Some Methods in the Calculus of Variations


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1 CHAPTER 6 Some Methods in the Clculus of Vritions 6. If we use the vried function ( α, ) α sin( ) + () Then d α cos ( ) () d Thus, the totl length of the pth is d S + d d α cos ( ) + α cos ( ) d Setting ( ) u, the epression for S becomes S α cosu + α cos u du (4) The integrl cnnot be performed directl since it is, in fct, n elliptic integrl. Becuse α is smll quntit, we cn epnd the integrnd nd obtin + 8 du S α cos u α cos u α cos u α cos u If we keep the terms up to co s u nd perform the integrtion, we find (5) which gives S 6 + α (6) 65
2 66 CHAPTER 6 Therefore S α 8 α (7) S α (8) nd S is minimum when α. 6. The element of length on plne is ds d + d () from which the totl length is If S is to be minimum, f is identified s ( ) d S d + d + (, ) d, d () f d + d Then, the Euler eqution becomes d d d d + (4) where d. (4) becomes d or, d d + + (5) constnt C (6) from which we hve Then, C C constnt (7) + b (8) This is the eqution of stright line.
3 SOME METHODS IN THE CALCULUS OF VARIATIONS The element of distnce in threedimensionl spce is ds d + d + dz () Suppose,, z depends on the prmeter t nd tht the end points re epressed b ( ( t), ( t), z( t ) ), ( ), ( ), ( ) ( ) t t z t. Then the totl distnce is The function f is identified s t t d d dz S + + dt dt dt dt () f + + z f f f Since, the Euler equtions become z from which we hve d dt d dt d dt f f f z constnt C + + z constnt C + + z z constnt C z (4) (5) From the combintion of these equtions, we hve C C C z C 3 If we integrte (6) from t to the rbitrr t, we hve (6)
4 68 CHAPTER 6 C C z z C C 3 On the other hnd, the integrtion of (6) from to t gives from which we find the constnts t C C z z C C 3 C, C, nd C 3. Substituting these constnts into (7), we find z z (9) z z This is the eqution epressing stright line in threedimensionl spce pssing through the,,,, z. two points ( z ), ( ) (7) (8) 64. z φ ρ ds The element of distnce long the surfce is ds d + d + dz () In clindricl coordintes (,,z) re relted to (ρ,φ,z) b ρ cos φ ρ sin φ z z () from which d ρsin φ dφ d ρcos φ dφ dz dz
5 SOME METHODS IN THE CALCULUS OF VARIATIONS 69 Substituting into () nd integrting long the entire pth, we find φ S ρ dφ + dz ρ + z dφ (4) φ where dz z. If S is to be minimum, dφ f ρ + z must stisf the Euler eqution: f f z φ z (5) f Since, the Euler eqution becomes z z φ ρ + z (6) from which or, z ρ + z constnt C (7) Since ρ is constnt, (8) mens C z C ρ (8) dz constnt dφ nd for n point long the pth, z nd φ chnge t the sme rte. The curve described b this condition is heli The re of strip of surfce of revolution is z (, ) ds (, ) da ds d + d () Thus, the totl re is
6 7 CHAPTER 6 d () A + where d. In order to mke A minimum, f d f + f + + must stisf eqution (6.39). Now Substituting into eqution (6.39) gives d d d + d ( )( ) + dd + + Multipling b Integrtion gives + nd rerrnging gives d d ( + ) ln + ln ln + where ln is constnt of integrtion. Rerrnging gives Integrting gives or ( ) b cosh + which is the eqution of ctenr. cosh b
7 SOME METHODS IN THE CALCULUS OF VARIATIONS θ θ (, ) (, ) If we use coordintes with the sme orienttion s in Emple 6. nd if we plce the minimum point of the ccloid t (,) the prmetric equtions re ( cosθ ) + ( θ + sin θ) Since the prticle strts from rest t the point (, ) 6.9] Then, the time required to rech the point (, ), the velocit t n elevtion is [cf. Eq. ( ) v g () is [cf. Eq. 6.] + t g( ) d Using () nd the derivtives obtined therefrom, cn be written s θ + cosθ t dθ g cosθ cosθ θ (4) () Now, using the trigonometric identit, θ θ + cos cos, we hve t g θ θ cos d θ θ θ cos cos g θ θ cos d θ (5) θ θ sin sin Mking the chnge of vrible, z sin θ, the epression for t becomes t g θ sin dz (6) θ sin z The integrl is now in stndrd form:
8 7 CHAPTER 6 Evluting, we find d sin (7) Thus, the time of trnsit from (, ) the strting point. t (8) g to the minimum point does not depend on the position of 67. θ θ n n (n > n ) The time to trvel the pth shown is (cf. Emple 6.) b d ds t v θ v v + v c n c n d () Although we hve v v(), we onl hve dv d when. The Euler eqution tells us d d v + Now use v c n nd tn θ to obtin () n sin θ const. This proves the ssertion. Alterntivel, Fermt s principle cn be proven b the method introduced in the solution of Problem To find the etremum of the following integrl (cf. Eqution 6.) (, ) J f d we know tht we must hve from Euler s eqution f This implies tht we lso hve
9 SOME METHODS IN THE CALCULUS OF VARIATIONS 73 J f d giving us modified form of Euler s eqution. This m be etended to severl vribles nd to include the imposition of uilir conditions similr to the derivtion in Sections 6.5 nd 6.6. The result is J g + λ j when there re constrint equtions of the form i j ( i ) j ( ) g, ) The volume of prllelepiped with sides of lengths, b, c is given b j i V b c () We wish to mimize such volume under the condition tht the prllelepiped is circumscribed b sphere of rdius R; tht is, + b + c R 4 We consider, b, c s vribles nd V is the function tht we wnt to mimize; () is the constrint condition: Then, the equtions for the solution re { } g, b, c () V g + λ V g + λ b b V g + λ c c (4) from which we obtin bc + λ c + λb (5) Together with (), these equtions ield b + λc Thus, the inscribed prllelepiped is cube with side b c R (6) 3 3 R.
10 74 CHAPTER 6 b) In the sme w, if the prllelepiped is now circumscribed b n ellipsoid with semies, b, c, the constrint condition is given b b c (7) 4 4b 4z where, b, c re the lengths of the sides of the prllelepiped. Combining (7) with () nd (4) gives Then, b c (8) b c, b b, c c (9) 69. The verge vlue of the squre of the grdient of (,, 3) is epressed s I φ V ( ) d d d3 φ within certin volume V φ φ φ v V + + d d d3 () 3 In order to mke I minimum, must stisf the Euler eqution: If we substitute f into (), we hve which is just Lplce seqution: f φ φ φ f f φ i i φ () i 3 φ i i i φ (4) Therefore, φ must stisf Lplce s eqution in order tht I hve minimum vlue.
11 } SOME METHODS IN THE CALCULUS OF VARIATIONS This problem lends itself to the method of solution suggested in the solution of Problem 68. The volume of right clinder is given b V R H The totl surfce re A of the clinder is given b bses side ( ) A A + A R + RH R R+H () We wish A to be minimum. () is the constrint condition, nd the other equtions re where g V R H. The solution of these equtions is A g + λ R R A g + λ H H R H (4) () 6. R θ ds The constrint condition cn be found from the reltion ds Rdθ (see the digrm), where ds is the differentil rc length of the pth: ( ) ds d + d Rdθ () which, using, ields + 4 d Rdθ () If we wnt the eqution of constrint in other thn differentil form, () cn be integrted to ield ( ) A+ R θ ln where A is constnt obtined from the initil conditions. The rdius of curvture of prbol,, is given t n point (,) b r. The condition for the disk to roll with one nd onl one point of contct with the prbol is R< r ; tht is,
12 76 CHAPTER 6 R < (4) 6. The pth length is given b nd our eqution of constrint is () + + s ds z d ( ) g,, z + + z ρ () The Euler equtions with undetermined multipliers (6.69) tell us tht d dg λ λ d + + z d with similr eqution for z. Eliminting the fctor λ, we obtin This simplifies to d d z d z zd z ( ) ( ) ( ) ( ) z + + z + z z z + + z z + z z nd using the derivtive of (), ( zz ) z z ( + zz ) z z + + (6) ( ) ( ) z z z (7) This looks to be in the simplest form we cn mke it, but is it plne? Tke the eqution of plne pssing through the origin: A+ B z (8) nd mke it differentil eqution b tking derivtives (giving A + B z nd B z ) nd eliminting the constnts. The substitution ields (7) ectl. This confirms tht the pth must be the intersection of the sphere with plne pssing through the origin, s required. (4) (5) 63. For the reson of convenience, without lost of generlit, suppose tht the closed curve psses through fied points A(,) nd B(,) (which hve been chosen to be on is O). We denote the prt of the closed curve bove nd below the O is s ( ) nd ( ) respectivel. (note tht > nd < ) The enclosed re is ( ) J(, ) ( ) d ( ) d ( ) ( ) d f(, ) d
13 SOME METHODS IN THE CALCULUS OF VARIATIONS 77 The totl length of closed curve is { } ( ) ( ) + ( ) + + ( ) + ( ) + + ( ) K, d d d g, d Then the generlized versions of Eq. (6.78) (see tetbook) for this cse re f d f g d g d + λ λ () d d d + ( ) f d f g d g d + λ λ d d d + ( ) () Anlogousl to Eq. (6.85); from () we obtin ( ) ( ) from () we obtin ( ) ( ) A + A λ B + B λ (4) where constnts A s, B s cn be determined from 4 initil conditions ( ±, ) nd ( ±, ) We note tht < nd >, so ctull nd (4) ltogether describe circulr pth of rdius λ. And this is the sought configurtion tht renders mimum enclosed re for given pth length It is more convenient to work with clindricl coordintes (r,φ,z) in this problem. The constrint here is z r, then dz dr ( ) ds dr + r dφ + dz dr + r dβ φ where we hve introduced new ngulr coordinte β In this form of ds, we clerl see tht the spce is dimensionl Eucliden flt, so the shortest line connecting two given points is stright line given b: r r r cos ( β β ) φ φ cos this line psses through the endpoints (r, φ ± ), then we cn determine unmbiguousl the shortest pth eqution cos r( φ) φ cos nd z r
14 78 CHAPTER 6 Accordingl, the shortest connecting length is dr dφ r sin dφ l d d I [ ] d ) Treting I[] s mechnicl ction, we find the corresponding EulerLgrnge eqution d ( ) d Combining with the boundr conditions (, ) nd (, ), we cn determine unmbiguousl the functionl form of ( ) (sin ) (sin). b) The corresponding minimum vlue of the integrl is d I [ ] d dcos cot ().64 d sin c) If then I[] ( 3 ) ) S is rc length d dz d 9 S d + d + dz d + + d + + Ld d d d 4 Treting S nd L like mechnicl ction nd Lgrngin respectivel, we find the cnonicl momentum ssocited with coordinte d δl p d d δ 9 d d d Becuse L does not depend on eplicitl, then EL eqution implies tht p is constnt (i.e. dp d ), then the bove eqution becomes 9 3 d + 4 p 9 9 p d + A + d p p + B 4 4 where A nd B re constnts. Using boundr conditions (, ) nd (, ) one cn determine the rc eqution unmbiguousl
15 SOME METHODS IN THE CALCULUS OF VARIATIONS ( ) nd z 3 b) z ) Eqution of ellipse which implies + b b becuse + b b so the miml re of the rectngle, whose corners lie on tht ellipse, is This hppens when b) The re of the ellipse is A M[A] M[4] ib. nd b b ; so the frction of rectngle re to ellipse re is then M[ A] A 68. One cn see tht the surfce z is locll smmetric with respect to the line z where >, <, z <. This line is prbol. This implies tht if the prticle strts from point (,,) (which belongs to the smmetr line) under grvit idell will move downwrd long this line. Its velocit t ltitude z (z < ) cn be found from the conservtion of energ. vz ( ) gz ( + )
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