VARIATIONS, Calculus of. 1. It has been observed (Maxima and Minima, vol. xv. p. 643) that the origin of the calculus of variations may be traced to John Bernoulli’s celebrated problem, published in 1696 in the Acta Eruditorum of Leipsic, under the following form, Datis in plano verticali duobus punctis A et B, assignare mobili Μ viam AMB per quam gravitate sua descendens, et moveri incipiens a puncto A, brevissimo tempore perveniat ad punctum B. This problem introduced considerations entirely different from those hitherto involved in the discussion of curves, for in its treatment it is necessary to conceive a curve as changing its form in a continuous manner, that is, as undergoing what is styled deformation. This change of form can be treated analytically as follows.
Suppose y = f(x) to represent the equation of a curve, and let us write y=f(x)+aψ(x) (1), where α is an infinitesimal quantity, and ψ (x ) any function of x subject only to the condition of being finite for all values of x within the limits of the problem. Then equation (1) represents a new curve indefinitely close to the curve y =f(x) ; and by varying the form of ψ(x) we may regard (1) as representative of any continuous curve indefinitely near to the original curve.
2. Again aψ(x) is the difference of the y ordinates of the two curves for the same value of x; this indefinitely small difference is called the variation of y, and is denoted by δy. It may be regarded as the change in y arising solely from a change in the relation connecting y with x, while x itself remains unaltered. In general, if y be a function of any number of independent variables x 1 , x 2 ,. .. x n , then δy represents any indefinitely small change in y arising solely from a change in the form of the function, while x 1 , x 2 , &c., are unchanged. Thus the variable y may receive two essentially distinct kinds of increment,—one arising from a change in one or more of the variables, the other arising solely from a change in the relation which connects y with these variables. The former increments are those contemplated and treated of in the ordinary calculus; the latter are those principally considered in the calculus of variations.
We shall follow Strauch, Jellett, Moigno, and the principal modern writers on the subject, by restricting, in general, the symbol δ to the latter species of increment. In many problems both kinds of increment have place simultaneously. Thus, if y =f(x 1 , x 2 , x 3 ), and if ∆y denote the total increment of y, we have = + (2); and so on in all cases. g y g
3. It is readily seen that = and in general jn y ç» β that δ y j ζ = ¾Λ The last equation may be written δD n y=D n δy . (3), where D stands for the symbol of differentiation, d/dx .
4. We shall adopt Newton’s notation and write y for dy/dx, ÿ for d 2 y/dx 2 , y(n) for d n y/dx n and proceed to consider the variation of the general expression V =f(x, y,y,ÿ,... y n ), in which the form of the function f is given, while that of y in terms of x is indeterminate. Here, considering x as unchanged, we have
Now let then we have δ V~= Pδy + P v Dδy + P⅛D~δy + . . . +P n D n δy (4).
5. Next let us consider the variation of the definite i xι integral Z7 = I Vdx, where V does not contain either of the limits x 0 and x 1 . Here, when the limits are unchanged, we have 8U= I δ Vdx, and, when the limits undergo change, J *0
δU=δ∣ 1 Kdx= ∣δVdx+ V 1 dx 1 - V 0 dx 0 (5).
J xo J xo
6. We shall here bring in a new symbol, called the sign of substitution, which was introduced by Sarrus. Thus, if F be any function of x, y, &c., the result of substituting x 1 for x in F —the other variables being unchanged—is
∕ xl F in the form of notation adopted by ∕ x l
Moigno and Lindelöf. The difference between ∕ F and ∣ x ^F is represented by J 1 F. Employing this notation, (5) is written δU= ∣ δVdχ+ ∕ Vdx (6).
J xo ∕ XO
7. We now proceed to transform the last equation, commencing with the case in which V is a function of x, y, and y solely.
∕⅛1 ∕≈ι
Here δU= I <J>δy + P v Dδy}dx+I Vdx, -h F - d ·^ + /y 9+ /^ m
Next let V be a function of x, y, ÿ, and ÿ, then
δ U= Γ (Pδy + Pι∑>δy + P 2 B 2 δy)dx + ∕ Vdx.
Hence, observing that we get W=∫^(i>-⅛ i +⅛)⅛'fe+∕ j .' r *'
+ ∕^ λ -S) s ≈' + ∕> λ (8) ·
In general, if V be a function of x, y, y, ÿ,. . . y n we have δU= Γ∖pδy + P 1 Pδy + . . . + P n D n δy)dx + ∕ 1 Vdx. J a⅛ ' a ⅛ . .
Now let D 1 denote the differentiation of δy, and D 2 the differentiation of P 1 , P 2 , &c.; we may write P m D m δy - ( - 1) m δyD m P m =(D 1 m - ( - 1) m D 2 m )P m δy = (D 1 + D 2 )(D 1 m-1 -D 1 m-2 D 2 + .... ±D 2 m-1 )P m δy
Hence [ 11 p m D m δydx={- l) m f 1 ^^δydx J x 0 J x 0 dx + <- ι )"-⅛^⅛)∙ Applying this to the different terms in the value of δ U given above, " egβt
√>-⅛÷∙∙∙÷<-υ"-≤⅜> ÷∕>S÷∙∙∙÷<-^)⅛ √>5⅛t∕> w ∙
This result may be written
ΛT 1 ,x 1 δU= ∕ {P)δydx+ ∕ Vdx J X 0 'Xo +∣ x ∖{P 1 )δy+{P 2 ')δy+ ... +(P n )δ√ n - 1 >} (10).
when, (P)=P-⅛√⅛- .... (P,)= λ -¾ + ⅛- ..„and so on.
8. This expression consists of three parts,—(1) the de-
P 1 finite integral I (P)δydx, which depends on δy, the change J XO Vdx, x 0 which depends solely on the change in the limiting values of x; and (3) the quantity ∕ 1 {(Z∖)δy + (P 2 )δ^ + · · · } , ' ⅜
which depends on the variations of y, y, &c., at the limits. It is often convenient to write (10) in the abbreviated form δU=L + f Mδydx (ll).
J xo
9. The principal applications of the calculus of variations have reference to the determination of the form of one or more unknown functions contained in a definite integral, in such a manner that the integral shall have a maximum or minimum value. For instance, to determine the form of the function y which renders f Vd jc a maximum or a minimum, we have as above δU-L + ^ 1 Jfδydx. Here, as in the differential calculus (see Infinitesimal Calculus, § 64), for a maximum or minimum value of U it is readily seen that we must have δ U= 0; this leads to Γx 1 L = 0 and I Jfδydx = 0. Now the latter integral cannot J xo . be zero for all indefinitely small values of δy unless M = 0 for all values within the limits of integration. Hence we get the differential equation in y,
<>’>·
10. We here suppose that there is no restriction on δy, so that for any value of x the increments + δy and - δy are equally compatible with the conditions of the problem. The reasoning consequently will not apply if the conditions render this impossible. For instance, if a curve be restricted to lie within a given boundary, then for all points on the boundary the displacements must be inwards, and the opposite displacements are impossible. In this case the curve satisfying a required maximum or minimum condition consists partly of portions of the boundary and partly of portions of a curve satisfying the equation M= 0.
11. The equation M =0 is in general a differential equation of the degree 2n; accordingly its solution usually contains 2n arbitrary constants. The values of these constants are to be determined by aid of the equations deduced from L =0 in combination with the given conditions at the limits. For example, if V be a function of x, y, dP and ÿ solely, the solution of P--^=0 is in general of the form y=f(,x, c 1 , c 2 ). Now, suppose that the limiting points are restricted to the curves y 0 =f o (x o ) and y 1 = f 1 (x 1 ), then we get at the limits δy0+y0dx0=f 0 '(x 0 )dx 0 and δy 1 + y l dx l =f 1 '(x 1 )dx 1 . Hence, substituting in L = 0 the values of δy 0 and δy 1 derived from these equations, we get, since dx 0 and dx 1 are arbitrary,
F"o + (∙Pι)oC∕o ’ (⅜) - ⅛ ’ o) = θ∖ ∕'i¾'(
P 1 +(Λ)ι(ΛX¾)→ι)=oΓ∙∙÷ ( λ
We have therefore six equations for the determination of the six quantities x 0 , y 0 , x 1 , y 1 , c 1 , and c 2 .
The integration of the equation M =0 is much simplified in particular cases. -
(1) Let V be a function of x and y solely, then we have =0, therefore P 1 = c (14).
(2) If V be a function of y and ÿ solely, P=⅛⅜, also ⅛ V) = Pÿ + P 1 jf dP dx dx
= -τ⅛' + P 1 y. Hence, integrating, we get V=P 1 y + c (15), where c is an arbitrary constant.
12. For example, let us consider Bernoulli’s problem of the curve of quickest descent under the action of gravity. Take the axis of x vertical and that of y horizontal, and suppose the particle to start from the point x 0 , y 0 , with the velocity due to the height h; then, if v be the velocity at any point, we have v 2 =2g(x + h-x 0 ), also dt= —= ..... Here, neglecting a constant factor, we may v √2<7√z + ⅛-ar 0 & J write Z7=; and by ( 14 ) we get - =c V¾ + hx„. jχ o ^dx + h-x o , ∖L+y
Now, writing a for l∕c 2 and assuming x + h - x 0 = a sin 2θ , we find dy =tan θdx=a(1 - cos2θ)dθ ; therefore y=a(θ - sin θ cos θ) + K, and we infer that the curve is a cycloid.
(a ) If we suppose the upper limit to be fixed and the lower restricted to the curve y 1 = f 1 (x 1 ), we get from (13)
l + tanθ 1 f 1 '(x 1 )=0 (16); accordingly the curve intersects the bounding curve orthogonally. (β ) Next let the upper limit be restricted to the curve y 0 = f 0 (x 0 ); then since V contains x 0 , we have an additional term in δU arising from C x ^d V the change in x 0 , viz., dx 0 I dx ; consequently the coefficient of J x o clx o dx 0 in δU is f'⅛⅛x - {F o + (PM ’ { χ o) ~ ^o)}, f x ιdV and we have J ^d χ = f r o + (Λ)o(∕o'(a ⅛) ~ ⅛ ’ o) (17)∙
If we substitute for V its value, we get without difficulty from this equation 1 + tan θ 1 f 0 '(x 0 ) = 0. Comparing this with (16), we see that the tangents to the bounding curves at the extremities of the trajectory are parallel (Moigno, op. cit., p. 230).
13. More generally, consider the curve for which U— ί μds rχι . J xo = Ι ΛV1 + i> 2 dx is a maximum or a minimum, where μ rs a given
* ∙^0 d∣JL U.Ï/ function of x and y. Here P= ~r∖∕l + z∕ 2 , P- l =— ==λ- , and we get ⅜ 1 √Γ+Z 6 dμ Λ ίdμ .dμ∖ d ( ÿ ∖ + ÿ2= √i7FΛ^ ⅞∕ ; hence' if ρ be the radius of curvature, we have
Γ-J(cos ⅛ +c,,s ^t) < 18 >’
where α, β are the angles which the normal makes with the coordinate axes respectively. Jellett remarks that, if the proposed /ds ·
• , we should then have arrived at the equation
- = -i cos a, c ^ + cosS⅛Y and consequently have seen that the two P μ∖ dx dyj curves contained under the equation ρ 2 =f(x,y,y) are such that, if one renders J^ μds a maximum or a minimum, the other possesses the same property with regard to (—.
14. Next, to find the curve such that the surface generated by its revolution round a given line shall be a minimum. Here, neglecting a constant multiplier, we may write U = I yVl + y i dx;
J XQ
accordingly by (15) we have y =c√1+ y 2 . This gives -, —- = . ∖∕y--c i c , , x+b 1 ∕ ∖⅞ 2 -c 2 ∖
and hence —= log (yd —— 1, or y = c -^+b∙)∣c +e -(x+b)l^ (19), which represents the common catenary.
15. We now proceed to generalize the results in § 11. If V be a function of x, ÿ, ÿ, . .. y {n not containing y explicitly, we have by integration
Λ-S÷⅛-∙÷<-υ-^ -
where c is an arbitrary constant. Again, if V does not contain x explicitly, ^( Γ)= ÿP+yP 1 + ... + ^ ft+1 ^P n ∙ Also, from the equation 3f= 0 we see that P=^- 1 -⅛⅛ + ... - ( -1)" ςτ~p; hence dx dx i dx n =(D 1 + Z) s X⅛P 1 ) + (2√ - Λ 2 )(^<s) +... + (Λ" - ( - 1 WW,), where D 1 represents the differentiation of y, ÿ, &c., and Z> 2 that of ∕∖, P 2 , &c. Consequently, by integration
Γ=c + 2 ∕(P 1 ) + ÿ(P 2 ) + . ·. + √ 7l >Λ 1 (21), a differential equation of the order 2n - 2 at the highest. Again, if V contains neither x nor y explicitly, we may substitute c' for (P 1 ), and consequently we have
V=c + c'⅛+y(P 2 )+ ... +√ n >P n (22).
16. As an example, let us investigate the curve for which the area between the curve, its evolute, and the extreme radii of curva-fs 1 ∕∙¾(1 + λ2)2 ture shall be a minimum. Here U= pds= — -d x. Hence J √ J χ 0 y by (22), since yP 2 = - V, we have P^=c 1 + c i f∣, or p = . C1 - -1—-=; ds . Vl + ÿ 2 VI + y 1 therefore 5 - = c 1 cos <∕> + c 2 sin ≠, where φ is the angle made by the dφ tangent with the axis of x. This leads to s= -c 1 sin ≠ + c 2 cos φ + c 3 (23), which shows that the curve is a cycloid. Again, (1) if the extreme points be fixed, the equations at the limits are (P 2 ) 0 = 0, (P 2 ) 1 = 0. This shows that the radius of curvature vanishes at both limits and the curve must be a complete cycloid. (2) If the extremities lie on the curves y 0 -∕o(⅞)> Vι=fι( x ι) ! then the equations furnished by L =0 are Po + (P])ol∕o'(⅜) - ⅞i = θ> (Ρ2)ο = θ1 ι<)i∖ i r ι + (P 1 ) 1 W(a∙ 1 )~⅛ι} =0, (P 2 ) 1 = 0 f
Hence the extreme points are cusps on the cycloid; moreover, since in this case V o = 0 and V 1 = 0, we have y 0 -∕ 0 '(a⅛) = θ> ⅛ 1 -∕ 1 '(αq) = 0, which show that the cycloid at each of its extremities touches these curves, and that the line joining the extreme points is normal to each of the bounding curves.
It is easily seen that the minimum area is four times that of the circle which generates the cycloid. vι
17. Next let us consider the variation of £7= I Vdx, J Xo where V is a given function of x, y, 2, ÿ, 2,y,2,... y^ n ∖ zt m ∖ and y, z are undetermined functions of x. τ . λ dV n dV r ,
Ut ' Q= li ,Q t =- 3i ,l u1 ., then SU= ∕ 1 ^Pdx+ ί (Pδy + P 1 Dδy +.. ,+P n IPSy)dx
'Xo J x o
Γ x ι
+ I (Qδz+Q 1 Dδ2 + . . . +Q m D δz)dx. J zθ
Proceeding as in § 7, we find δU= Γ∖p)δydx + Γ 1 (Q)δzdx + ∕ xi Vdx
J Xn J Xλ I x a
+ ∕ xι {(P 1 )δy+(ΛW+· · · + 7 ’ n δ√ n - υ }
∕ x 0 + / X1 {(Qι)Sz+(Q a )δz + ...+ Q m δz<∙ m - 1 >} (25), ' Xo where (7 1 ), (Z ’ 1 ),... have the same meaning as in § 7, and (Q ), (Q 1 ),.. . are the corresponding functions for the variable z . r χ ι
18. The determination of y and z where 1 Vdx is a χ o maximum or a minimum leads, as in § 8, to the equations (P ) = 0, (Q)=0, along with the equation at the limits ∕Vd⅛+ ∕* 1 {(P 1 ')δy + ...P n δ∕ n ~ γ >} + ∕ x ∖(Q 1 )δz+ ...Q m δJ m ~V} =0. I Xo ' Xo ,χ o
The mode of treatment is similar to that for a single dependent variable.
19. In the discussion of a curve which possesses a maximum or minimum property, if we limit the investigation to all curves of a given length or which satisfy some other condition, we have a distinct class of problems, which originated in the isoperimetrical problems of James Bernoulli. They were originally styled isoperimetrical, but are now called problems of relative maxima and minima. Thus, let it be proposed to determine the form of y which renders c χ ι U = I Vdx a maximum or a minimum, and which also J Xo f x l satisfies the condition V = I Vdx = constant, where V and j Xo V are given functions of x, y, ÿ, &c. It is obvious that, if U is a maximum or a minimum, so also is U+ aU', where a is any arbitrary constant. Accordingly the problem reduces to the determination of the maximum or minimum value of the integral f (V + aV')dx, regarding a as a constant, whose value is to be determined by aid of the given value of U'.
20. Another class of problems closely allied to the preceding is that in which the variables are connected by one or more equations of condition: for instance, when y, ÿ, ÿ, & c., z, z , &c., are connected by a relation W= 0, we investigate the maximum or minimum values of U=[ xι (Γ+λJV)dx (26), J xo where λ is an indeterminate function of x. We will illustrate these principles by examples.
21. To find the curve connecting two fixed points such that the surface generated by its revolution round a fixed line shall be given, and the volume of the generated solid shall be a maximum. Taking the fixed line as the axis of x, we may write U= (y 2 + ay∖ll + y∙ i )dx. J X o
Hence by (15) we have 1∕ 2 + αy√l+⅛ 2 =c + ^p==,
ΛV
or ^ + √γ⅞f= c -' ' 2i, ∙
If the curve meets the fixed axis we have c =0, and the curve is a circle whose centre lies on the axis.
The further integration of (27) depends on elliptic functions; it can, however, be shown without difficulty, as was proved by De-launey, that the curve is that generated by the focus of an ellipse or hyperbola which rolls on a fixed right line.
22. As an illustration of the method of § 20 we shall consider again the problem of § 13, taking the arc for independent variable. Let ⅛ = - r , ÿ = ⅛ then we have the relation x i + ÿ 2 - 1 = 0; and we write ds ds U= P {μ + ⅜λ(⅛ 2 + y 2 -l)} ds = p Vds, where F= μ + ⅛λ(i 2 + ÿ 2 -1 ). J Sq y Sq
Hence for a maximum or minimum we have
⅛=⅛ ω >- ⅛=> 1 < 28 > i , 1 - dμ .dμ dμ dλ therefore - r =x - i - + y - τ - = - r ∙, ds dx dy ds or μ = λ + c (29).
Again, the equation at the limits is ! μds+ I λ(xδx+yδy') = 0 (30);
∕ s 0 ’ s 0 also at each limit we have dx=δx+xds, dy=δy + yds ; hence (30) transforms into ∕ (μλ)ds + ∕ λ(xdx + ÿdy) = 0,
' «o '»ο
∕sι
or cd(s 1 -s 0 ) + ∕ λ(xdx+ydy')=0 (31).
' so
Now if the length of the curve be given, the first term in (31) vanishes; whereas, if the length be not given, we must have c =0. Consequently in the isoperimetrical problem we have μ=λ + c, and in the general case λ =μ, If we substitute μ for λ in (28) we can readily deduce the results already given in § 13.
23. More generally, let μ be a function of x, y, x, and ÿ, then we have V = μ + ½λ(x 2 + y 2 -l). Here we readily deduce r =*÷<÷< and the equation at the limits becomes crf(¾ - ⅞) + 1 g (^dx + ~⅜) = 0.
This shows, as before, that when the length of the curve is not given we have c =0.
24. The criterion for distinguishing between maximum and minimum solutions was investigated by Legendre, Lagrange, and other eminent mathematicians, for whose contributions to the solution of this problem, see Todhunter’s History of the Calculus of Variations. The full analytical investigation was, however, first given by Jacobi, of whose results we here give a brief discussion.
In this investigation, as in ordinary problems of maxima and minima, it is necessary to take into consideration the squares and the higher powers of our indefinitely small variations. Thus, if 17= ί Vdx, the complete variation of U may be written δU+½δ 2 U+
2 ⅞
&c., where δU depends on the first powers of δy, δy, &c., and δ 2 U on their second powers, he. Again, we will suppose that the limiting values of y and of its first n-1 differential coefficients are fixed, then δl7= ί δp r dx= ∕ Mδydx, and δ 2 U= ∣^ δMδydx, The solutions of
J x 0 J x 0 J x 0 the problem, as already observed (§ 8), are given by the equation M =0. Also for a true maximum it is necessary that δ 2 U should be constantly negative, and for a minimum that it should be positive. If δ 2 U=0, or if it change its sign between the limits of integration, the result is in general neither a maximum nor a minimum. As before, we suppose V a function οΐ x, y, y. . . y n and "= i -5 + ⅛-∙∙∙ + <- ι >"S ≈(⅛-∙⅛÷^--) fi therefore 8Jf = ( ≠ - D ⅛ + 2)⅛ - ... W
∖dy dy dy )
When this is developed, we see that every term, disregarding sign, is of the form 2/ —- π — τ D m δu. Also, if we combine this with
⅜<0 dyW J the corresponding term, and make l=m + r, we readily see that w =∑( - ι⅛"{^^ 5i +( - ("X
Hence δM is reducible to the form δM=(A a +DA 1 D +D 1 A 2 D 2 + . . . + D n A n D n }δy ...(35), provided it can be shown that D r u + ( - Y) r uD r is reducible to an operation of such a form. Now this is readily seen, for Du - uD ≡ Ù, therefore D 2 u - DuD ≡ Dû and DuD - uD 2 ≡ ÙD ; consequently D i u + uD 2 ≡2DuD + ii. In like manner 7fu-uD 3 ≡2DuD + u, and the proposed result can be readily shown by induction. Hence we infer that we may write δM in the form (35).
25. We have in the next place to show that the symbolic operator u(A 0 + DA 1 D+ . .. +D n A n D n )u may be written in the form B 0 +DB 1 D+ ... + D n B n D n . To establish this, it is sufficient to show that uD r A r D r u can be transformed into the shape in question. By Leibnitz’s theorem we have D r u≡ uD r + ru-D r ^ 1 + ⅛~⅛iDΓ~ 2 + ... and uD r ≡ D*u - rD r ~ λ u + ^~^ 2Γ⅞ - ...
ι · z
But these may be written D r u ≡ uD r +L 1 D r ~ 1 + L 2 D r ~ 2 +.. +L r , uD r ≡ D r u - D r ~ 1 L 1 + D r ~ 2 L 2 - ..., therefore uD r A r D r u, ≡ (D r u - D r ~ 1 D i + .. .)A r (uD r +L 1 D r ~ 1 + ..). Hence by § 24 we readily see that uD r A r D r u is of the required form. Accordingly we may write 2 '(∙^o ^f^ DA^O + ·..+22 A n D )uv = (2?θ + DB- i D +... + D n B n D n )v{36}. If uv=δy, we get from (35)
uδM=(B 0 + DB 1 D+ ... +D n B n D n fä (37); u again, if v =l in (36), we have u{A< i + DA i D+ ... +D n A n D n }u = B fi ......... (38).
Hence, if w 1 be a solution of (A 0 + DA 1 D+ ... +D n A n D ’ ")u=0, that is, if it be a solution of δM = 0, the corresponding value of B o is 0. Consequently we have from (36)
u 1 δM= D(B 1 + DB 2 D +...+ D n ~ γ B n D n -'')D⅛.
1 ω ι
If u 1 = —, this becomes z ι δJf=z 1 D(B 1 + DB 2 D+ .. . D n ~ 1 B n D n ~ 1 )Dz 1 δy (39).
Again, the symbolic operator B 1 +DB 2 D+ . . . + D n ~ 1 B n -ιD n ~ 1 can be transformed into z 2 D(C 1 + D(7 2 D+ . . . + D n ~ 2 C n ^ιD n ~ 2 )Dz 2 , and so on. Finally we get δM=z 1 Dz 2 D... z n-1 Dz n Dz n ^ x ... Dz 2 Dz λ δy (40).
Thus we obtain b~ U= ί δyz 1 Dz. i D ... Dz n D.. .Dz 2 Dz 1 δydx (41 ).
J «o
If we integrate by parts, then, since the limiting values of y, #...
2 C 1 are fixed, we get !U=- (Dz 1 δy)(z 2 Dz 3 . . . z 2 Dz 1 δy)dx, and after J x 0 n successive integrations S n U= ( -1)” f ∑ n (Dz n ~ 1 Dz n - 2 · . D 1 zδy) 2 dx (42).
J x 0 n cP V
Again, from (33) we see that J n = (-1) — rv ,; also from (40) we d↑f n > i find without difficulty
-'4 n ≡ z n(2 l s 2∙ · · 2 n-l) 2 (43).
Hence we get finally δ 2 U= f∖ n ( ⅝>-ιP zn - ii ∙ ∙^ sy ∖ 3 dx (44),
J Xq X Zn-l%n-2 · · · ¾ ∕
where Q = — 7 - λ5 . From this it follows that the sign of δ 2 U de- ⅜W 2 d 2 jr pends in general on that of Q n or ^^^^∙ Accordingly for a maximum or minimum solution it is necessary that Q n should have the same sign for all values of x between the limits of integration, dx being supposed always positive. The reader will find no difficulty in applying this criterion to any of the examples which we have hitherto given.
26. A new and complete discussion of the criteria for the discrimination of maximum and minimum solutions has been given by Mr Culverwell in Trans. Roy. Soc., vol. clxxviii. (1887). Owing to want of space we can only make an allusion to this remarkable memoir, which contains an elementary investigation of the criteria for maxima and minima not only in the case of one but for any number of dependent variables, as also for multiple integrals.
27. We now proceed to the application of the calculus of variations to multiple integrals, commencing with the
∏y 1 Vdxdy, where V is a function of Vo 1 , . 1 . , dz dz d?z x > y, ¾ P, r, s, and t, in which j p = - 7 -, ç = — r =
<∕ 2 z d 2 z t (dd 1 dy ’ dx 2 S = dfcdff t = d^∙ , * n accor ^ ancθ with the ordinary notation.
, m ∕∙≈ι
Let V 1 = ∕ Pdy, then U^= V 1 dx ; therefore by (6) we have
2 .j/ο J x 0 δU=[ xi δ V 1 dx + f cχ Vχdx
J xo! xo rxχ ry 1 rx 1 cyχ f x 1 ∣∙y 1 = δ Vdxdy + ∕ ∕ Vδydx + dr I P 1 dy. j X<A Vθ 2 xj y 0 I Xo J 2/0
Again δ V= Nδz + Pδp + Qδq + Rδr + Sδs + Tδt, where dz j Q= *T 1.% g .<Γ T^. Hence ff‰⅛, dp’ * dq dr ds dt J z 0 J y 0
= P ΡΝδζ + Pδp + Qδq + Rδr + Sδs + Tδt)dxdy......... (45). J W Vo
In the transformation of this expression we shall employ the following formulae, which are easily established,
^XS' i →==∕CX ,,,i ≈'jχ⅛ z m '
m ,ι rp⅛≈⅛=r∕‰ ο?).
J x 0 J y 0 dy J √ y 0
Hence we get j J P^^dxdy aiso iX ρ ≡" i, = √X⅜ 5λ ⅛ + ⅛> λ ∙ Next, to transform ∣ dxdy, we have
Consequently, by subtraction, - ^δz)dxdy
<<(p S-⅛÷-∕Z(⅛⅞)>
Again, in transforming the latter integral it is necessary to observe that y 1 and y 0 are functions of x, and accordingly that in that integral we have ^=-^-(δz)-^ hence dx dx dx dy ’ als ° X⅛⅛⅛* -≡⅜
γxwj>mw⅛> ÷rχ{⅛÷^÷^)> ÷Γ√>(I)>÷∕X⅛ <“>
Next, to transform ∖ 1 [ 1 S^-^-dxdy. Here J √ y 9 dxd V j
÷CM*-∞aWB9* "
Finally, to transform P-^dxdy, we have rx('3∙w⅛⅛>-r√2-s- md JX(¾ ⅜ + ⅛ ss ) , "≈' =J2∕7J⅛ β * ax, c ° nsβqumt ⅛ m rvι Combining the preceding results, we get δU=δ I V^dxdy
J χ 0 J y 0 C χ ι pnf dP dQ ι d i R ι d?S ι rf 2 T∖ s . , JxJy 0 ∖~ dx dy + dx i+ dxdy + dy 2 ) * y + p/ V V Q . P ^ +lt ^ + 2 ^ d ^(^_ d S_^ ôsdx J ¾√ a/ο × dx dx dx dx d y∖ dχ J dχ d vJ
+∕X^<∕X⅛> :
28. In many applications the limits of x and y are determined from a single equation. Thus, suppose the integral V^dxdy to express the sum of the elements Vdxdy for all values of x and y which satisfy either φ(x, y) > 0 or φ(x, y) < 0, then the limits are given by the equation φ(x, y) = 0. In this case the preceding value of δ U becomes much simplified, for y 0 , y 1 are determined from the equation φ=0; and the extreme values of x are found by eliminating y between the equations φ=0 and dφ∣dy = 0. But these are the conditions that φ = 0 should have equal roots in y ; we consequently infer that, when either x=x 0 or x=x 1 we have y 0 = y 1 . Hence we observe that ∣ udy=0 and ∕ udy=0 for all values of u, so that the last four terms in the expression for δU in (51) disappear in this case.
The methods and results of this and the preceding section can without difficulty be extended to triple and higher multiple integrals.
29. The method of application of the calculus of variations to the determination of maximum or minimum values of multiple definite integrals proceeds on the same principles as those already considered in § 9 for single integrals.
We shall limit the discussion to a brief consideration cf the double integral U^= Vdxdy, in which V is a function of x, y, z, p, and q solely, and the limits are determined, as in § 28, by an equation of the form ≠(λj, y) = 0. Such problems readily admit of geometrical interpretation.
Here (51) reduces to
-=ΓXF≡ p -f)^
+ X{^- p <a) fc+r M ,fe <52) ·
Consequently, as in the case of single integrals, for a maximum or minimum value of ι7^ we must have
^-g-⅛ ρ =o «
and ∕ 2zi {(f 2- 7 ⅛) fo +^}= 0 < δ4 )∙
The former is in this case a partial differential equation of the second order, whose solution, whenever it can be obtained, consists of an equation between x, y, z having two arbitrary functions. The form of these functions in each case is to be determined by aid of (54) combined with the given limiting conditions of the problem. Thus, let us suppose the upper limit restricted to points in the surface z=f 1 (x, y), and let p ’ =^, q ’ =^', thenδ∕ z = ∕ q'δy,
.y x .y x dx dy ll'
and/ {δz + qδy)=! q ’ ξ y . This gives ∕ y, {s≈+(q-q' y )Sy} =0 (55).
Substituting in (54), we get
∕ w {r÷(e-p⅛)(i '-,)}= 0 (56).
Again, along the limiting curve we have dz=pdx+qdy and dz=p'dx + q ’ dy; hence we get ⅛= Substituting in (56),
CIX (J — (J
we have along the limiting curve
F+Λy-p) + ‰'- 9 ) = 0 (57).
A similar equation holds if the lower limit be restricted by another surface.
By aid of these equations the arbitrary functions involved in the solution of (53) are determinable. For example, to find the surface such that μdS shall be a minimum, dS being an element of the surface and μ a given function of the co-ordinates x, y, z. Here MM M^7
r=μ√l+ y > + ΛF=^^,e=√ 1+ ^ τji ∙ Hencetheequa- „ dP dQ n , , i d f p ∖ , d f q tιon Nj— -v- = 0 leads to - i -( . .... . -= 1 + -7-1 - 7- - —= j I
cte dy cta∖√ι +jp 2 +? 2/ (⅛∖√ny + s -V
_ 1 ί dμ dμ dμ∖ ~ μ∖∕l+p sι + q 2 ∖dz P dx ^dy) ’ ll 1/ dμ n dμ dμ∖ ., oλ or Λ + 2i=-√ cos ¾ + ∞ 3 ⅛ +c ∞ 7 i) (58) ’
where R and R' are the principal radii of curvature of the surface at the point, and α, β, γ the angles which the normal to the surface makes with the co-ordinate axes. Again, if we suppose the surface limited by two fixed surfaces, the equation V+P(p' -p) + Q(q' - q) = 0 reduces to l+pp' + qq'=0 (59).
This shows that the surface intersects each of its bounding surfaces orthogonally. As a particular case of (58) we infer that the surface of minimum area is in general such that its principal radii of curvature at every point are equal and of opposite signs. A remarkable investigation of the equation of this surface was given by Bonnet in Liouville's Journal, 1860. Again, in the surface o f given area and of maximum volume we have V = z - a√ l +p 2 +q 2 . The equation N- 0 readily leads in this case to + —,=oe.
30. The principles and method of the calculus of variations are largely employed in all branches of mathematics, both pure and applied. Want of space prevents our entering into any account of such application. The reader will find the history of this calculus fully detailed in Todhunter’s treatise, already mentioned. The most important recent work in English is Mr Carll’s Calculus of Variations, New York and London, 1885. (b. w.)
