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CALCULUS VARIATIONS

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Originally appearing in Volume V27, Page 919 of the 1911 Encyclopedia Britannica.
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CALCULUS VARIATIONS OF of points such as (xo, yo) and. (6o, 11o) were afterwards called con- jugate points by Weierstrass. The proof that the in.. CoaJutegral cannot be an extremum if the arc of the curve te between the fixed end points contains a pair of conjugate gapaints. points was first published by G. Erdmann (1878). Examples of conjugate points are afforded by antipodal points on a sphere, the conjugate foci of geometrical optics, the kinetic foci of analytical dynamics. If the terminal points are a pair of conjugate points, the integral is not in general an extremum; but there is an exceptional case, of which a suitably chosen arc of the equator of an oblate spheroid may serve as an example. In the problem of the catenoid a pair of conjugate points on any of the catenaries, which are the stationary curves of the problem, is such that the tangents to the catenary at the two points A and A' meet on the axis of revolution (fig. 2). When both the end points of the required curve move FIG. 2. on fixed guiding curves Co, Cl, a stationary curve C, joining a point Ao of Coto a point Ai of Cl, cannot yield an extremum unless it is cut transversely by Co at Ao and by CI at Al. The en- velope of stationary curves which set out from Co towards Cl, and are cut transversely by Co at points near Ao, meets C at a point Do; and the envelope of stationary curves which proceed from Co to Cl, and are cut transversely by Cl at points near Al, meets C at a point Di. The curve C, drawn from Ao to Al, cannot yield an extremum if Do or Di lies between Ao and Al, or if Do lies between Ai and Di. These results are due to G. A. Bliss (1903). A simple example is afforded by the shortest line on a sphere drawn from one small circle FIG. 3. to another. In fig. 3 Do is that pole of the small circle AoBo which occurs first on great circles cutting AoBo at right angles, and proceeding towards AiBi; Di is that pole of the small circle A,B1 which occurs first on great circles cutting at right angles, and drawn from points of AoBo towards A,BI. The arc AoAi is the required shortest line, and it is distinguished from BoBI by the above criterion. Jacobi's introduction of conjugate points is one of the germs from which the modern theory of the calculus of variations has sprung. Another is a remark made by Legendre (1786) in regard to the solution of Newton's problem of the sources solid of least resistance. This problem requires that a otWeien curve be found for which the integral strass's fYY"(I +y")—1dy theory. should be a minimum. The stationary curves are given by the equation yy',(1+y'2)_'=eonst., a result equivalent to Newton's solution of the problem; but Legendre observed that, if the integral is taken along a broken Une, consisting of two straight lines equally inclined to the axis of x in opposite senses, the integral can be made as small as we please by sufficiently diminishing the angle of inclination. Legendre's remark amounts to admitting a variation of Newton's curve, which is not a weak variation. Variations which are not weak are such that, while the points of a curve are but slightly displaced, the tangents undergo large changes of direction. They are distinguished as strong variations. A general theory of strong variations in connexion with the First Problem, and of the conditions which are sufficient to secure that the integral taken along a stationary curve may be an extremum, was given by Weierstrass in lectures. He delivered courses of lectures on the calculus of variations in several years between 1865 and 1889, and his chief discoveries in the subject seem to have been included in the course for 1879. Through these lectures his theory became known to some students and teachers in Europe and America, and there have been published a few treatise* and memoirs devoted to the exposition of his ideas. In the First Problem the following conditions are known to be necessary for an extremum. I. The path of integration must be a stationary curve. II. The expression 02F/ay" or the expression denoted by fi in the application of the parametric sacesmethod, must not change sign at any point of this curve ditioas. between the end points. III. The arc of the curve between the end points must not contain a pair of conjugate points. All these results are obtained by using weak variations. Additional (cos'(x, v) a-ar av au'ds' +) [Z cos(x, s)cos(y, v) a —cos(x, v)-:7c (af) ]wds'. In forming the first term within the square brackets we then use the relations ,as Cos(x, v) _ -- ,cos(y, v),S cos(y, v) =P 'cos(x, v), as' a = —cosy, v)ax a -+cos(x, v)a- ar , where p' denotes the radius of curvature of the curve s'. The necessity of freeing the calculus of variations from dependence upon the notion of infinitely small quantities was realized by Lagrange, and the process of discarding such quantities was partially carried out by him in his Theorie des functions analytiques (1797). In accordance with the interpretation of differentials which he made in that treatise, he interpreted the variation of an integral, as expressed by means of his symbol S, as the first term, or the sum of the terms of the first order, in the development in series of the complete expression for the change that is made in the value of the integral when small finite changes are made in the variables. The quantity which had been regarded as the The variation of the integral came to be regarded as the first second variation, and the discrimination between maxima and variation. minima came to be regarded as requiring the investigation of the second variation. The first step in this theory had been taken by A. M. Legendre in 1786. In the case of an integral of the form f 'F(x, y, y')dx Legendre defined the second variation as the integral Cx,t a2F a2F a2F xo —sy20Y)'+2sy - i6yay~+sy,2(Sy)2 dx. To this expression he added the term [1a(by)2t, which vanishes identically because Sy vanishes at x=xo and at x=xi. He took a to satisfy the equation a2F 1a2F da a2F 2 ay ,2 aye+dx) = ayay +a) and thus transformed the expression for the second variation to fx, a2F xoay2(5Y' +may) 2dx, where a2F a2F m , a yayay From this investigation Legendre deduced a new condition for the existence of an extremum. It is necessary, not only that the varia- tion should vanish, but also that the second variation Le- should be one-signed. In the case of the First Problem nnot h ccoond tion. has the same i ds gn at aall points of the stat unary cu e between the end points, and that the sign must be +for a minimum and —for a maximum. In the application of the perametric method the function which has been denoted by fi takes the place of a2F/ay'2. The transformation of the second variations of integrals of various types into forms in which their signs can be determined by inspection subsequently became one of the leading problems of the calculus of variations. This result came about chiefly through the publica-Jacobl, tion in 1837 of a memoir by C. G. J. Jacobi. He trans- formed Legendre's equation for the auxiliary function a into a linear differential equation of the second order by the substitution a2F __ _ a2F I dw ay y +a ay 2 w dx and he pointed out that Legendre's transformation of the second variation cannot be effected if the function w vanishes between the limits of integration. He pointed out further, that if the stationary curves of the integral are given by an equation of the form y=4(x, a, b), where a, b are arbitrary constants, the complete primitive of the equation for w is of the form w =Aa-+Bad as ab' where A, B are new arbitrary constants. Jacobi stated these pro-positions without proof, and the proof of them, and the extension of the results to more general problems, became the object of numerous investigations. These investigations were, for the most part, and for a long time, occupied almost exclusively with analytical developments; and the geometrical interpretation which Jacobi had given, and which he afterwards emphasized in his Vorlesungen fiber Dynamik, was neglected until rather recent times. According to this interpretation, the stationary curves which start from a point (xo, yo) have an envelope; and the integral of F, taken along such a curve, cannot be an extremum if the point (io, no) where the curve touches the envelope lies on the arc between the end points. Pairs results, relating to strong as well as weak variations, are obtained by a method which permits of the expression of the variation of an integral as a line integral taken along the varied curve. Let A, B be the end points, and let the stationary curve AB be drawn. If the end points A, B are not a pair of conjugate points, and if the point conjugate to A does not lie on the arc AB, then we may find a point A', on the backward continuation of the stationary curve BA beyond A, so near to A that the point conjugate to A' lies on the forward continuation of the arc AB beyond B. This being the case, it is possible to delimit an area of finite breadth, so that the arc AB of the stationary curve joining A, B lies entirely within the area, and held no two stationary curves drawn through A' intersect within otsta- the area. Through any point of such an area it is possible to draw one, and only one, stationary curve which passes curves through A'. This family of stationary curves is said to con- stitute a field of stationary curves about the curve AB. We suppose that such a field exists, and that the varied curve AQPB lies entirely within the delimited area. The variation of the integral f F(x, y, y')dx is identical with the line integral of F taken round a contour consisting of the varied curve AQPB and the stationary curve AB, in the sense AQPBA. The line integral may, as usual, be replaced by the sum of line integrals taken round a series of cells, the external boundaries of the set of cells being identical with the Q P given contour, and the internal boundaries of ad-B jacent cells being traversed twice in opposite senses. We may choose a suitable Let Q, P be points on the varied curve, and let A'Q, A'P be the stationary curves of the field which pass through Q, P. Let P follow Q in the sense
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