Some Methods in the Calculus of Variations

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

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.

SOME METHODS IN THE CALCULUS OF VARIATIONS 67 6-3. The element of distnce in three-dimensionl 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 3 + + 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)

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 three-dimensionl spce pssing through the,,,, z. two points ( z ), ( ) (7) (8) 6-4. 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

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. 6-5. The re of strip of surfce of revolution is z (, ) ds (, ) da ds d + d () Thus, the totl re is

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

SOME METHODS IN THE CALCULUS OF VARIATIONS 7 6-6. 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:

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 6-7. 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 6-8. 6-8. 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

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.

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 3 3 3 (9) 6-9. 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 + + 3 3 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.

} SOME METHODS IN THE CALCULUS OF VARIATIONS 75 6-. This problem lends itself to the method of solution suggested in the solution of Problem 6-8. 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 4 + + ln + 4 + 4 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,

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) 6-3. 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

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. 6-4. 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

78 CHAPTER 6 Accordingl, the shortest connecting length is dr dφ r sin dφ l + 6-5. d d I [ ] d ) Treting I[] s mechnicl ction, we find the corresponding Euler-Lgrnge 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 ).667. 6-6. ) 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 + + 4 d Becuse L does not depend on eplicitl, then E-L 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

SOME METHODS IN THE CALCULUS OF VARIATIONS 79 3 8 9 ( ) + 3 3 8 4 nd z 3 b) z.75.5.5.75.5.5.75.5.5 6-7. ) 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 6-8. 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 ( + )

8 CHAPTER 6