In general it is not necessary that a field of stationary curves should consist of curves which pass through a fixed point. Any family of stationary curves depending on a single parameter may constitute a field. This remark is of importance in connexion with the adaptation of Weierstrass’s results to the problem of variable limits. For the purpose of this adaptation A. Kneser (1900) introduced the family of stationary curves which are cut transversely by an 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 ∫Fdx, 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 ∂F/dy′ and F−y′(∂F/∂y′) are continuous at the corners. In the parametric method ∂f/∂ẋ′ and ∂f/∂ẏ′ must be continuous at the corners. This result limits Very considerably mscom the possibility of such discontinuous solutions, though it Discontinuous solutions. 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. |
(fig. 8). A broken line like AA′B′B is excluded. Discontinuous solutions have generally been supposed to be of special importance in cases where the required curve is restricted by the condition of not crossing the boundary of a certain limited area. In such cases part 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 times the theory has been much extended by C. Carathéodory.
In any problem of the calculus of variations the first step is the formation of the principal equation or equations; and the second step is the solution of the equation or equations, in accordance with the assigned terminal or boundary conditions. If this solution cannot be effected, the methods of the calculus fail to answer the question of the existence or nonexistence 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
∂2V∂x2 + ∂2V∂y2 + ∂2V∂z2＝0
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 the integral
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 Dmch be, among the functions which have the prescribed ers boundary values, some one which gives to I the smallest principle. possible value. This unsound argument was first exposed by Weierstrass. He observed that precisely the same argument would apply to the integral ∫x2y′dx taken along a curve from the point (−1, a) to the point (1, b). 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 −1≤x≤1 if y has different values at the end points. On the other hand, the integral can be 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 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.
Bibliography:—The literature of the subject is very extensive, and only a few of the more important works can be cited here. The earlier history can be gathered from, M. Cantor's Geschichte d. Math. Bde. 1–3 (Leipzig, 1894–1901). I. Todhunter’s History of the Calculus of Variations (London, 1861) gives an account of the various treatises and memoirs published between 1760 and 1860. E. Pascal’s Calcolo delle variazioni . . . (Milan, 1897; German translation, Leipzig, 1899) contains a brief but admirable historical summary of the pre-Weierstrassian theory with references to the literature. A general account of the subject, including Weierstrass's theory, is. given by A. Kneser, Ency. d. math. Wiss. ii. A 8; and an account of various extensions of Weierstrass's theory and of Hilbert's work is given by E. Zermelo and H. Hahn, Ency. d. math. Wiss. ii. A Sa (Leipzig, 1904). The following treatises may be mentioned: L. Euler, Methodus im/eniendi lineas curvas maximi miniinive proprietate gaudentes . . . (Lausanne and Geneva, 1744); J. H. Jellett, An Elementary Treatise on the-Calculus of Variations (Dublin, 1850); E. Moigno and L. Lindelöf, “Leçons sur le calc. diff. et int., ” Calcul des variations (Paris, 1861), t. iv.; L. B. Carll, A Treatise on the Calculus of Variations (London, 1885). E. Pascal's book cited above contains a brief systematic treatise on the simpler parts of the subject. A. Kneser, Lehrbuch d. Variationsrechnung (Brunswick, 1900); H. Hancock, Lectures on the Calculus of Variations (Cincinnati, 1904); and O. Bolza, Lectures on the Calculus of Variations (Chicago, 1904), give accounts of Weierstrass's theory. Kncser has made various extensions of this theory. Bolza gives an introduction to Hilbert's theories also. The following memoirs and monographs may be mentioned: J. L. Lagrange, “ Essai sur une nouvelle méthode pour déterminer les max. et les min. des formules intégrales indéfinies, " Misc. Taur. (1760–62), t. ii., or Œuvres, t. i. (Paris, 1867);.A. M. Legendre, “Sur la manière de distinguer les max. des min. dans le calc. des var., " Mém. Paris Acad. (1786); C. G. J. Jacobi, " Zur Theorie d. Variationsrechnung.., ” J. f. Jllath. (Crellffli Bd. xvii. (1837), or Werke, Bd. iv. (Berlin, 1886);'M. Ostrogradsky, “Mém. sur le calc. des var. des intégrales multiples,” Mém. St Pétersburg Acad. (1838); J. Steiner, “Einfache Beweise d. isoperimetrischen Hauptsätze, " J. f. .Math. (Crelle), Bd. xviii. (1839); O. Hesse, “Über d. Kriterien d. Max. u. Min. d. einfachen Integrale, " J. f. Math. (Crelle), Bd. liv. (1857); A. Clebsch, “Über dieenigen Probleme d. Variationsrechnung Welche nur eine unablifingige Variable enthalten,” J. f. Math. (Crelle), Bd. lV.'(1858), and other memoirs in this volume and in Bd. lvi. (1859); A. Mayer, Beiträge 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 . . . les max . . . des intégrales multiples, ” Bull. St Pétersburg Acad. (1870), t. xv., and “Développements . . . pour . . . la discussion de la variation seconde des into rales . . . multiples,” Bull. d. sciences math. (1878); G. Frobenius, “Über adjungirte lineare Differential-ausdrücke,” 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, “Erläuterungen z. d. Anfangsgründen d. Variationsrechnung," Math. Ann. (1879), Bd. xv.; L. Scheeffer, “ Max. u. Min. d einfachen Int., ” Math. Ann. (1885), Bd. xxv., and “Über d. Bedeutung d. Begriffe Max., " Jllath. Ann. (1886), Bd. xxvi.; A. Hirsch, “Über e. charakteristische Eigenschaft d. Diff.-Gleichungen d. Variationsrechnung, " Ilfath. Ann. (1897), Bd. xlix. The following deal with Weierstrassian and other modern developments: H. A. Schwarz, “Über ein die Flächen kleinsten Flächeninhalts betreffendes Problem d. Variationsrechnung, " Festschrift on the occasion of Weierstrass's 70th birthday (1885), Werke, Bd. i. (Berlin, 1890); G. Kobb, “ Sur les max. et 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,” Göttingen Nachr. (1900), and “Über das Dirichlet’sche Prinzip, " Göttingen Festschr. (Berlin, 1901); G. A. Bliss, “Jacobi's Criterion when both End Points are variable,” Math. Ann. (1903), Bd. lviii.; C. Crathéodory, “Über d. diskontinuierlichen Lösungen i. d. Variationsrechnung,” Dissertation (Göttingen, 1904); and “Über d. starken Max . . .," Math. Ann. (1906), Bd. lxii. (A. E. H. L.)
VARICOSE VEINS (Lat. varix, a dilated vein), a condition of the veins which mostly occurs in those parts of the blood-stream which are farthest from the heart and occupy a dependent position. Thus they are found in the legs and thighs; in the lowest part of the bowel (piles; see Haemorrhoids), and in the spermatic cord (varicocele). Any condition which hinders the return of blood from the veins is apt to cause their permanent dilatation; thus is explained the occurrence of varicose
