1.9 C 2 inner variations


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1 46 CHAPTER 1. INDIRECT METHODS 1.9 C 2 inner vritions So fr, we hve restricted ttention to liner vritions. These re vritions of the form vx; ǫ = ux + ǫφx where φ is in some liner perturbtion clss P, for exmple, P = C c, b or P = C 1 [, b]. These vritions hve the dvntges tht the dependence on ǫ is simple nd explicit, nd there re wellunderstood useful liner perturbtion clsses vilble. Sometimes, especilly in the context of nonliner dmissibility conditions, it is more or less necessry to consider more generl vritions. I sy more or less for two resons. First, if one restricts ttention to the theory of the first vrition, then considertion of more generl vritions does not for the most prt yield ny essentilly new or different informtion. This point cn be mde little more precise. A generl vrition of u A is function v :, b ǫ 0, ǫ 0 R where ǫ 0 > 0 nd vx; 0 = ux, 1.28 the condition 1.28 being the essentil defining one. In this generlity, the question of how v depends on ǫ cn be very subtle, nd the regulrity of v in both vribles x nd ǫ requires creful ttention. On the other hnd, one hs much more flexibility, in generl, when it comes to dmissibility conditions. Returning to the point under considertion, the perturbtion ssocited with generl vrition v is usully defined to be 5 ϕx = v = v x; 0, 1.29 nd it cn be shown, for exmple, under rther generl hypotheses tht the first vrition with respect to generl vritions, defined by δf u = d dǫ F[vx; ǫ] depends only on the perturbtion ϕ nd is given by the formul δf u [ϕ] = [F z x, u, u ϕ + F p x, u, u ϕ ] dx. 5 More generlly, the prtil derivtive v/ defined t generl x; ǫ is sometimes clled the vrition vector.
2 1.9. C 2 INNER VARIATIONS 47 It will be noticed t once tht the first vrition formul for generl vrition is essentilly identicl to the one ppering in the condition 1.1 of Proposition 1 for liner vrition; the only difference is tht the liner perturbtion φ of 1.1 is replced by the generl perturbtion ϕ of The second reson generl vritions my be of somewht secondry interest is tht they re rrely used in pplied mthemtics, i.e., by pplied mthemticins. Even when the second vrition is considered to determine or define stbility of physicl system nd the use of generl vritions cn mke difference, pplied mthemticins tend to prefer the use of liner vritions nd the ssocited condition of liner stbility s opposed to the more nturl though more complicted notion of nonliner stbility ssocited with generl vritions. It my be noted, in this connection, tht the term stbility is used for both concepts in the literture, but liner stbility is wht most uthors nd especilly pplied mthemticins usully hve in mind. We now proceed to consider n interesting specil clss of generl vritions which gives some insight into the origin of the Erdmnn opertor nd the first integrl eqution; the discussion lso leds to generliztion clled Noether s eqution. A C 2 prmeter vrition of the intervl [, b] is function stisfying x The prtil derivtive ξ C 2 [, b] ǫ 0, ǫ 0 > 0, ξ; ǫ, ξb; ǫ b, nd 1.30 ξx; 0 = id [,b] x ψ = = x; is clled the prmeter perturbtion or inner perturbtion. Given C 2 prmeter vrition nd nd extreml u C 1 [, b], nd inner vrition or C 2 inner vrition is generl vrition of the form vx; ǫ = uξx; ǫ When considered s generl vrition, n inner vrition hs the peculirity tht the ssocited perturbtion depends on the vried function: ϕx = u x x; 0 = u xψx.
3 48 CHAPTER 1. INDIRECT METHODS For this reson, the first vrition functionl ssocited with n inner vrition is given different nottion by the uthors of BGH: F u [ψ] = d F x, v, v dx dǫ x = d F x, uξ, u ξ dx dǫ x. They lso give this expression specil nme: the first inner vrition of F in the direction ψ. If u hs dditionl regulrity, sy u C 2 [, b], then we cn write F u [ψ] = δf u [u ψ]. We will follow up on this sitution fter we estblish the min results on inner vritions from BGH Generlized Erdmnn s eqution We first estblish formul for the first inner vrition. Theorem 7 Proposition 1.14 in BGH. First inner vrition formul If u C 1 [, b] nd v = vx; ǫ is C 2 inner vrition of u with prmeter perturbtion ψ, then where F u [ψ] = F[u] = [Euψ F x x, u, u ψ] dx 1.34 Fx, u, u dx is the usul integrl functionl with C 1 Lgrngin nd Eu = u F p x, u, u Fx, u, u is the Erdmnn opertor ppering in the first integrl eqution. Proof: For ech fixed ǫ, the prmeter trnsformtion x ξx; ǫ hs n inverse, which we denote ξ 1 = ξ 1 η; ǫ. Differentiting the reltion with respect to x we find ξ 1 ξx = x 1.35 ξx x = 1 or x = 1 ξx.
4 1.9. C 2 INNER VARIATIONS 49 Strting with the definition of the first inner vrition, we chnge vribles using η = ξx; ǫ so tht dη = x 1 dx nd dx = η dη. Thus, F u [ψ] = d dǫ = d dǫ F x, uξ, u ξ dx x F ξ 1, uη, u η 1 dη. The dependence on ǫ in the integrnd is now isolted in the ppernces of ξ 1, nd we hve b F u [ψ] = {[F x ξ 1, uη, u η 1 F p ξ 1, uη, u η 1 u η 2 ξ F Since ξx; 0 = id [,b] x, we hve lso ξ 1, uη, u η 1 2 ξ 1 } dη ξ 1 η; 0 = id [,b] η nd η; 0 1. Furthermore, we cn differentite 1.35 with respect to ǫ to obtin + 1 Evluting t ǫ = 0, this mens = 0 or = 1. = ψη.
5 50 CHAPTER 1. INDIRECT METHODS Similrly, 2 ξ 1 η; 0 = ψ η. Mking these substitutions in our expression for F u [ψ], we hve F u [ψ] = {[F x η, uη, u η ψ F p η, uη, u ηu η ψ ] + F η, uη, u η ψ } dη. Renming the vrible of integrtion η to x nd rerrnging the terms, we see this is Theorem 8 Proposition 1.16 in BGH 6. If u C 1 [, b] nd F u [ψ] = 0 for ll C 2 inner vritions, then there is some constnt c such tht Eu = u F p x, u, u Fx, u, u = c x F x η, uη, u η dη nd d dx [u F p x, u, u Fx, u, u ] = F x x, u, u. The first eqution, involving Erdmnn s first order opertor, generlizes the theorem of the first integrl. The second eqution is clled Noether s eqution. As with the first integrl eqution, this eqution, presumbly, should not be considered equivlent to the EulerLgrnge eqution, but rther second order eqution with possibly mny more solutions thn the EulerLgrnge eqution. Proof: For ny ψ C c the prmeter vrition ξx; ǫ = x + ǫψ is welldefined for ll ǫ smll enough. Tht is, uξx; ǫ is n inner vrition with inner perturbtion x; 0 = ψx. 6 There seems to be sign error in the sttement of this result in BGH; if the function Φ used in BGH is replced with wht we hve clled the Erdmnn opertor, the sttement seems to be correct.
6 1.9. C 2 INNER VARIATIONS 51 According to 1.34 we hve shown [Euψ F x x, u, u ψ] dx = 0 Integrting by prts we get for ll ψ C c, b. [Eu + g]ψ dx = 0 for ll ψ C c, b where gx = x F x η, uη, u η dη. Notice tht g C 1 [, b]. It now follows from the Lemm of DuBoisReymond tht there is some constnt c for which Eu = c gx. This estblishes the first ssertion of the theorem. Becuse the right side c gx is C 1 function, the left side is differentible too, nd Noether s eqution follows from differentition Inner vritions with higher regulrity We used chnge of vribles in the clcultion of the first inner vrition formul which we then used to derive the generlized first integrl reltions. If we hve dditionl regulrity, we should be ble to mke this clcultion without the chnge of vribles. Let us verify tht this is the cse. If u C 2 [, b] nd ξ C 2 [, b] ǫ 0, ǫ 0 is prmeter vrition with ssocited inner vrition v = uξx; ǫ, then the formul for the first inner vrition becomes F u [ψ] = d F x, v, v dx dǫ x [ = u F z x, u, u + u F p x, u, u ] + u F p x, u, u 2 ξ dx x = [ u F z x, u, u + u F p x, u, u ψ + u F p x, u, u ψ ] dx.
7 52 CHAPTER 1. INDIRECT METHODS Notice tht d dx Fx, u, u = F x x, u, u + u F z x, u, u + u F p x, u, u. By dding nd subtrcting the integrl F x x, u, u ψ dx, integrting by prts, nd pplying the lemm of DuBoisReymond, we rrive t the sme conclusions of Theorem 8 under the dditionl regulrity ssumptions. See Exercise 22 Exercise 22. Crry out the detils of the proof of Theorem 8 suggested bove under the dditionl regulrity ssumption u C 2 [, b]. Exercise 23. An inner vrition is sid to be Cc I, b such tht ξ x. [,b]\ī if there is some intervl Give condituions on the prmeter vrition ξ nd the function u under which there is some ϕ Cc, b such tht the inner vrition v = uξx; ǫ stisfies the formul [ ] F u x; 0 = δf u [ϕ] Vritionl constrints; Lgrnge multipliers It is quite common to encounter vritionl problem with n integrl constrint of the form G[u] = Gx, u, u dx = g 0 constnt. For exmple, rther thn looking for the shortest grph connecting, 0 to b, 0 in the plne, which is esily seen to be given by the grph of ux 0, one my look for the shortest grph connecting these two points mong grphs enclosing long with the segment long the xis between the two