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AQPB

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Originally appearing in Volume V27, Page 920 of the 1911 Encyclopedia Britannica.
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AQPB in which the varied See also:

curve is described. Then See also:General the See also:contour consisting of the stationary curve A'Q, See also: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 See also:cell (fig. 4). Let us denote the integral of F Integral. taken along a stationary curve by See also: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-See also:hand member can be expressed as a See also:line integral taken along the varied curve AQPB. To effect this transformation we seek an approximate expression for the See also:term (A'Q) +[QP] _ (A'P) when Q, P are near together. Let As denote the arc QP, and 1G the See also:angle which the tangent at P to the varied curve, in the sense from A to B, makes with the See also:axis of x (fig. 5). Also let di be the angle which the tangent at P to the stationary curve A'P, in the sense from A' to P, makes with x the axis of x. We evaluate (A'Q) — (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 See also: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 See also:equation (A'Q) — (A'P) = —As.See also:cos ¢ F(x,y,p)+(tan Ili— p)a } p=tan (ji which is correct to the first See also: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 See also:

function E takes the See also: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 See also:original curve, is expressible as the line integral fEds function. taken along the varied curve. We can therefore See also:state a sufficient (but not necessary) See also: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 See also: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 See also: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 See also:drawn from Q to a point of the arc AB, and the arc PB of the stationary curve AB (fig. 6). The sign of the variation is then the same as that of the 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 See also:

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 See also:special problems it is generally permissible to identify A' with A, and to regard QP as straight. The direction of QP must be such that the integral of F taken along it is finite and real. We shall describe such directions as admissible. In the statement of the sufficient condition, and the new necessary condition, it is of course understood that the direction specified by 4, is admissible. The excess function generally vanishes if 4,=0, but it does not change sign. It can be shown without difficulty that, when \ is very nearly equal to the sign of E is the same as that of tan 1/.--tan (a2F '2 ( / ~)2 cos ay y,=tan and thus the necessary condition as to the sign of the excess function includes See also: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 See also: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 See also:principal equation is example y2x2,r =See also:coast., of the and the stationary curves include the axis of x, straightlines excess parallel to the axis of y, and the See also:family of exponentialcurves f°°ctioa. y=ae".

A field of stationary curves is expressed by the equation y=yo exp {c(x—xo)}, 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 See also:

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 (cote¢—3 coed) +2 tan coed)). The directions 4,=o and '4, = ir are inadmissible. On putting+,=2ir we get 2y2cot3¢; and on putting >G= See also: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 See also: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 See also: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 See also:

para- of sta- See also:meter may constitute a field. This remark is of See also:im- tionary portance in connexion with the See also: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 See also: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 See also:system of See also: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 See also: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 F—y'(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 See also:

solution of the problem is presented by the broken line See also:ACDB (fig. 8). A broken line like AA'B'B is excluded. Discontinuous solutions have generally been sup- posed to be of special im- portance in cases where the required curve is re- stricted by the condition C D of not See also:crossing the boun- axis of x dary of a certain limited of the boundary may have to be taken as See also:part of the curve. Problems of this See also:kind were investigated in detail by J. See also:Steiner and I. See also:Todhunter. In See also:recent times the theory has been much extended by C. Caratheodory. 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 See also:accord- ance with the assigned terminal or boundary conditions. erica- If this solution cannot be effected, the methods of the theorems. calculus fail to See also:answer the question of the existence or non-existence of a solution which would yield a maximum or minimum of the integral under See also: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 _ See also:

art ay- + a:.2 — ° at all points within a closed See also:surface S, and assumes a given value at each point of S. The See also:differential equation is the principal equation answering to (theeiintegral (V1 —JJJ \ax)2+(a ) 2+\a.:l2}dxdydz taken through the See also: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 See also: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 See also: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 See also:interval —I<x<i if y has different values at the end points. On the other hand, the integral can he made as small as we please by a suitable choice of y. Thus the argument fails to distinguish between a minimum and an inferior limit (see FUNCTION). In order to prove Dirichlet's principle it becomes necessary to devise a See also:proof that, in the See also: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 See also:pro prietate gaudentes (See also:Lausanne and See also:Geneva, 1744) ; J.

H. Jellett, An Flementary See also:

Treatise on the -Calculus of Variations (See also:Dublin, 1850); E. Moigno and L. Lindelof, " Lecons sur le calc. See also:duff. et int., " Calcul See also:des variations (See also:Paris, r86r), t. iv.; L. B. Carll, A Treatise on the Calculus of Variations (See also:London, 1885). E. See also:Pascal's See also:book cited above contains a brief systematic treatise on the simpler parts of the subject. A. Kneser, Lehrbuch d. Variationsrechnung (See also:Brunswick, 1900); H. See also:Hancock, Lectures on the Calculus of Variations (See also:Cincinnati, 1904) ; and O.

Bolza, Lectures on the Calculus of Variations (See also:

Chicago, 1904), give accounts of Weierstrass's theory. Kneser has made various extensions of this theory. Bolza gives an introduction to Hilbert's theories also. The following See also:memoirs and monographs may be mentioned: J. L. See also:Lagrange, " Essai sur une nouvelle methode pour determiner Ies max. et See also:les See also:min. des formules integrales indefinics," Misc. Taur. (, 76o-62), t. ii., or Euvres, t. i. (Paris, 1867) ; A. M. Legendre, " Sur la maniere de distinguer les max. des min. dans le calc. des See also:var.," Mein. Paris Acad.

(1786) ; C. G. J. See also:

Jacobi, "Zur Theorie d. Variationsrechnung . . . ," J. f. Math. (Crelle), Bd. xvii. (1837), or Werke, Bd. iv. (See also:Berlin, 1886); M. Ostrogradsky, " Mein. sur le calc. des var. des integrales multiples," Mean St See also:Petersburg Acad.

(1838); J. Steiner, " Einfache Beweise d. isoperimetrischen Hauptsittze," J. f. Math. (Crelle), Bd. xviii. (1839) ; O. See also:

Hesse, " Uber d. Kriterien d. Max. u. Min. d. einfachen Integrale," J. f. Math. (Crelle), Bd. liv. (18J7); A.

Clebsch, " Uber See also:

die'enigien Probleme d. Variationsrechnung welche nur eine unabhangige Variable enthalten," J. f. Math. (Crelle), Bd. lv.'(1858), and other memoirs in this volume and in Bd. lvi. (1859); A. See also:Mayer, Beitrdge z. Theorie d. Max. u. Min. einfacher Integrale (See also: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," See also:

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. See also:

Erdmann, " Zur Untersuchung d. zweiten Variation einfacher Integrale," Zeitschr. Math. u. Phys. (1878), Bd. See also:xxiii. ; P. Du Bois-Reymond, " Erlauterungen z. d.

Anfangsgrunden d. Variationsrechnung," Math. See also:

Ann. (1879), Bd. xv.; L. Schecffer, " Max. u. Min. d einfachen Int.," Math. Ann. (1885), Bd. See also:xxv., and " Uber d.. Bedeutung d. Begriffe Max ... ," Math. Ann.

(1886), Bd. See also:

xxvi. ; A. See also:Hirsch, Uber e. charakteristische Eigenschaft d. Diff.-Gleichungen d. Variationsrechnung," Math. Ann. (1897), Bd. xlix. The following See also:deal with Weierstrassian and other See also:modern developments: H. A. See also: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.," See also:

Annals of Math. (1901), vol. ii., also, " On the Existence of a Minimum ... ," and " On a Fundamental See also:Property of a Minimum ... ," Amer. Math. See also:Soc.

Trans. (1901), vol. ii.; D. Hilbert, " Math. Probleme," See also:

Gottingen Nachr. (1900), and "Uber das Dirichlet'sche Prinzip," Gottingen Festschr. (Berlin, 1901); G. A. See also: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.

End of Article: AQPB

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