it is shown tht one gets two Euler equtions x d dt hẋ = 0, d dt hẏ These re not independent, but re connected by differentil constrint. 3.4. Severl Independent Vribles = 0. (3.19) This cse includes most ppliction problems involving continuous medi in two or three spce dimensions. The ssocited functionls involve multidimensionl integrls. The Euler equtions re prtil differentil equtions (PDEs) rther thn ODEs. The boundry conditions re now given over sets of boundry points forming curves, surfces or surfce-time boundries. The essentil tool for pssing from the vritionl to the wek form is the Green-Guss theorem in multiple dimensions (lso clled the divergence theorem), which is generliztion of the integrtion by prts formul. For definiteness, in this section we consider the simplest functionl of this form, with one independent vrible z nd two independent vribles x, y: J = F(z, z x, z y, x, y) d (3.20) 3 5

Chpter 3: GENERALIZATIONS OF THE BASIC FUNCTIONAL where z = z(x, y) is C 1 function tken over the two-dimensionl domin. Then ( z h + h x + ) h y d (3.21) z x z y where δz = h, δz x = h x, δz y = h y, the vrition h being function of x, y stisfying essentil B.C. Integrtion by prts gives d h x d = hd + cos(n, x)hd (3.22) z x dx z x z x z x h y d = d dy z y hd + z y cos(n, y)hd (3.23) Adding: ( h x + ) ( dpx h y d = z x z y dx + dp ) y hd + p n hd (3.24) dy where p x = z, p y = x z, p n = p x n x + p y n y. This is specil cse of the n-dimensionl y Green-Guss theorem (divergence theorem) covered in the next section. Substituting into δ J yields ( z d dx p x d ) dy p y hd + p n hd. (3.25) The Euler eqution is E = F z d dx p x d dy p y = 0 in. (3.26) To enuncite the boundry conditions, it is convenient to split the boundry s : e n, where e nd n re the portions of over which essentil nd nturl B.C.s, respectively, re given. These prts re disjoint, i.e. e n : 0. Mthemticlly: 3.5. Green-Guss Theorem in m Dimensions essentil: h = 0 or z =ẑ on e nturl: p n = 0 on n (3.27) In the sequel, denotes n m-dimensionl domin referred to rectngulr Crtesin system x = (x 1, x 2,...x m ). [In prcticl pplictions n 4]. The domin is bounded by boundry with unit externl norml n = (n 1,...n m ). Let h = h(x 1,...x m ) = h(x) be C 1 sclr function in, nd p i = p i (x 1...,x n ) = p i (x) re n sclr C 1 functions. Then Green-Guss theorem in m dimensions (lso clled the divergence theorem) reds in component nottion with summtion convention over i = 1,...m implied: (p i h) d = p i n i hd = p n hd, (3.28) x i where p n = p i n i. Expnding the domin term: p i x i hd + h p i d = p n hd. (3.29) x i 3 6

In vector nottion divp hd + 3.5GREEN-GAUSS THEOREM IN M DIMENSIONS p T grdhd = p n hd. (3.30) This theorem is the bsis for the vrition-homogeniztion step in constructing wek forms. For exmple, consider the cse of one dependent vrible u = u(x, y, z) nd three independent vribles (x, y, z): J = F(u, u x, u y, u z, x, y, z) d, (3.31) where is three-dimensionl domin. The first vrition is ( u h + h x + h y + ) h z d. (3.32) u x u y u z Now pply Green-Guss s theorem for m = 3, with x 1 = x, x 2 = y, x 3 = z, p 1 = / u x, p 2 = / u y, nd p 3 = / u z : ( ) ( h h h p1 p 1 + p 2 + p 3 d = + p 2 + p ) 3 hd p n hd. (3.33) x 1 x 2 x 3 x 1 x 2 x 3 Then ( d dx u x d dy d ) hd + p n hd, (3.34) u y dz u z which immeditely yields the Euler eqution nd the essentil/nturl boundry conditions. The topic of multiple independent vribles is briefly sketched in Chpter 5 (Chpter 1) of Gelfnd-Fomin, nd retken in full generlity in their Chpter 7. Lnczos trets the subject in his Chpter XI entitled Mechnics of Continu, but this dd-on to the initil editions is wek nd not up to the stndrds of his previous Chpters. 3 7

Chpter 3: GENERALIZATIONS OF THE BASIC FUNCTIONAL Homework Assignment for Chpter 3 EXERCISE 3.1 Find first integrl of the Euler eqution (7.4) if the functionl (7.1) is function of y only, tht is F = F(y ). Hint: the Euler eqution (d 2 /dx 2 )/ = 0 cn be immeditely integrted twice in x.) EXERCISE 3.2 (Gelfnd-Fomin) Find the extremls of the functionl subject to the boundry conditions π/2 J = (y 2 y 2 + x 2 ) dx 0 y(0) = 1, y (0) = 0, y(π/2) = 0, y (π/2) = 1 Prtil nswer: y = 1 (1 + 2 e π/2 )e x + 3 more terms. EXERCISE 3.3 (Gelfnd-Fomin) Find the extremls of π/2 J = (y 2 + z 2 + 2yz) dx 0 where y = y(x) nd z = z(x), subject to the boundry conditions y(0) = 0, y(π/2) = 1, z(0) = 0, z(π/2) = 1 (Hint: exploit symmetry in the ppernce of y nd z). EXERCISE 3.4 (Gelfnd-Fomin; tougher). Find the Euler eqution nd the boundry term tht gives essentil/nturl boundry conditions for the two-dimensionl functionl J = F(u, u x, u y, u xx, u xy, u yy, x, y) d, (E3.1) where u = u(x, y) is C 2 function over the domin with boundry : Hint: use the integrtion-by-prt formuls given in Exercise 4 of Chpter 7 of Gelfnd-Fomin, nd the hint for Exercise 5. The bbrevitions p x = / u x, p y = / u y, p n = p x c + p y s, m xx = / u xx, m xy = / u xy, m yy = / u yy, m nn = q xx c 2 + q yy s 2 + 2q xy sc, m tt = q xx s 2 + q yy c 2 2q xy sc, etc. with c = cos(n, x) = dx/d, s = cos(n, x) = dy/d, my be useful. [This result will be used lter for bending of thin elstic pltes, where u represents the trnsverse displcement, F the strin energy, p the plte slopes, nd m the bending moments. If you despertely need help, red pp. 88 91 of Timoshenko s Pltes nd Shells.] 3 8