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AQPB in which the varied curve is described. Then General the contour consisting of the stationary curve A'Q, express from A' to Q, the varied curve QP, front Q to P, and vtarin for iatlon the stationary curve A'P, from P to A', is the .boundary ofan of a cell (fig. 4). Let us denote the integral of F Integral. taken along a stationary curve by round brackets, thus (A'Q), and the integral of F taken along any other curve by square brackets, thus [PQ]. If the varied curve is divided into a number of arcs such as QP we have the result [AQPB](AB) = ~[ (A'Q) +[QP] (A'P) 1, and the right-hand member can be expressed as a line integral taken along the varied curve AQPB. To effect this transformation we seek an approximate expression for the term (A'Q) +[QP] _ (A'P) when Q, P are near together. Let As denote the arc QP, and 1G the angle which the tangent at P to the varied curve, in the sense from A to B, makes with the axis
axis
(A'P) approximately by means of a result which we obtained in connexion with the problem of variable limits. Observing that the angle here denoted by 4, is it equivalent to the angle formerly denoted by +w (cf. fig. t), while tan is equivalent to the quantity formerly denoted by y', we obtain the approximate equation (A'Q) (A'P) = As.cos F(x,y,p)+(tan Ili p)a } p=tan (ji which is correct to the first order in As. Also we have [QP] =0s . cos 4,F(x,y, tan +,) correctly to the same order. Hence we find that, correctly to the first order in As, (A'Q)+[QP](A'P)=E(x,y, tan 0, tan 4,)Os, where E(x, y, tan 4,, tan IG) =cos , F(x,y, tan ) F(x,y,p) (tan tt p)- tan $. When the parametric method is used the function E takes the form (oaf +/zay)=,5,=w(oaf + a )z=l,5, =m where X, si are the direction cosines of the tangent at P to the curve AQPB, in the sense from A to B, and 1, m are the direction cosines of the tangent at P to the stationary curve A'P, in the sense from A' to P. The function E, here introduced, has been called Weierstrass's excess function. We learn that the variation of the integral, that Wekr- is to say, the excess of the integral of F taken along the stress's varied curve above the integral of F taken along the excess original
function. taken along the varied curve. We can therefore state a sufficient (but not necessary) condition for the existence of an extremum in the form: When the integral is taken along astationary curve, and there is no pair of conjugate points on the arc of the curve terminated by the given end points, the integral is certainly an extremum if the excess function has the same sign at all points of a finite area containing the whole of this arc within Sufflctevt it. Further, we may specialize the excess function by identifying A' with A, and calculating the function for a and point P on the arc AB of the stationary curve AB, and an necessary On- arbitrary direction of the tangent at P to the varied curve. dCltlons. This process is equivalent to the introduction of a particular type of strong variation. We may in fact take, as a varied curve, the arc AQ of a neighbouring stationary curve, the straight line QP drawn
function E(x, y, tan 4,, tan ), where (x, y) is the point P, 4, is the angle which the straight A line QP makes with the FIG. 6. axis of x, and di is the angle which the tangent at P to the curve APB makes with the same axis. We thus arrive at a new necessary (but not sufficient) condition for the existence of an extremum of the integral fFds, viz. the specialized excess function, so calculated, must not change sign between A and B.The sufficient condition, and the new necessary condition, associated with the excess function, as well as the expression for the variation as fEds, are due to Weierstrass. In applications to special
change sign. It can be shown without difficulty that, when \ is very nearly equal to the sign of E is the same as that oftan 1/.--tan (a2F '2 ( / ~)2 cos ay y,=tan and thus the necessary condition as to the sign of the excess function includes Legendre's condition as to the sign of a1F/3y'2. Weierstrass's conditions have been obtained by D. Hilbert from the observation that, if p is a function of x and y, the integral f) F(x,y,p)+(y'p) (ay) y'=n f dx, taken along a curve joining two fixed points, has the same value for all such curves, provided that there is a field of stationary curves, and that p is the gradient at the point (x, y) of that stationary curve of the field which passes through this point. An instructive example of the excess function, and the conditions connected with it, is afforded by the integral f y2y'2dx or f y2x3y-2d. The first integral of the principal equation is example y2x2,r =coast., of the and the stationary curves include the axis of x, straightlines excess parallel to the axis of y, and the family of exponentialcurves fctioa. y=ae". A field of stationary curves is expressed by the equation y=yo exp {c(xxo)}, and, as these have no envelope other than the initial point (xo, yo), there are no conjugate points. The function fl is 6xy-4, and this is positive for curves going from the initial point in the positive direction of the axis of x. The value of the excess function is y2cos ,f (cote3 coed) +2 tan coed)). The directions 4,=o and '4, = ir are inadmissible. On putting+,=2ir we get 2y2cot3; and on putting >G= fir we get 2y2cot34,. Hence the integral taken along AQ'PB is greater than that taken along APB, and the integral taken along AQPB is less than that taken along APB, when Q'Q are sufficiently near to P on the ordinate of P (fig. 7). It follows that the integral is neither a maximum eY nor a minimum. i It has been proved by Weierstrass that the excess function cannot be one-signed if the function f of the parametric method is a rational function of x and y. This result includes the above example, and the problem of the solid of least resistance, ` for which, as Legendre had FIG. 7. seen, there can be no solu- tion if strong variations are admitted. As another example of the calculation of excess functions, it may be noted that the value of the excess function in the problem of the catenoid is 2y sin2i(SG~) Developments connected with the excess function. In general it is not necessary that a field of stationary curves should consist of curves which pass through a fixed point. Any Field family of stationary curves depending on a single para- of sta- meter
tionary portance in connexion with the adaptation of Weierstrass's curves results to the problem of variable limits. For the purpose and of this adaptation A. Kneser (1900) introduced the family trans- of stationary curves which are cut transversely by an versals. assigned curve. Within the field of these curves we can construct the transversals of the family ; that is to say, there is a finite area of the plane, through any point of which there passes one stationary curve of the field and one curve which cuts all the stationary curves of the field transversely. These curves provide a system of curvilinear co-ordinates, in terms of which the value of fFdx, taken along any curve within the area, can be expressed. The value of the integral is the same for all arcs of stationary curves of the field which are intercepted between any two assigned transversals. In the above discussion of the First Problem it has been assumed that the curve which yields an extremum is an arc of a single curve, which must be a stationary curve. It is conceivable that the required curve might be made up of a finite number of arcs of different stationary curves meeting each other at finite angles. It can be shown that such a broken curve cannot yield an extremum unless both the expressions aF/ay' and Fy'(aF/ay') are continuous at the corners. In the parametric method of/ax and of/ay must be continuous at the corners. This result limits very considerably Discon. the possibility of such discontinuous solutions, though it t/nuous does not exclude them. An example is afforded by the solutions. problem of the catenoid. The axis of x and any lines parallel to the axis of y satisfy the principal equation; and the conditions here stated show that the only discontinuous solution of the problem is presented by the broken line ACDB (fig. 8). A broken line like AA'B'B is excluded. Discontinuous solutions have generally been sup- posed to be of special
portance in cases where the required curve is re- stricted by the condition C D of not crossing the boun- axis of x dary of a certain limited of the boundary may have to be taken as part of the curve. Problems of this kind were investigated in detail by J. Steiner and I. Todhunter. In recent
In any problem of the calculus of variations the first step is the formation of the principal equation or equations; and the second Exist- step is the solution of the equation or equations, in accord- ance with the assigned terminal or boundary conditions. erica- If this solution cannot be effected, the methods of the theorems. calculus fail to answer the question of the existence or non-existence of a solution which would yield a maximum or minimum of the integral under consideration. On the other hand, if the existence of the extremum could be established independently, the existence of a solution of the principal equation, which would also satisfy the boundary conditions, would be proved. The most famous example of such an existence-theorem is Dirichlet's principle, according to which there exists a function V, which satisfies the equation a2V a2V a2V _ art ay- + a:.2 at all points within a closed surface S, and assumes a given value at each point of S. The differential equation is the principal equation answering to (theeiintegral (V1 JJJ \ax)2+(a ) 2+\a.:l2}dxdydz taken through the volume within the surface S. The theorem of the existence of V is of importance in all those branches of mathematical physics in which use is made of a potential function, satisfying Laplace's equation; and the two-dimensional form of the theorem is of fundamental importance in the theory of functions of a complex variable. It has been proposed to establish the existence of V by means of the argument that, since I cannot be negative, there must Dirkh- be, among the functions which have the prescribed let's boundary values, some one which gives to I the smallest principe. bysWeie stra s. He observed that precisely the same argument would apply to the integral fx2y'1dx taken along a curve from the point (1, a) to the point (1, h). On the one hand, the principal equation answering to this integral can be solved, and it can be proved that it cannot be satisfied by any function y at all points of the interval I order to prove Dirichlet's principle it becomes necessary to devise a proof that, in the case of the integral I, there cannot be a limit of this kind. This has been effected by Hilbert for the two-dimensional form of the problem. lineas curvas maximi minimive pro prietate gaudentes (Lausanne and Geneva, 1744) ; J. H. Jellett, An Flementary Treatise on the -Calculus of Variations (Dublin, 1850); E. Moigno and L. Lindelof, " Lecons sur le calc. duff
Beitrdge z. Theorie d. Max. u. Min. einfacher Integrale ( Leipzig
1866), and " Kriterien d. Max. u. Min... , J. f. Math. (Crelle), Bd. lxix. (1868); I. Todhunter, Researches in the Calc. of Var. (London, 1871) ; G. Sabinine, " Sur . . . Ies max . . . des integrales multiples," Bull. St Petersburg Acad. (187o), t. xv., and Developpements . . . pour . . . la discussion de la variation seconde des integrales ...multiples," Bull. d. sciences math. (1878); G. Frobenius, " Uber adjungirte lineare Differentialausdrucke," J. f. Math. (Crelle), Bd. lxxxv. (1878) ; G. Erdmann, " Zur Untersuchung d. zweiten Variation einfacher Integrale," Zeitschr. Math. u. Phys. (1878), Bd. xxiii. ; P. Du Bois-Reymond, " Erlauterungen z. d. Anfangsgrunden d. Variationsrechnung," Math. Ann. (1879), Bd. xv.; L. Schecffer, " Max. u. Min. d einfachen Int.," Math. Ann. (1885), Bd. xxv., and " Uber d.. Bedeutung d. Begriffe Max ... ," Math. Ann. (1886), Bd. xxvi. ; A. Hirsch, Uber e. charakteristische Eigenschaft d. Diff.-Gleichungen d. Variationsrechnung," Math. Ann. (1897), Bd. xlix. The following deal with Weierstrassian and other modern developments: H. A. Schwarz, " Uber ein die Flachen kleinsten Flacheninhalts betreffendes Problem d. Variationsrechnung," Festschrift on the occasion of Weierstrass's loth birthday (1885), Werke, Bd. i. (Berlin, 1890) ; G. Kobb, " Sur les max. at les min. des int. doubles," Acta Math. (1892-93), Bde. xvi., xvii.; E. Zermelo. " Untersuchungen z. Variationsrechnung,"Dissertation (Berlin, 1894) ; W. F. Osgood, " Sufficient Conditions in the Calc. of Var.," Annals of Math. (1901), vol. ii., also, " On the Existence of a Minimum ... ," and " On a Fundamental Property of a Minimum ... ," Amer. Math. Soc. Trans. (1901), vol. ii.; D. Hilbert, " Math. Probleme," Gottingen Nachr. (1900), and "Uber das Dirichlet'sche Prinzip," Gottingen Festschr. (Berlin, 1901); G. A. Bliss, " Jacobi's Criterion when both End Points are variable," Math. Ann. (1903), Bd. lviii. ; C. Caratheodory, Uber d. diskontinuirlichen Losungen i. d. Variationsrechnung," Dissertation (Gottingen, 1904) ; and " Uber d. starken Max ... ," Math. Ann. (1906), Bd. lxii. (A. E. H. L.) End of Article: AQPB If you wish, you can link directly to this article.
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