Calculus of variations. I = F(y, y,x) dx (1)

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1 MT58 - Clculus of vritions Introuction. Suppose y(x) is efine on the intervl, Now suppose n so efines curve on the ( x,y) plne. I = F(y, y,x) x (1) with the erivtive of y(x). The vlue of this will epen on the choice of the function y n the sic prolem of the clculus of vritions is to fin the form of the function which mkes the vlue of the integrl minimum or mximum (most commonly minimum). The sort of question which gives rise to this kin of prolem is exemplifie y the Brchistochrone prolem, solve y Newton n the Bernoullis (the nme comes from the Greek for shortest time ). This consiers prticle sliing own smooth curve uner the ction of grvity n poses the question s to wht curve minimises the time for the prticle to slie etween fixe points A n B. A Clerly the time will nee to e foun y clculting the spee t ech point then integrting long the curve. Other exmples rise in vrious res of physics in which the sic lws cn e stte in terms of vritionl principles. For exmple in optics Fresnel s principle sys tht the pth of light ry etween two points is such s to minimise the time of trvel etween the two points. The Euler-Lgrnge eqution. B First recll the conition uner which n orinry function y(x) hs n extremum. If we expn in Tylor series

2 y(x + δx) = y(x) + δx y (x) + 1 δx y (x) +... then the conition is tht the term proportionl to δx must vnish, so tht if the secon erivtive is non-zero the ifference etween y(x + δx) n y(x) will lwys hve the sme sign for smll δx. The sme principle pplies to our present prolem. Wht we o is consier smll chnge in the function y(x), replcing it with y(x) + η(x). (Note tht ll the functions we introuce re ssume to hve pproprite properties of ifferentiility etc, without prticulr comment eing me.) We then prouce chnge in the integrl, which cn e expne in powers of η. We emn tht the term proportionl to η vnishes. Sustituting into (1) we get I(y + η) = F(y + η, y + η,x)x F = F(y, y,x)x + y η + F η x +O(η ) so tht wht we wnt is F y η + F η x =. () Integrting the secon term y prts gives F y F x η(x)x = (3) In otining this we hve ssume tht η() = η() =, ie the perturtion vnishes t the en points, leving the en points A n B of the curve unchnge, s shown elow. y A B x The unperture curve (full line) n the perture curve ( otte line) Since this must hol for ll η(x)we otin

3 F y F x =. (4) This is the Euler-Lgrnge eqution, the sic eqution of this theory. It is ifferentil eqution which etermines y s function of x. Exmples. () Fin the curve which gives the shortest istnce etween two points on plne. If the curve is y = y(x) then the element of length is so we wnt to minimise l = x + y = 1 + y x 1 + x (where n re the x-coorintes of the points of interest). The integrn is inepenent of y so we just get 1 + x = giving = const, or y = const. As expecte this just gives stright line 1 + y = mx +c with the constnts fixe y the positions of the en points. () Fin the curve which minimises (y + y )x The Euler-Lgrnge eqution for this is x ( y ) y = y = n if we multiply y we get first integrl y = const. Assuming this constnt to e positive n equl to we get the solution y = sinh(x +). If the constnt is negtive we cn tke it to e n get the solution y = cosh(x +). In oth cses is constnt n n nee to e foun using given en points. This is firly simple, rtificil exmple, ut it illustrtes more generl point. Note tht we coul esily fin first integrl n reuce the prolem to first orer DE. The existence of first integrl like this turns out to e generl property of the Euler- Lgrnge eqution whenever the integrl hs no explicit epenence on x.

4 Uner these circumstnces, if we multiply the E-L eqution y y we get or y x F F y = F x F F y =. Since F oes not contin x explicitly, the lst two terms comine to give F, the totl x erivtive. So, we get the first integrl F F = const (5) As more interesting exmple we return to the rchistochrone prolem mentione erlier. Suppose two points A n B re connecte y smooth rmp long which prticle cn slie, strting t rest t A. Tking A t the origin n the y irection verticlly ownwrs, then t point (x,y) on the curve, the prticle spee is given y v = gy (with g the ccelertion ue to grvity). The time to move n increment (x,y) long the curve is t = x + y / v = 1 + So, the integrl which we nee to minimise is 1 + n the first integrl of the Euler-Lgrnge eqution s erive ove (Eq. (5)) is This simplifies to (with k = 1 /c ) or y(1 + y ) y y y(1 + x 1 + y = k y(1 + y ) = k y = k y = c. This cn e integrte y mking the sustitution y = k sin θ, giving y. x gy

5 which hs the solution y x = k sin θ cos θ θ x = cos θ sin θ x = k ( θ sin θ) + K. Putting = k / n φ = θ we get prmetric equtions for the curve in the form x = (φ sin φ) + K y = (1 cosφ). As illustrte y the igrm elow, these represent cycloi, the curve trce out y point on the circumference of wheel of rius rolling long the x xis. A x φ B y Since the curve psses through the origin, K =. The vlue of is etermine y the conition tht the curve psses through B. More thn one epenent vrile. Suppose F = F(y 1, y 1,y, y,y 3, y 3,...) with ech y i = y i (x)n gin we re looking for n extremum of Fx. The nlysis procees s efore, replcing ech y i with y i + η i. Since ech η i cn e chosen inepenently, we must let the coefficient of ech in the integrn vnish. We en up with system of Euler-Lgrnge equtions

6 F = F y i x y i. (6) It hs, of course, een ssume tht the en points re fixe, s efore. Exmple: Fin the curve which minimises 1 ( y + z + y )x n which joins the points (,, ) n (1,1,1). The E-L equtions re with generl solutions x ( y ) y = x ( z ) = y = cosh x + sinh x z = cx +. Imposing the en point conitions gives the curve y = cosh 1 1cosh x Hmilton s Principle z = x. ( ) n Suppose conservtive ynmicl system is escrie y coorintes q 1,q,...,q n the rtes of chnge of these re &q i ( i = 1,...,n ). Then the kinetic energy of the system is, in generl T(q 1,...,q n ; &q 1..., &q n )n the potentil energy is V(q 1,...,q n ). The Lgrngin is then efine y L = T V n Hmilton s principle sttes tht long the prticle orit the integrl t t 1 Lt hs n extremum. This gives rise to the set of equtions of motion L t &q L q =, usully known in this context s Lgrnge s equtions. These cn e erive from Newton s lws of motion n then Hmilton s principle ecomes euction from them. For complicte systems Lgrnge s equtions re usully esier to hnle thn ny ttempt to work out the equtions of motion irectly from Newton s equtions.

7 Exmple: Fin the equtions of motion for the oule penulum system shown elow. θ φ m M The height of the top o ove its equilirium position is (1 cos θ) n the height of the lower o ove its equilirium is (1 cos θ) + (1 cos φ). So V = mg(1 cos θ) + Mg (1 cos θ) +(1 cosφ) The horizontl component of velocity of the top o is θ & cos θ n the verticl component θ & sin θ. For the lower o the corresponing components re θ & cos θ + φ & cosφ n θ & sin θ + φ & sin φ n so T = 1 m θ & + 1 M ( θ & cos θ + φ & cosφ) + ( θ & sin θ + φ & sin φ) = 1 m θ & + 1 M θ & + φ & + θ & φ & sin(θ + φ) From Lgrnge s equtions we then get t m θ & + M &θ + M φ & sin(θ + φ) ( & t M φ + M θ & sin(θ + φ) ) + Mg sin φ =. ( ) + mg sin θ + Mg sin θ = Prolems with constrints. Recll tht to fin the extremum of function of severl vriles with constrints impose we use Lgrnge s metho of unetermine multipliers. An exct nlogy hols in the cse of clculus of vritions. Suppose we wnt to fin the extremum of

8 suject to the conition I = F(y,,x)x H = G(y, y,x)x = const. Then we pply the Euler Lgrnge equtions to F λg (or F + λg if you prefer) with λ n unetermine multiplier which is etermine y the constrint n the en points. Exmple; A hevy chin with constnt mss/unit length is suspene etween two points. Wht curve oes it tke up in equilirium? The equilirium conition is such s to minimise grvittionl potentil energy. x With the geometry shown this mens tht we minimise yl where the element of length is given y l = 1 + y x. There is lso the constrint tht the totl length is fixe, so we must minimise suject to y y 1 + x 1 + x = const. Thus we pply the Euler-Lgrnge eqution to y λ ( ) 1 + y. Since this hs no explicit epenence on x we cn use the result we otine lrey (Eq. (5)) to get first integrl, nmely, from which we get y ( y λ) 1 + y λ ( ) 1 + = y λ 1. k y = k

9 If we mke the sustitution y λ = k cosh z this gives z = 1 so tht z = x +c n we otin y = λ + k cosh(x +c). The three constnts k, c n λ re otine from the coorintes of the en points n the length. This curve is clle ctenry. If there is more thn one constrint then we introuce more thn one multiplier. Exmple: In sttisticl mechnics, the istriution of energy of system of prticles is escrie y proility istriution function f (E). In equilirium, theory sys tht this istriution shoul e such s to mximise the function suject to the conitions f log fe f (E)E = 1 Ef (E)E = E. The first of these is just the stnr conition on proility function. The secon, with E given constnt, sys tht the verge energy per prticle, or equivlently the totl energy of the system, is fixe. The Euler-Lgrnge eqution with these constrints is ( f log f λf µef ) = f with λ n µ the two multipliers corresponing to the two constrints. This yiels f = Ce µe where C is constnt into which λ hs een incorporte. Using the two constrints gives µ = 1 / E, C = E. This is the Boltzmnn istriution n E is proportionl to the temperture of the system. The isoperimetric prolem - fin the shpe which hs mximum re for given perimeter. Suppose the prmetric equtions of the require curve re x = x(t) y = y(t) t t t 1 with x(t ) = x(t 1 ), y(t ) = y(t 1 ), so tht we hve close curve. The length is n is fixe. The re enclose is L = t 1 t A = ( &x + &y ) 1/ t xy A

10 which, y mens of Green s theorem, cn e expresse s the line integrl t 1 A = 1 (x &y y &x)t. t So, we construct the Euler-Lgrnge equtions for the two vriles x n y from the function φ(x,y, &x, &y) = 1 (x &y y &x) λ( &x + &y ) 1/. These equtions re t y λ &x &x + &y x t λ &y &x + &y These cn e integrte immeitely to give ( ) 1/ ( ) 1/ y + &y = &x =. λ &x ( &x + &y ) = A x λ &y ( &x + &y ) = B Multiplying the first of these y &x n the secon y &y n ing gives &x(x B) + &y(y A) = This hs the integrl ( x B) + ( y A) = const. so tht the require curve is circle. Geoesics If G(x,y,z) = efines surfce in three imensionl spce, then the geoesics on this surfce re the curves which prouce the shortest istnce etween points on the surfce. So, s we hve seen, the geoesics on plne re just stright lines. We cn cst the prolem of fining geoesics on surfce into vritionl prolem with constrint s follows. If x = x(t),y = y(t),z = z(t) re prmetric equtions for curve on the surfce, then long ny curve on the surfce t 1 ( ) G x(t),y(t),z(t) t = (8) t

11 Since the element of length long curve is &x + &y + &z t, the prolem is to minimise t 1 t &x + &y + &z t suject to the constrint (8). This gives &x G λ t F x = plus similr equtions with for y n z, with F = &x + &y + &z. As prticulr exmple consier geoesics on the sphere, for which G = x + y + z R n so the equtions for the geoesic re &x &y &z t F t F t F = = = λ. x y z Expning the erivtives in the first eqution gives xf && &x F & yf = && &y F& xf yf which cn e rerrnge into y&& x x&& y y &x x &y = F & F. In similr wy, z&& y y&& z z &y y &z = F & F. We now equte these two expressions for & F / F n write the result in the form which integrtes to give Writing this in the form we cn integrte gin to get (y &x x &y) (z &y y &z) t = t y &x x &y z &y y &z y &x x &y = C 1 (z &y y &z). &x +C 1 &z x +C 1 Z = &y y x +C 1 z = C y.

12 This is the eqution of plne pssing though the origin. So, the geoesics on sphere re the curves forme y the intersection of the sphere n plnes through its centre. These re the gret circles on the sphere. Estimte of n eigenvlue using vritionl metho. Suppose we hve prolem x (p(x) y ) + q(x)y = λr(x)y y()=y(1)=. (7) This will oviously hve trivil solution y =, ut for certin vlues of λ (the eigenvlues ) there will e non-trivil solutions. If we consier the clculus of vritions prolem of minimising suject to the conition tht 1 { } I = p(x) q(x)y x 1 J = r(x)y = const. n the given ounry conitions on y, then we otin the ove eqution from the Euler-Lgrnge equtions n the metho of multipliers. The lowest possile eigenvlue is then the minimum possile vlue of I/J. Since J is constrine to e constnt this is just equivlent to the prolem of minimising I suject to J eing constnt. The stnr pproch to this les ck to the DE n we my pper to e going roun in circles. The usefulness of this pproch is tht if we use ny function y(x) then the resulting vlue of λ is greter or equl to the minimum possile, so we otin n upper oun on the lowest eigenvlue. With choice of y which is resonle pproximtion to the solution we cn get goo estimte. Exmple: Use this technique to fin n estimte of the lowest eigenvlue of the prolem y + λy = y() = y(1) = This is, of course prolem to which we know the solution, nmely tht the eigenvlues re given y λ = n π, so the lowest is π = (corresponing to the solution y = sin(πx)). We wnt tril function with the require en vlues, n preferly one which is esily integrte (though numericl integrtion is reily one with pckge like MAPLE). Let us tke y = x(1 x). Then p(x) = 1 q(x) = r(x) = 1 n so

13 1 I(x) = (1 x) x = 1 3 c 1 J = x (1 x) x = 1 3 giving n upper oun of 1, which is ctully firly goo pproximtion to the lowest eigenvlue. Vrints on the ounry conitions re possile, for exmple in the following. Exmple: Fin the lowest eigenvlue for the prolem with y () = y(1) =. x (x y ) + λxy = A simple function stisfying the ounry conitions is y = 1 x. For which 1 I = x( x ) = J = x(1 x ) x = 1 giving n upper oun of 6. One wy of mking this proceure more ccurte is to introuce one or more unknown prmeters into the ssume function. For, exmple here we coul tke y = (1 x )(1 +cx ), which retins the correct ounry conitions. Then I J = c + 1 c c c The point of the exercise is tht this gives n upper oun for ny vlue of c. So, if we minimise this expression with respect to c we will get the est possile estimte for this form of y. Differentiting with respect to c we get the conition for minimum tht 4c + 15c + 3 =, n the root which gives minimum is c =.35. The corresponing vlue of I/J is The eqution hs solution which is Bessel function n from this the eigenvlue cn e clculte to e The technique of introucing unknown prmeters (the Ryleigh-Ritz metho) n minimising with respect to them is very useful technique when the minimum vlue of n integrl is neee. Note tht even in this simple exmple the lger is teious, so tht use of computer lger system is ig help. In mny prolems (eg fining the groun stte energy in quntum system) the lowest eigenvlue is ll tht is neee. It is possile to exten this technique to get higher eigenvlues, ut we shll not pursue this here..

14 Vrile en points Suppose we relx the conition tht the vlues of y e fixe t the en points ut inste ssume tht they re llowe to vry freely. Then, following the proceure which le to Eq. (3) we otin, s well s the term in (3), n extr contriution F η F η. x = x = Since, the extremum, if it exists, must e the extreme vlue for whtever en points turn out to e suitle, the integrl term must vnish s efore. Otherwise slightly greter or smller vlue for the integrl coul e otine y tking ifferent curve with the sme en points. Also, this extr term must vnish which, since η is ritrry, mens tht F y = t oth en points. As simple exmple we cn consier the prolem of minimising the istnce etween x = n x = 1 without fixing y t the en points. Here F = 1 + n the solution of the E-L eqution is stright line s efore. The extr conitions yiel = 1 + t oth ens. The erivtive must then e zero everywhere, so we rrive t the expecte result tht the shortest line etween x = n x = 1 is stright line prllel to the x xis. Another vrint of the prolem is to consier cse where the en points re constrine to lie on given curve. For simplicity, let us ssume tht the lower en point is fixe while tht the upper en point hs to lie on the curvey = g(x). Suppose tht the extremum hs its upper limit t x =, while the lower limit x = is fixe. Then, if y is replce with y + η s efore, there is chnge in the upper limit of the integrl to + x, sy.

15 y+η y y(x) y=g(x) x + x The corresponing chnge in y is y = y( + x) + η( + x) y() x y () + η(). However, there is lso the constrint tht the en point lies on the given curve, which gives y g () x. Putting these reltions together we get η() x = g () y (). Now, look t the chnge in the integrl - + x I = F(y + η, y + η,x)x F(y, y,x)x + x η F y + η F + x x + F(y, y,x)x. In the first integrl here, which contins the smll perturtion η, we cn neglect the chnge in the upper limit. Then if we integrte y prts we get the usul integrl contining the E-L expression, plus contriution F η. x = The secon integrl is pproximtely η() F(y, y,) x = F(y, y,) g () y ().

16 For I to vnish for ritrry η we require tht the E-L eqution e stisfie n lso tht F y + F g y = (9) t the upper limit. If the lower limit h een constrine lso to lie on curve rther thn eing fixe, conition nlogous to (9) woul pply there lso. Exmple: Fin the curve connecting the origin to curve y = g(x) n which hs the shortest length. Here, s we hve seen efore, F = 1 + n the solution of the E-L eqution gives stright line. For this cse, the conition (9), which must e stisfie where the stright line meets the given curve, reuces to g y = 1, implying tht the stright line giving the shortest istnce to the curve must e orthogonl to the curve t the point of intersection. Some further comments In the cse of the simple prolem of fining locl mximum or minimum of ifferentile function we know tht vnishing of the erivtive is necessry conition, ut tht it is not sufficient. The erivtive cn vnish ut the point cn e point of inflection rther thn turning point. For clculus of vrition prolems the sitution is similr, in tht the Euler-Lgrnge eqution is necessry, ut not sufficient, conition for n extremum. In the cse of function, the nture of the criticl point is esily etermine y ientifying the lowest non-zero erivtive. The nlysis of the clculus of vritions prolem is, however, rther complicte n will not e pursue here. The sic ies iscusse here cn e extene in vrious wys, for exmple to integrns which involve higher erivtives of y or to prolems which involve minimising multiple integrl over some given omin. Further Reing Clculus of Vritions R Weinstock Vritionl Clculus in Science n Engineering M J Forry An Introuction to the Clculus of Vritions L A Prs